Linguistic Graphs and their Applications

Linguistic Graphs and their Applications

W. B. Vasantha Kandasamy Ilanthenral K Florentin Smarandache

Editorial Global Knowledge 2022

Copyright2022by Editorial Global knowledge andthe Authors

Thisbookcanbeorderedfrom:

Editorial Global Knowledge 848 BRICKELL AVE STE 950 MIAMI, FLORIDA, US 33131. info@egk.ccgecon.us

This work is licensed under a Creative Commons AttributionNonCommercial-NoDerivatives 4.0 International License.

Peerreviewers:

GiorgioNordo,MIFT-DepartmentofMathematicalandComputerScience, PhysicalSciencesandEarthSciences,MessinaUniversity,Italy. MohamedElhoseny,AmericanUniversityintheEmirates,Dubai,UAE, LeHoangSon,VNUUniv.ofScience,VietnamNationalUniv.Hanoi, Vietnam.

Manybookscanbedownloadedfromthefollowing DigitalLibraryofScience: http://www.gallup.unm.edu/eBooks-otherformats.htm

ISBN-13:978-1-59973-749-2 EAN:9781599737492

PrintedintheUnitedStatesofAmerica

2

CONTENTS

Preface 4 Chapter One BASIC CONCEPTS 7

1.1Introduction 7

1.2LinguisticVariablesasWordsorTerms 9

1.3OrderingorPartialOrderingonLinguisticSets/Terms12 1.4minandmaxOperationsonLinguisticSets 17 1.5.SuggestedProblems 20

Chapter Two

BASIC LINGUISTIC GRAPHS 28

2.1Introduction 28 2.2LinguisticGraphsinGeneral 29 2.3Linguistic-directedgraphs 59 2.4Linguisticedge-weightedgraphs 115 2.5SuggestedProblems 140

3

ADJACENCY LINGUISTIC MATRICES OF LINGUISTIC GRAPHS AND THEIR APPLICATIONS

149

3.1Introduction 149

3.2Adjacencylinguisticmatricesoflinguisticgraphs 150

3.3LinguisticBipartiteGraphsandtheirProperties 191

3.4SuggestedProblems 216

FURTHER READING 228 INDEX 233 ABOUT THE AUTHORS 238

PREFACE

In this book, authors, systematically define the new notion of linguistic graphs associated with a linguistic set of a linguistic variable. We can also define the notion of directed linguistic graphs and linguistic-weighted graphs. Chapter two discusses all types of linguistic graphs, linguistic dyads, linguistic triads, linguistic wheels, complete linguistic graphs, linguistic connected graphs, disconnected linguistic graphs, linguistic components of the graphs and so on. Further, we definethenotionoflinguisticsubgraphsofalinguisticgraph.

However, like usual graphs, we will not be able to arbitrarily connect any two linguistic words of a linguistic set associated with a linguistic variable. They can be related or adjacentdependingonthelinguisticvariableassociatedwiththe linguistic set. This is an exceptional feature of a linguistic graph.

Further, the direction in general of a linguistic-directed graph is special. For if the linguistic set corresponds to the linguisticvariableheightofapersonandifwewishtorepresent itbyalinguistic-directedgraph,thenitismandatorythatwecan have a directed line from short to tall and never a directed line from the node tall to node short for in nature the height of a personneverdecreasesfromtalltoshortonlyadirectedlinecan be drawn from short to tall. This is yet another striking feature oflinguistic-directedgraphsingeneral.

Finally, we discuss the linguistic adjacency matrices associated with the linguistic graphs. As the entries of the linguistic adjacency matrix must have linguistic notations, we use in the case of linguistic-undirected graphs the linguistic

terms {e, }, where e denotes the linguistic edge exists, and  denotes that there is no linguistic edge between two linguistic nodes.

{e, }  {1, 0}. 1 is represented by e in the linguistic graph (that is, there is an edge) 0 is represented by , which is noedge.

Thus the entries of the linguistic adjacency matrices associated with the undirected linguistic graph will be only e and ; in this case, the matrices will be symmetric. In the case of the linguistic-directed graph, the linguistic adjacency matriceswillhaveentriesfrom{e, }but will not besymmetric as in the case of an undirected linguistic graph. This is the only difference between directed linguistic graphs and undirected linguistic graphs. However, in the case of linguistic edgeweighted graphs, the linguistic adjacency matrices will have entries as linguistic terms together with the  the empty edge. However, in the case of undirected weighted linguistic graphs, the linguistic adjacency matrices will be symmetric with entries from a linguistic set together with {}. However, in the case of directed linguistic edge-weighted graphs, the linguistic adjacency matrices will have linguistic entries from a linguistic set together with . The third chapter also introduces the notion of Linguistic Cognitive Maps (LCMs) analogous to Fuzzy CognitiveMaps(FCMs).

These Linguistic Cognitive Maps (LCMs) can be applied to problems whose data is unsupervised, but the values are linguistic. Illustrative examples to this effect are given in the last chapter of this book. In each chapter of this book, the authors suggest some problems for the reader to solve. Finally, thisbookhasover200illustrativelinguisticgraphs.

So that the reader has a clear idea about these linguistic graphs when they try to apply them in LCMs models, these LCMmodelswillbeaboontonon-mathematiciansatlarge.

We acknowledge Dr K.Kandasamy with gratitude for his continuoussupport.

W.B.VASANTHAKANDASAMY K.ILANTHENRAL FLORENTINSMARANDACHE

BASIC CONCEPTS

1.1Introduction

Inthischapter,wedefinethenewnotionoflinguisticsets associated with a linguistic variable. We use only linguistic wordsintheselinguisticsets.

We do not use the notion of linguistic sentences to be present in our linguistic words (or sets or terms). For linguistic sentencesmaycombinetwoormorelinguisticwords,forwetry to build structures like graphs etc. We do not, at this stage use the notion of linguistic sentences for the linguistic sets which weassociatewithanylinguisticvariable.So, witha verysimple modification from Zadeh [26-7] definition of a linguistic variable, we define the linguistic variable as a variable whose associatedvaluesarewordsinanartificialoranaturallanguage. We do not include as per the researcher Zadeh the linguistic variable as a variable whose values are sentences or words or a fuzzymembership.

We see height is a linguistic variable, age is a linguistic variable and so on. We wish to state if someone wants a rough estimate of the age of a person who is just in front of us or in

shop or a park so on. One can easily use any linguistic words, young or very young or just young or old or very old or middle agedorsoon.

No one is natural senses say 35 years and 2 months or 45 years or10yearsorsoon.So,whatfirstoccursinthemindofa person, be it illiterate or a literate to describe the age only in linguistic term or words. So undoubtedly, linguistic terms are morenaturalthannumericalvalues.

Here we do not say numerical values should not be used or it is in any way less than linguistic values, what we wish to emphasize is in general linguistic values are in many places handyincomparisonwithnumericalvalues.

So, an honest trial is made how the study of linguistic terms can influence the human thinking while studying or analyzing a problem. Or to be more precise, we try to build linguistic value as a possible analogue to numerical values whenevertheuseofthemisexplicit.

So, we build simple concepts on linguistic sets (terms) like ordering operations like min or max [24]. Using these, we further develop structural concepts like linguistic graphs and theirapplication.

Thischapterhasfoursections.Sectiononeisintroductory in nature. Section two describes the notion of linguistic variables as a linguistic set or words or terms. The ordering operations are defined on these linguistic sets in section three. The final section defines some operations on these linguistic sets.

1.2

LinguisticVariablesasWordsorTerms

Here for the first time, we describe linguistic variables onlybywords,developanddefinethem.

We illustrate linguistic variables in terms of linguistic wordsbysomeexamples.

Example 1.2.1. Let us consider the linguistic variable “speed of a vehicle on road”. Let S be the linguistic set associated with thisvariable,

S = {very fast, fast, just fast, fastest, medium speed, slow, just medium, very slow, just slow, } here the term  means the empty linguistic word/term corresponding to zero 0 of the numbersystem,thatisnowords.

Here if we take any two linguistic terms from S we can comparethemlikeslowandfastwhere “slowisalesserspeedthanfast”; slow  fast mediumandjustslow justslow  mediumandsoon.

Example 1.2.2. Let us consider the linguistic variable “colour” andnotshadesofaparticularcolour.

Let S be the linguistic set associated with the linguistic variablecolour. S={black,blue,green,white,brown,red,orange}.

Wecannotcompareanyofthetwocolours.

Comparison of colours is possible only if shades of a coloursaylikedarkblue,blue,lightblue,whiteshadedblueand soon.

Next,wegiveyetanotherexample.

Example 1.2.3. Let us consider the linguistic variable height of persons. The linguistic set or linguistic terms associated with heightisasfollows.

S = {tall, very tall, very very tall, just tall, short, shortest, justshort,veryshort,medium,justmedium}.

We study the linguistic set S later and define operations onthem.

Example 1.2.4. Let us consider the linguistic variable overgrowth(height,colourofleavesand soon)of paddyplants. LetSbethelinguisticsetassociatedwithit.

S = {good, average, very good, bad, not up to the mark, verybad,justbad,justgoodandsoon}.

Example 1.2.5. Let us consider the linguistic variable temperatureofwaterwhileheatingfromicestate.Thelinguistic termsSassociatedwiththisvariableisasfollows.

S = {high, very low, highest, lowest,  (corresponds to 0 degree),low,justlow,andsoon}.

Now we can also have linguistic variables in social problems like school dropouts, unemployment as a societal problem,symptomdiseasemodel,etc.

Example 1.2.6. The social problem of unemployment among the educated youth is the linguistic variable. The linguistic termsassociatedwithitare

S = {educated criminals, frustration, taking up to drugs, taking toviolence,distress,mentalstress,emptinessoflifeandsoon}.

Example 1.2.7. Let us consider the linguistic variable school dropoutsinchildren.LetSbethelinguisticterm,

S = {bonded labourers, runaways from home, family problems (parents in fault) rag pickers, bonded labourers}. This social problemcanbegivenlinguisticrepresentation.

Example 1.2.8. Now we deal with the diagnostic problem of diseasesymptom modelforlunginfection,whichistakenasthe linguisticvariable. LetSbethesetoflinguisticterms.

S = {l1 - temperatureintheevening l2 - coldandfever l3 - onlycough l4 - coughandfever l5 - breathlessness

l6 - reductionofweight

l7 - paininthechest

l8 - coldandcoughwithphlegm l9 - tiredness} describingthelinguisticvariable“lunginfection”.

1.3 Ordering or Partial Ordering on Linguistic Sets / Terms

In this section, we introduce the notion of partial or total order on linguistic sets [23, 24]. It is important to note that linguistictermsorsetsareof3types.

i) Notpartiallyortotallyorderable

ii) Onlypartiallyorderable

iii) Totallyorderable

We will define and describe the notion of all the 3 concepts.

Whenwesaynotorderable,we cannot compare evenany twoofthem.Fortakethelinguisticset,

S = {Black, Brown, Green, Blue, Red, White, Orange, Pink, Violet}

related with the linguistic variable colours and not shades of colour.

So, this set S is not orderable for we do not have even a singlepairofdistinctelementswhicharecomparable.

So, working with not orderable linguistic sets is a problem in such cases one can seek fuzzy membership techniques or making a power set using that S so that the linguistic power set P(S) of S becomes a partially orderable set undertherelationofcontainmentrelation.

Nowwediscussaboutpartiallyorderedlinguisticsets.

Suppose we have a linguistic variable colours and shades of colours of the persons / people all over the world. The correspondinglinguisticsetorterms

S = {white, black, dull black, dark black, brown, light brown, dark brown, very dark brown, very light brown, yellow, wheatishyellowandsoon}.

NowweseeS isa partiallyorderablesetandnot atotally orderableset.

We see white and black cannot be ordered under shades of colour whereas dull black, black and dark black can be ordered as dull black is some shades less than black so we can orderas

dullblack  black

Similarly black is somewhat some shades less than dark black sothatwecanorderitas

black  darkblack

Now we see dull black is many shades less than dark black so dullblack  darkblack.

Infact,wehave dullblack  black  darkblack

Thus, we see in this linguistic set S some of the linguistic terms are not comparable and some of them are comparable we call thislinguisticsetasapartiallyorderedset.

Wewillgivethedefinitioninatechnicalway.

Definition 1.3.1. Let Sbe alinguistic set (set oflinguisticterms) associated with some linguistic variable. We say this linguistic set S is partially orderable if there exist at least two distinct linguistic terms in S which are comparable, and we generally denote that comparable relation by ‘’. We call (S, ) as a partially orderedlinguisticset.

We give one more example of a linguistic set which is onlyapartiallyorderablelinguisticset.

Example 1.3.2. Let S be a linguistic set associated with some linguistic variablewhichisnot orderable.Let P(S) bethepower set of S which we call as linguistic power set of S. We show P(S) is a partially orderable set under the order; inclusion of subsetsofasetS.

Let S= {s1,s2,s3,s4}bethe linguisticset.LetP(S) bethe linguisticpowersetofS;

P(S) = {, {s1}, {s2}, {s3}, {s4}, {s1, s2}, {s1, s3}, {s1, s4}, {s2, s3}, {s2, s4}, {s3, s4}, {s1, s2, s3}, {s1, s2, s4}, {s1, s3, s4}, {s2, s3, s4},S={s1,s2,s3,s4}}.

We see {s1, s3, s4}  S = {s1, s2, s3, s4}, {s1}  {s1, s3}, {s1,s3}  {s1,s3,s4}.

So{s1}  {s1,s3}  {s1,s3,s4}  {s1,s3,s2,s4}.

However,{s1,s2}and{s,s3}cannotbeorderedunderany containmentrelation.

{s4, s3} cannot be ordered under the containment relation with{s1,s2,s3}andsoon.

Thus{P(S), }isonlyapartiallyorderedlinguisticset.

Now we provide an example of a totally ordered linguisticsetS.

Example1.3.3. Let

S = {tall, short, very very short, very tall, just tall, just short, veryshort,tallestshortest,justmedium,medium}

bealinguisticsetassociatedwiththeheightofsomepersons.

Nowwhenwevisualizetheheightofapersoningeneral.

We knowinterms of numbers or numerical values height ofapersoningeneral variesfromonefoottosayamaximumof 8feet.Wedenote(say)thepersonwithheightasonefoot asthe

shortestandthepersonwhomeasures8feetasthetallestandall otherpersonsvaryinthatinterval[1,8]thiscanbecategorically putas

{veryveryshort, veryshort,short,justshort,medium,tall, very tall,veryverytallandsoon}.

Since the linguistic set S associated with height and the numerical values of height are comparable or totally orderable, wecansayanytwotermsinSarecomparableunderthe‘’less thanorequaltorelation.

In view of this we can order this S related with height as follows.

shortest  veryveryshort  veryshort  short  justshort  justmedium  medium  justtall  tall  verytall  tallest I

We see every pair of elements in S is linguistically comparable. In fact we can form a chain of the form given in I. We call this setasthetotallyorderedlinguisticset.

Wedefinethetotallyorderedsetasfollows.

Definition 1.3.2 S be a linguistic set associated with some linguistic variable. If every pair of elements in S is comparable, that is for any a, b  S, a b (or b a) where ‘’ means they are linguistically comparable then we define (S, ) tobe atotallyorderedset.

In fact, every totally ordered set S has a chain associated withitwhichisunique.

Now having seen the concept of partial order, total order and not ordered we proceed on define some operations on linguisticsetsinthefollowingsection.

1.4minandmaxOperationsonLinguisticSets

In this section, we define the notion of two operations min and max on totally ordered set and linguistic power set of a linguistic set S. We will first illustrate this situation by some examples.

Example 1.4.1. S = {old, oldest, very old, just old, young, very young,youngest,justyoung,middleage,adult,justadult}

be the linguistic terms associated with the linguistic variable age.

We see S is totally ordered set the linguistic chain associatedwithSisasfollows.

youngest  just young  young  very young  just adult  adult  middleage  justold  oldveryold  oldest I WedefineminandmaxonthelinguisticsetS.

Supposewetake{justadult,veryold}; justadult  veryold. min{justadult,veryold}=justadultand

max{justadult,veryold}=veryold.

SowecandefineminandmaxoperationsonS.

Thus, S is closed with respect to both operations min and max.

Nowonapartiallyorderedlinguisticsetwecannotdefine min or max operations for min or max is defined on every pair oflinguisticterms.

Now we define on the linguistic power set P(S) of a linguistic set S the notion of  and , where S is only a linguisticsetwhichisnotatotallyorderedlinguisticset.

Example 1.4.2. Let S = {fast, slow, medium speed, very slow} be a linguistic set associated with the linguistic variable speed ofacarinahighway.

Let P(S) = {{}, {fast}, {slow}, {medium speed}, {very slow}, {fast, slow}, {fast, medium speed}, {fast, very slow}, {slow, medium speed}, {slow, very slow}, {medium speed, very slow}, {fast, slow, medium speed}, {fast, slow, very slow}, {fast, medium speed, very slow}, {slow, very slow, mediumspeed},S={fast,slow,mediumspeed,veryslow}} bethelinguisticpowersetofthelinguisticsetS.

Now P(S) is closed under both the operations  and  P(S) is only a partially ordered set under the set inclusion relation‘’.

{P(S), }isaclosedoperationonthelinguisticpowerset P(S).Fortaketwosubsetsfrom P(S)say

{fast,mediumspeed},{fast,veryslow,slow}  P(S), wesee {fast,mediumspeed}  {fastveryslow,slow}={fast}.

Wefind{fast,mediumspeed}  {fast,veryslow,slow} ={fast,slow,veryslow,mediumspeed}

Consider  and{slow,fast,veryslow}  P(S)   {slow, fast, veryslow} = {slow, fast,veryslow} and   {slow,fast,veryslow}= 

We see {P(S), } and {P(S), } are closed under the operations  and .

Now we define two other operations min and max on linguisticsubsetsofP(S).

Let A={slow,fast,veryslow} and B={medium,slow}  P(S) Wefind

min {A, B} = min{{slow, fast, very slow}, {medium, slow}} = {min{slow, medium}, min{slow, slow}, min{fast, medium}, min{fast, slow}, min{very slow, medium}, min{very slow, slow}}={slow,medium,veryslow} I

Nowwefind max{A,B}=max{{slow,medium},{slow,fast,veryslow}}

={max{slow,slow},max{slow,fast}

max{slow, very slow}, max{medium, slow}, max{medium,fast},max{medium,veryslow}}

={slow,fast,medium} II

Clearly, I and II are distinct they are very different operationsonP(S).

In fact, ,  min and max are 4 distinct different operationonP(S).

HavingdefinedoperationsonP(S)andSwenowproceed on to suggest some problems for the reader in the following section.

1.5.

SuggestedProblems

In this section, we suggest a few problems on linguistic sets associated with the linguistic variable, total order or partial

order on linguistic sets and finally, four operation min, max,  and  onlinguisticsets.

1.5.1 Find the linguistic set S associated with the linguistic variable ‘weightofpeople’.

1.5.2 For the linguistic variable prediction of rain for a week, givetheassociatedlinguisticsetS.

1.5.3 For the linguistic variable colour of eyes of people describetheassociatedlinguisticsetS.

i) Is S a totally ordered set or a partially order set or neither?

1.5.4 Let S be a linguistic set associated with the linguistic variablegrowthofplantsinaragifield.

i) Is S a totally ordered linguistic or a partially ordered linguisticset?

Justifyyourclaim.

1.5.5 Let the symptoms suffered by a patience be the linguistic variable. The linguistic set S associated with the patience problemsareasfollows.

S= (a1) - fever (a2) - vomiting (a3) - chillsonandoff

(a4) - sweating (a5) - noappetite

(a6) - painontherightsideofabdomen

i) ProveSisnottotallyorderable. ii) IsSapartiallyorderableset? iii) Find the linguistic power set P(S) of S and define thetwooperations  and .

iv) IsitpossibletodefineminormaxonP(S).

1.5.6 Let S = {good, bad, fair, very bad, just bad, just good, best, very good} be the linguistic set related with the linguistic variableperformanceofanemployeeinafactory.

i) IsSatotallyorderedset? ii) Is P(S) the linguistic power set of S a partially orderedsetortotallyorderedset?

1.5.7 Let A={good,bad,verygood,justbad}. B={goodbest,justgood,fair,verybad}  P(S)

Find (a) A  B (b) A  B (c) min{A,B}and

(d) max{A,B}

Aretheydistinctorsomeofthemareidentical?

i) DoesShavealinguisticchainoflength8?

ii) How many linguistic chains of length 7 exist in P(S)?

iii) What is the length of the largest linguistic chain in P(S)?

iv) Does the sub set P = {{}, {best}, {good, best}, {bad, best, good}, {best, good, bad, just bad}, {best, good, bad, just bad, very bad}, {best, good, bad,justbad,verygood,verybad}  P(S)

formalinguisticchain?

ObtainthechainformedbythesubsetP  P(S).

1.5.8 Letyieldofacertaincropconsidermaizebethelinguistic variable

i) FormthelinguisticsetSformedbyit ii) Isitatotallyorderedset? iii) Isitapartiallyorderedset? iv) IsSafiniteoraninfinitelinguisticset?

v) Can we sayforming the linguistic set S is a flexible one depending onthe expert? Justifyor substantiate yourclaim.

1.5.9 Let S be the linguistic set associated with the linguistic variableintelligenceofstudentsinaclassroom.

i) IsSatotallyorderedset?

ii) IsSapartiallyorderedset?

iii) Find the linguistic power set P(S) of the set S and prove P(S) is closed under the four operations, , ,minandmax.

iv) Prove all the four operations mentioned in (iii) are distinct.

v) Obtain any other special features associated with thisproblem.

1.5.10 Let S be the linguistic set associated with the linguistic variableallshadesofthethreecoloursyellowblueandred.

i) IsSafiniteorinfiniteset?

ii) ProveSisnotatotallyorderableset.

iii) ProveSisapartiallyorderedset.

iv) Find P(S) the powerset ofS and prove {P(S), min} and{P(S),max}arenotclosedoperationsonP(S)

v) Prove{P(S), }and{P(S), }areclosedoperation onP(S).

1.5.11 Give one example of a linguistic variable other than colourwhoseassociatedlinguisticsetisnotorderable.

1.5.12 Can we say all time dependent linguistic variables are alwaystotallyorderable?Justifyyourclaim.

1.5.13 Can be say time independent linguistic variables have linguisticsetswhicharenotpartiallyorderable?

1.5.14 Canwe saythelinguisticvariabletheintensityofrainin the rainy season will give a linguistic set which is a totally orderedset?

1.5.15 Can we say the emotions of a person as a linguistic variableyieldsalinguisticsetwhichisatotallyorderedset?

1.5.16 List out all emotions related with a novel and order the related linguistic set S. Will S be of infinite cardinality or be onlyoffinitecardinality.

1.5.17 Give some linguistic set which is associated with emotionsofapersonoffinitecardinality.

1.5.18 Can we order the linguistic term angry and happy by someorder?Justifyorsubstantiateyourclaim.

1.5.19 Can the feelings depressed and unhappy be ordered? Justifyyourclaim.

1.5.20 Suppose we study the growth of a plant for a period of onemonth.

i) Can we say the growth can always be an increasing orderofheight?Justifyit.

ii) Doapracticalstudyofthisproblem?

1.5.21 Findoutamethodbywhichonecantacklethelinguistic setswhicharenotordered.

1.5.22 Spell out some very special features enjoyed by the linguisticset.

1.5.23 Will linguistic set concept be easily understood by a illiteratepersonthanthenumericalvalues?

Substantiateyourclaim!

1.5.24 Can this study make a non-mathematician more confidentthanoperationswithnumbersystem?

1.5.25 Which do you think is in the easy grasp of school children under 10 linguistic set approach or number theoretic approach?Justifyyourclaim.

1.5.26 Can we say operations like min and max on linguistic sets which are totally orderable easier than sum and product of numbersintheagegroupslessthan7years?

1.5.27 Can we say this linguistic approach will make them like mathematicsthannumericalapproach?

1.5.28 Will this linguistic approach build more confidence in schoolchildren?

1.5.29 Will linguistic approach make a researcher in fields like sociology etc. (that is non mathematicians at large) a easy way toapplytheminrealworldproblem?

1.5.301 Obtain any other advantages of linguistic sets in the placeofnumbersystems.

1.5.31List outthedisadvantagesandlimitationsoflinguisticset approach.

BASIC LINGUISTIC GRAPHS

2.1 Introduction

In this chapter we for the first time introduce basic concepts about linguistic graphs. These linguistic graphs are different from linguistic line, linguistic planes and curves built using linguistic terms. Basically, linguistic graphs like classical graphs will be associated with a set of linguistic vertices and relations connecting linguistic vertices. When we say linguistic nodes / vertices they are nothing but elements from the linguistic set of words or which we call as linguistic terms associatedwithalinguisticvariableL.

These basic notions about linguistic variables and the associated linguistic terms and the notion of ordering etc, etc; has been introduced in chapter I of this book to make this book aselfcontainedone.

Before we proceed onto define the concept of linguistic graphsweproceedontodescribethembysomeniceexamples.

We will describe three types of linguistic graphs in this chapter. The chapter has five sections. Section 1 is introductory in nature. Section 2 introduces linguistic graphs which are not directed and the edges are arbitrary. Section 3 introduces the notion of directed linguistic graphs. The notion of linguistic weighted linguistic graphs are introduced in section four. The final section supplies some problems which will make the student / researcher well versed with the concept of linguistic graphs.

2.2 LinguisticGraphsinGeneral

In this section we introduce the notion of linguistic graphs using the linguistic set which is associated with the linguisticvariableL.

We find the number of possible linguistic graphs given a finitelinguisticsetS.

We discuss various properties associated with linguistic graphs in general and in particular the linguistic subgraphs associated withthem.

Example2.2.1.Let S={good,best,bad,verygood,verybad,fair} be the linguistic set / term or word associated with thelinguistic variable;performanceofastudentintheclassroom.

We give them some possible linguistic graphs associated withS.

Figure2.2.1

If wehavea edgeor arelationbetweenany twolinguistic terms / words we denote it by e. If there is no edge or a relation between twolinguistic terms/ words we denote it by  meaning emptylinguisticedge/relation.

good bad fair best good bad e bad best e good best e best fair e

Figure2.2.2

These are some of the linguistic graphs with two nodes. Usually,theywillbecalledaslinguisticdyads.

In fact using the set S we can have 6C2 = 15 such linguistic dyads. For from the linguistic set of order nwe can in generalhavenC2 numberoflinguisticdyads.

A few of the linguistic this dyads using set S given in example2.2.1isgiveninFigure2.2.2.

Figures 2.2.3(a) to 2.2.3(d) represents empty linguistic graphs.

best bad good good fair very good

Figure2.2.3(a)

Figure2.2.3(b)

good very goodfairvery good best fair

Figure2.2.3(c) Figure2.2.3(d)

We do not include in these empty linguistic graphs or the null linguistic graphs with only linguistic node / term / word withnodelinguisticrelationoredge.

There are 6 + C62 + 6C3 + 6C4 + 6C5 + 1 number of empty linguistic graphs with one linguistic node, two linguistic nodes, three linguistic nodes, four linguistic nodes, five linguisticnodesandsixlinguisticnodesrespectively.

Next, we give examples of linguistic triads and disconnectedlinguisticgraphs.

best bad fair bad

bad best very good

bestvery good good

fair very good good

best very bad

Figure2.2.4

The Figures 2.2.4 gives us some of the linguistic graphs oforderthreewithlinguisticnodesfromthelinguisticsetS.

Some of them are linguistic triads or linguistic complete graphs if all the three nodes are connected by linguistic edges /

relation. If one linguistic edge is left and there are only two linguisticedgeswecallthemaslinguisticincompletetriads.

There can be several such in complete linguistic triads however we have only 6C3 = 20 number of complete linguistic triadsusingthelinguisticsetS.

We say linguistic graphs as in case of usual graphs to be disconnectedif wehaveatleastonelinguistic nodewhichisnot connected with any of the other linguistic nodes for the given setofnodesorinthelinguisticgraph.

Definition 2.2.1. A linguistic graph G is disconnected or not connected, if we have at least one linguistic node which is not connectedwith anyoftheotherlinguistic nodesforthegivenset ofnodesorinthelinguisticgraph.

We will illustrate this situation by some examples using thelinguisticsetS.

good best very good

bad

Figure2.2.5(a)

goodbad very good good very good very good

bad fair goodbadbest fair

Figure2.2.5(b) Figure2.2.5(c) Figure2.2.5(d)

goodbad best

fair very good very bad good bad

best very bad

best

Figure2.2.5(e) Figure2.2.5(f) Figure2.2.5(g)

fair

very good fair good very good

very good bad

The Figures 2.2.5 (a) to 2.2.5 (g) give several linguistic graphs which are disconnected linguistic graphs using the linguisticsetS.

Now we provide some illustrations of complete linguistic graphs.

Before we give examples, we provide the abstract definitionofthecompletelinguisticgraph.

Definition 2.2.2. A linguistic graph G is said to be complete if every linguistic node in that linguistic graph is connected with every otherlinguistic nodeofthatlinguistic graph.

Wegivesomeexamples.

good bad

Figure2.2.6

This a linguistic dyad which is a complete linguistic graphofordertwoorwithonlytwonodes.

The dyads are foundation of any graph so are the linguisticdyadsincaseoflinguisticgraphs.

There are 6C2 = 15 such complete linguistic graphs of order2.

Considerthefollowinglinguisticgraphs.

good best bad very bad

bad best fair very bad very good

fair very good best good very good best

Figures2.2.7

All these linguistic triads are complete linguistic graphs of order three. In fact, there are 6C3 = 20 such linguistic completegraphsoforderthree.

Now we provide examples of linguistic complete graphs oforderfour.

good fair best

badgood bad very good

fair very good fair good

very badgood best very bad

Figures2.2.8

fair

The linguistic graphs given in Figures 2.2.8 are complete linguistic graphs of order four. There are 15 linguistic complete graphsoforderfour.

Next,weproceedontodescribelinguisticgraphsoforder fivewhichiscompleteinthefollowing.

bad fair best

good very bad very good bad good

very bad fair

Figures2.2.9

Figures 2.2.9 gives the linguistic complete graphs with 5 linguisticnodesusingthelinguisticsetS.

In fact, we have 6 such complete linguistic graphs with 5 linguisticnodesusingthelinguisticsetS.

Now we see there is only one linguistic complete graph usingallthesixlinguistictermsfromS.

ThisisgiveninthefollowingFigure2.2.10.

bad very good

Figure2.2.10

best

good fair very bad

Now we proceed onto give examples of linguistic subgraphsofagraph.

We use the linguistic set S associated with the age of a person.

Example2.2.2.Let

S = {old, young, very young, just young, youngest, very old, just old, middle age, just middle age, not that young, not that old,veryveryyoung,oldest}

be the linguistic set / terms associated with the linguistic variableage.

We give some linguistic graphs using them and obtain theirlinguisticsubgraphsinthefollowing.

Let G be the linguistic graph with its linguistic nodes takenfromthelinguisticsetS.

old very young

G= Figure2.2.11

just old young oldest

middle age

very old veryvery young notthat young

WefirstgivelinguisticsubgraphsofG.

The notion of linguistic subgraph of G is just as in case of classicalsubgraphsofagraphG.

Consider H1 =

old very young

just old oldest middle age

young

Figure2.2.12

isalinguisticsubgraphofthelinguisticgraphG.

It isnotedforthesubsetoflinguisticverticestakenbythe linguistic vertices of G all the edges which are connected in G oughttoremainconnectedinH1 also.

Consider H2 =

old

young

middle age very young

Figure2.2.13

just old oldest

Clearly from Figure 2.2.13 is not a linguistic subgraph of Gastheedgesconnecting

youngmiddleoldyoung age to , to , very young toyoungandmiddle age toyoung

Figure2.2.14 aremissing;hencetheclaim.

Now consider the linguistic subgraph given by the followingFigure. H3=

old young very young

just old middle age oldest

Figure2.2.15

ConsidertheFigure2.2.15,H3 isnotalinguisticsubgraph ofGastheedgesfrom

Figure2.2.16 arenotintheoriginallinguisticgraphG. Thus we define a linguistic subgraph H to be a linguistic graph G if denoted by G = ( VL, EL ) where VL denotes the linguistic nodes / vertices / terms from a linguistic set S and EL denotes the linguistic relation (e or ) from the linguistic vertices, then H = (WL, FL) where WL is a proper subset of VL (WL  VL)andFL isalsoapropersubsetofEL (FL  EL).

NowweillustratesomemorelinguisticsubgraphsofG. Let P1 =

just old

middle age very young middle age tojustold to andmiddle age tooldest old very young young

Figure2.2.17

P2 =

P1 isalinguisticsubgraphofGoforderfour.

old very young young

Figure2.2.18

P2 isalinguisticsubgraphofGoforder8. Consider P3 the linguistic subgraph of G of order 5 given bythefollowingFigure.

P3 =

very young very old oldest

just old middle age very old veryvery young oldest young veryvery young

Figure2.2.19

P4 is a linguistic subgraph of G of order 3 which is not a completetriadgivenintheFigure2.2.20.

P4 =

Figure2.2.20

Let P5 be a linguistic subgraph of G given by the followingFigure.

P5 = Figure2.2.21

just old young middle age very young middle age very old veryvery young

P5 isjustadisconnectedlinguisticsubgraphofordertwo. P6 =

Figure2.2.22

P6 is a linguistic subgraph of G of order two which is a dyad.

ConsiderP7 thefollowinglinguisticsubgraph.

P7 = Figure2.2.23

very old notthat old veryvery young old very youngyoung

just old oldest very old notthat old

middle age

P7 isacompletelinguistictriadofthelinguisticgraphG. P8 = Figure2.2.24

P8 is a linguistic subgraph of the linguistic graph G of order8giveninFigure2.2.24.

P9=

Figure2.2.25

P9 is a linguistic subgraph of G of order three in fact a linguisticlinesubgraphofG.

We give some more types of linguistic graphs using the sameB. Let W=

just young

old oldest notthat old very young youngest

young oldest

old very old

Figure2.2.26

justold

WecallWasthelinguisticstargraphoforder8.

Once in the linguistic subgraph W the term oldest is removedwegetthefollowinglinguisticsubgraphV1 V1 =

very old youngest

just young old

Figure2.2.27

Consider the linguistic subgraph V2 of W given by the followingFigure2.2.28.

very young very old

just old young very young oldest youngest

Figure2.2.28

old just old

V2 isalsoastarlinguisticgraphbutitsorderissix.

Now consider the linguistic graph given by the following Figure2.2.29.

just middle age very old

just young youngest

X= Figure2.2.29

middle age

oldest

very young old just old

WeseeXisawheellinguisticgraphoforder9.

Let Y1 be a linguistic subgraph of X given by the Figure 2.2.30.

Y1 =

middle age

just middle age very bad

very young old just old

Figure2.2.30

oldest

Clearly Y1 is a circle linguistic subgraph of order 8. It is notalinguisticwheel.

Let Y2 be a linguistic subgraph of X given by the Figure 2.2.31.

Y2 =

just young youngest very old just old

old

Figure2.2.31

oldest

Y2 is not a linguistic wheel graph or a linguistic circle graph.

Next, we prove some example of linguistic wheel graph using the same linguistic set S associated with the linguistic variableageofpeople.

young

R= Figure2.2.32

oldest justoldjust young young just young old just old

old middle age

R is called as the linguistic circle graph. The linguistic subgraphs T1, T2, T3 and T4 of R are given in the following Figure2.2.33andFigures2.2.34. T1 =

Figure2.2.33

oldest youngold just old middle age young

young middle age just old

T2 = T3= T4 = Figures2.2.34

old just youngjust old

middle age

NoneofthelinguisticsubgraphsofRarelinguisticcircles.

Some of them are linguistic connected subgraphs and some of them disconnected linguistic subgraphs given in Figure 2.2.34.

Nextweproceedontodefineedgelinguisticsubgraphs.

Wecallalinguisticsubgraphof alinguistic graphHtobe a linguistic edge subgraph if from the linguistic graph G some oftheedgesareremoved.

Wewillillustratethesituationbysomeexample.

Example2.2.3. Let Lbealinguisticvariableassociated withthe speedofthecaronroad.

Let S = {fastest, fast, just fast, slow, very slow, very fast, medium speed, just medium speed, very very slow,  ( = not runningsoemptyspeed),justslow,slowest}

bethelinguisticset/word/termassociatedwithspeedofacar.

Let K be the linguistic graph given by the following Figure2.2.35.

K=

slow very slow very fast

fast veryvery slow

medium

Figure2.2.35

L1 =

ThelinguisticedgesubgraphsofKisasfollows.

slow veryvery slow

very fast

Figure2.2.36

We have the edge linguistic subgraph got by removing theedges

L2 = Figure2.2.37 (which we will denote by (fast) (medium) (very very slow) mediumand(veryslow)medium.

When we remove the three edges we see the resultant edgelinguisticsubgraphalsoloosestwolinguisticnodesviz

L3 =

very slow fastmedium fastmediumand

Figure2.2.38

Let L2 be the edge linguistic subgraph of K got by removing the edges (very fast) (slow), (very slow) (slow) and (veryveryslow)(slow)fromK.

L4=

very slow veryvery slow

fast medium

Figure2.2.39

When the 3 edges are removed from K we get the edge linguisticsubgraphwith5verticesgiveninFigure2.2.39.

We now give the edge linguistic subgraph L5 got by removing the edges (very slow) (very fast), (very fast) (slow), (veryslow)(medium)and(veryveryslow)(slow).

L5 =

very fast slow medium

fast very slow

Figure2.2.40

Clearly L5 is a disconnected linguistic graph of order four,thishastwolinguisticdyads.

Thus, edge linguistic subgraphs play a vital role in the studyofsocialinformationnetworks.

We provide yet another example of a linguistic graph M andfinditsedgelinguisticsubgraphs.

The nodes of M are from the same linguistic set S associatedwiththelinguisticvariableage.

ThelinguisticgraphMisgiveninFigure2.2.41.

M=

old very young young just young very old

Figure2.2.41

just old

Clearly M is a complete linguistic graph of order six, so M is a connected linguistic graph. Removal of an edge will not result in a disconnected linguistic graph or with lesser number ofnodes.

Only a maximum of 5 edges can result in a linguistic subgraph of lesser order that too when all 5 edges are removed fromthesamenode.

However, to make a empty linguistic edge subgraph how manyedgesoneshouldremove.

Study in thisdirection is not difficult; simpleandis given asanexercisetothereader.

In the following section we describe directed linguistic graphs.

2.3 Linguisticdirectedgraphs

In this section we proceed onto define 3 types of linguisticdirectedgraphs.

i) Thedirectionsaregivenintheselinguistic graphs very arbitrarily (when linguistic nodes cannot be ordered).

ii) The directions are from lowest or least to largest that is increasing relation of the linguistic nodes (in case the set of linguistic nodes are totally orderable).

iii) The directions of the linguistic graph are from largest to least that is in decreasing order (in case thesetoflingnodesaretotallyorderable).

Wewilldescribe,developanddefinelinguisticgraphs.

Here we assume mostly and take either partially ordered set or totally ordered set to develop these linguistic directed graphs for if the set is unordered we can have only empty linguistic graph of (ii) and (iii) mentioned in the beginning of this section. However we can have linguistic directed graphs mentionedin(i).

Wewillfirstdescribethesesituationsbyexamples.

Example 2.3.1. Let S = {brown, black, red, blue, orange, white, green,yellow}

bealinguisticsetassociatedwithcolours.

ClearlySisneitherorderablenotpartiallyorderable.

We can only have linguistic graphs whose relations or edgesarearbitrarilymarked.

LetG1 bealinguisticgraphwithnodesfromthelinguistic setS.ThisG1 isgivenbytheFigure2.3.1.

G1 =

blue black white

orange yellow

Figure2.3.1

UsingthesamesetofvertexnodesasinG1 weget

G2 =

blue black white

Figure2.3.2

ClearlyG1 andG2 aredifferent.

Let G3 be the linguistic directed graph with same set of 5 linguisticnodes.

G3 =

orange yellow blue black white

orange yellow

Figure2.3.3

Let G4 be a linguistic directed graph with same set of 5 linguisticnodes.

G4 =

blue black white

Figure2.3.4

G5 be a linguistic directed graph with same set of vertices asthatofG1

orange yellow

G5 = Figure2.3.5

orange yellow blue black white

Let G6 be the linguistic directed graph using the same set oflinguisticnodesasinG1.

G6 = Figure2.3.6

blue black yellow

Let G7 be alinguistic directedgraphusingthesamesetof 5nodes.

G7 = Figure2.3.7

orange white blue black yellow

orange white

Let G8 be the linguistic directed graph with same set of verticesasinG1

G8 =

blue yellow orange

Figure2.3.8

Wegive yetanotherlinguisticdirectedgraphG9 usingthe samesetoflinguisticnodesasinG1

G9 =

black white blue orange

yellow white

black

Figure2.3.9

All the 9 linguistic directed graphs are distinct, but we have only empty linguistic directed graphs in case of increasing and decreasing relations between nodes as the linguistic nodes basically of the linguistic set S related to the linguistic variable colour which is neither totally ordered nor partially ordered. These are unordered linguistic set in this case. They do not fall under(ii)and(iii)mentionedinthebeginningofthissection.

Further finding the number of such directed linguistic graphscannotbeeasilyfound.

We leave it as a open problem to find out the number of suchlinguisticdirectedgraphsgivenanunorderedlinguisticset. However,weworkwith3linguisticnodes,viz., {black,white,blue}andenumerate alldirectedlinguisticgraphs inthefollowing.

black bluewhite

B1 = B2 = B1 isalinguisticemptygraph. B3 = B4 =

black bluewhite black bluewhite

black bluewhite

black bluewhite

B5 = B6= B7 = B8 = B9 = B10 = B11 = B12 = B13 = B14 =

black bluewhite black bluewhite

black bluewhite black bluewhite

black bluewhite black bluewhite

black bluewhite black bluewhite

black bluewhite

B15 = B16 =

black bluewhite

black bluewhite black bluewhite

B17 = B18= B19 = B20 = B21 = B22 =

black bluewhite black bluewhite

black bluewhite black bluewhite

black bluewhite

black bluewhite

B23 = B24= B25 = B26 = B27 = Figures2.3.10

black bluewhite black bluewhite black bluewhite

black bluewhite

Thus, we have mentioned the linguistic directed graphs using3linguisticnodes.

We have only one empty linguistic directed graph using increasinganddecreasingrelationoflinguisticnodes.

Now we provide an example of linguistic directed graph incaseofatotallyorderedlinguisticset.

Example2.3.2. Let

S = {good, bad, fair, very good, very bad, just good, just fair, justbad,best,worst} be the linguistic set associated with the linguistic variable performanceofateacherinaclassroom.

ClearlythelinguisticsetSistotallyorderable,weprovide thelinguisticincreasingchaingiveninthefollowing: worst  verybad  bad  justbad  justfair  fair  justgood  good  verygood  best …I Let W1 be a linguistic directed graph using the linguistic subset ofSgivenby T={good,bad,verybad,best,worst,verygood,fair} W1 =

bad best very bad

worst

fair very good fair

Figure2.3.11

We see W1 is a linguistic circle. However, we have not mappedallpossiblerelationsjustveryarbitrarybutsomesense!

Here we can have more number of such linguistic directedgraphsusingthelinguisticT  S.

Let W2 be a linguistic directed graph using the linguistic subsetTofS. W2=

good best very bad

bad very good fair

Figure2.3.12

W2 is also just a linguistic directed graph which is differentfromW1

Let V be the linguistic directed graph where the direction isintheincreasingrelationthatifbestandbadaretorelated.

worst bad best increasing

Figure2.3.13

Sowecannothave

Figure2.3.14

We have only one linguistic directed graph which is boundbytheincreasingrelation.

VisgiveninFigure2.3.15. V=

best

very good fair

worst

best bad decreasing good bad verybad

Figure2.3.15

Clearly V is a directed linguistic complete graph given in theincreasingdirection.

This graph is unique once given the linguistic set and the orderinwhichthedirectededgesoccur.

Let U be the linguistic directed graph in the decreasing directionusingthesamelinguisticsubsetTofS;

UisgivenbythefollowingFigure2.3.16. U= Figure2.3.16

good bad very bad

very good fair

worst

We see U is the only linguistic graph in the decreasing direction. They are unique once the set of linguistic nodes are supplied, unlike the linguistic directed graph whose edges are markedveryarbitrarily.

Nowtakethelinguisticsubset

R={good,justgood,verygood,bad}  S.

The linguistic directed graphs using the linguistic set R areasfollows

B1 = B2 =

best good badvery good

just goodgood badvery good

just good

B3 = B4 =

good badvery good

just goodgood badvery good

just good good badvery good

B5 = andsoon Figures2.3.17

We canhave severalsuchlinguistic directed graphs using thelinguisticsubsetR  S.

However, if we use the increasing or decreasing relation linguistic directed graphs are two both are distinct but we have oneandonlyonesuchlinguisticdirectedgraphinthatcase.

LetA1 bethelinguisticdirectedgraphwhichhasitsedges intheincreasingdirectionusingthelinguisticsetR.

A1 =

just good good badvery good

just good

Figure2.3.18

Let A2 be thelinguistic directedgraph where therelations oredgesareinthedecreasingdirection.

A2 isgiveninFigure2.3.19.

good badvery good

A2 = Figure2.3.19

Thelinguisticdirectedgraphwithdecreasingdirectionis unique and in relation it is just opposite of the linguistic directed graph withincreasingdirectionandviceversa.

A1 is also unique unlike the directed linguistic graphs whichenjoysnoconditionofeitherdecreasingorincreasing.

Consider the set T we canhave several directed linguistic stargraphsgiveninthefollowingFigures.

best very bad very good

bad

N1 = Figure2.3.20

just good good worstfair

bad

fair good worst

best very bad very good good worst very good

N2 = Figure2.3.21 N3= Figure2.3.22

bad

best very bad fair

worst

N4 = Figure2.3.23 N5 = Figure2.3.24

very good best bad

good very bad fair very good best bad

very bad worst good

fair

N6 =

very good

fair good best

Figure2.3.25 andsoonandsoforth.

All linguistic directed graphs with increasing or decreasing order of relations are only complete linguistic directed graphs andtheycanneverbelinguisticstargraphs.

Wecanfindlinguisticdirectedsubgraphsof all3typesof graphs.

We find some linguistic directed subgraphs of W2 which aregivenbytheFigure2.3.26.

C1 =

bad worst very bad worst

badgood very bad

Figure2.3.26

C2 =

best very bad

Figure2.3.27

C1 isacompletelinguisticdirectedsubgraphofW2

C3= Figure2.3.28

fair

badgood worst fair very good

best worst very bad

C3 isnotacompletelinguisticdirectedsubgraphofW2

badgood worstbest

LetC4 = Figure2.3.29

C5 =

Figure2.3.30

C6 = Figure2.3.31

bad best very good goodbad very bad best very good fair good very bad best very good fair

C7 = Figure2.3.32

good fairvery good

C8 = C9 =

Figure2.3.33 Figure2.3.34

C8 isaemptylinguisticdirectedgraph.

Now we see all the 9 linguistic directed subgraphs of the linguisticdirectedgraphgiveninFigure2.3.15isasfollows.

bad worstfair good worstvery bad

good fairvery good good very bad best

VL = Figure2.3.35 V2 = V3 = Figure2.3.36 Figure2.3.37

bad worst verygood

bestfair

good

bad very bad best

worst

bad good good

bestvery good

bad very bad fair

very bad very good

best good fair

V4 = V5 = Figure2.3.38 Figure2.3.39 V6= Figure2.3.40 V7 = Figure2.3.41

best

V8 =

Figure2.3.42 and V9 = Figure2.3.43

We have 9 linguistic directed subgraphs of V (increasing relation) and all of them are complete linguistic directed subgraphs. The Figures 2.3.35 to 2.3.43 gives 9 complete linguisticdirectedsubgraphsofVgiveninFigure2.3.15.

Now we give the same 9 linguistic directed subgraphs of U(decreasingdirectionoftheedges).

U1 =

good fair very good good fair very good good very bad best worst

bad fair

Figure2.3.44

U2 = Figure2.3.45

good very bad worst

U3 = Figure2.3.46

bad good very good

worst bad very bad worst

good

U4 = Figure2.3.47

bad best

U5 =

bad very good best very bad fair

Figure2.3.48 U6 = Figure2.3.49 U7 = Figure2.3.50

good fair

very bad very good

bad best

best

good very good

good very good fair

U8 = Figure2.3.51 and U9 = Figure2.3.52

fair

good very good

We see the nine linguistic directed subgraphs of V and U respectively are directed differently and further U8 and U9 (and V8 andV9)areidenticalthoughC8 andC9 aredifferent.

Recall we say two linguistic directed graphs are identical so they have same node set and the same set of directed edges (relations) V8 and V9 is an example of identical linguistic directed graphs similarly U8 and U9 are identical linguistic directedgraphs.

The edge linguistic directed subgraph in case of (increasingordecreasingorderrelation)isnotpossible.

Only in case of linguistic directed graph which has neither increasing nor decreasing edge / relation the concept of edgelingsubgraphispossible.

That is the directed linguistic graphs mentioned in (1) introductionofthissection2.3.

Example2.3.3. Let

S = {tall,short,just short,veryshort,justtall,verytall,medium height,veryveryshort,veryverytall,shortest} bethelinguisticsetassociatedwiththelinguisticvariableheight ofaperson.

Consider the linguistic directed graph G given by the followingFigure2.3.53.

G=

very short tallest

tall shortest

just short just tall

short very tall

Figure2.3.53

WegiveafewoftheGedgelinguisticdirectedsubgraphs inFigure2.3.54.

RemovethefollowinglinguisticedgesinG, edge(veryshort)(tall)

,edge (justshort)(tall)

, edge (tall)(short)

and edge (tallest)(shortest)

,edge (veryshort)(tallest)

just short

very short tallest

short just tall shortest

very tall

LetH1 betheedgelinguisticdirectedsubgraphG. H1 = Figure2.3.54

Let H2 be the edge linguistic directed subgraph of G by removingtheedges. (justtall)(veryshort)

, (justtall)(short)

, (justtall)(verytall)

(tallest)(justtall)

, (veryshort)(tallest)

H2 =

(tall)(short)

tall

and (veryshort)(tall)

just short

very short tallest

short shortest

Figure2.3.55

very tall

Let H3 be the edge linguistic directed subgraph given in thefollowing. Theedgesremovedare (justshort)(tall), (short)(justshort), (justshort)(verytall), (shortest)(justshort)

, (veryshort)(justshort)

, (justtall)(shortest)

(tallest)(shortest)

and (shortest)(verytall).

Theedgelinguisticdirectedgraphisasfollows.

short tall

just tall

very short tallest very tall

Figure2.3.56

In the case of linguistic directed graphs (either) edges increasing(relation)oredgesdecreasing(relation)).

We cannot have the edge linguistic subgraphs for we havethefollowingdefinitionofsubgraphs.

Definition 2.3.1. Let G(V, E) be a linguistic directed graph (either the edges or relations are in increasing order or decreasing order). H is defined as the linguistic directed edge subgraph of G if some of the edges e E are removed H = (V1, F) where V1  V and F  E (it is mandatory F is a proper subsetofE).

H may not in general be a linguistic directed graph in the usual sense for a relation should exist by definition of linguistic directedgraphbut ismissing.

Thus, with this limitations and thus understanding only we can define linguistic directed subgraphs (increasing / decreasing relations) which may not have some edges as per

definitions of linguistic directed graphs which are complete if thelinguisticsetunderstudyisatotallyorderedset.

If only partially ordered the definition may sometimes coincidewiththeclassicaldefinition.

Wewillillustratethissituationbysomeexamples.

Example 2.3.4. Let S be as in example 2.3.3. Let G be a linguisticdirectedgraph(increasingrelation)isgivenbelow.

G =

short tall very tall

just tall very short

just short

theresultinglinguisticdirectedsubgraphT1 asfollows;

T1 =

just tall very short

just short

Figure2.3.58

Clearly T1 is a linguistic directed subgraph in fact T1 is a completelinguisticdirectedsubgraphofG.

Consider T2 a linguistic directed subgraph of G got by removingtheedges (tall)(verytall)

and (justshort)(tall)

giveninFigure2.3.59. T2 =

short short very tall very short

, (veryshort)(short)

short short just short

just tall

Figure2.3.59

The edge linguistic directed subgraph T2 is not a completelinguisticdirectedsubgraphofG.

We proceed onto discuss linguistic directed graphs using partiallyorderedsetsbysomeexamples.

Let SE be any linguistic set; P(S) will denote the linguisticpowersetofS.

Clearly P(S) contain the empty linguistic set  and S the wholelinguisticset.

Example2.3.5. LetS={good,bad,fair,verygood} alinguisticsetassociatedwiththeperformanceofastudent.

P(S) = {, p1 = {good}, p2 = {bad}, p3 = {fair}, p4 = {very good}, p5 = {bad, good}, p6 = {bad, fair}, p7 = {bad, very good}, p8 = {good, fair}, p9 = {good, very good}, p10 = {fair, very good}, p11 = {bad, good, fair}, p12 = {bad, good, very good}, p13 = {bad, fair, very good}, p14 = {fair, good, very good},p15 ={good,bad,fair,verygood}=S} bethelinguisticpowersetofS.

Nowwegivelinguisticedgeweights,

E = {i - improvement, gi - good improvement, ni - no improvement, js - just the same, vgi - very good improvement, si-someimprovement}

forthelinguisticgraphswithlinguisticnodesfromP(S).

G1 =

p2 gi p4 p5

si p6

ni i ni ni p12

si ni

sigi p15 p9

Figure2.3.60

G1 is a linguistic graph (not directed) using partially orderedset.

Thus, we can have undirected linguistic graphs using the subsets of the linguistic power set P(S) of a linguistic set S as nodes.

Let G2 be yet another linguistic graph using the linguistic nodesofP(S).

G2 isalsonotadirectedlinguistic graphbutG2 is alsohas edgeweightorlinguisticedgeweightsfromE.

G2 =

p4 p9 si p6

vgi i ni si p8 p81

i i

i ni

gi p4 p7

Figure2.3.61

G2 isaedgeweightedlinguisticgraphoforder7.

Let G3 be another edge weighted linguistic graph given bythefollowingFigure2.3.62.

G3 =

si p14 si p12

vgi i gi gi

gip9 p10

ni i p9 i ni

Figure2.3.62

Now using the same linguistic set P(S) that is the linguistic power set of the linguistic set S which is only a partially ordered set we give a few linguistic directed graphs in thefollowing.

Let H1 be a linguistic directed graph using a subset of the linguisticpowersetP(S).

H1 =

i

p1 vgi p2 p3

ni p3 si p3

Figure2.3.63

p4p6 p9ivgivgi gi

ni gi p14 gi

p11 p10 p7

Let H2 be the linguistic directed graph given by the followingFigure2.3.64. H2 = Figure2.3.64

si

We see H2 is a linguistic directed star graph with p14 as thelinguisticcentralnode.

ni

ni

gi vgi si si

p2p1 p7 p6 p5

p9 si

p3 gi

vgi

p4

si i i si

Consider the linguistic directed graph H3 with vertex set fromthelinguisticpowersetP(S). H3= Figure2.3.65

Clearly H3 is a linguistic graph which has a linguistic subgraph that is a linguistic star graph with p9 as the central node.

Weseeitisnotawheelasthedirectionsarenotinorder.

Now let H4 be the linguistic directed graph with entries fromP(S)givenbythefollowingFigure2.3.66.

H4 =

p5 p1 p2

gi vgi

p14 gi

p4 gi

gi

p3

gi si ni ip6p8

p7 si i

Figure2.3.66

ni

We see H4 is a linguistic directed graph but is disjoint, it has2linguisticdirectedgraphsasitscomponents.

Now we have briefly described a few linguistic edge weightedsubgraphsoflinguisticedgeweightedgraphs.

Now we proceed onto give linguistic directed graph in increasing relations (here we take containment as the increasing relation if A  B A  B if B  A B  A). However, we do not give the edge (linguistic) weights by the following Figure 2.3.67.

W1=

p1 p10

p11 p2 p12 p3 p6

Figure2.3.67

We see unlike the linguistic directed graph which are built using totally ordered sets are always complete linguistic directedgraph(beitofincreasingorderorofdecreasingorder).

If totally ordered linguistic set is replaced by a partially ordered linguistic set we see the resulting linguistic directed graphingeneralmaynotbeacompleteone.

p1 p3

p2 p4

Nowwegivesomemoreexamplesofthem. W2 = Figure2.3.68

W2 isjustalinguisticdirectedemptygraphastheycannot berelatedaseachisalinguisticsubsetoforderone.

Let W3 be a linguistic directed graph given by the following Figure 2.3.69 using the linguistic subsets p1, p2, p3, p4 p15 and  ofP(S).

p15 p1

W3 =

W4 =

Figure2.3.69

Let W4 be a linguistic directed graph with entries from the linguistic power setP(S) (increasingorder relation)givenby thefollowingFigure2.3.70.

p10 p2 p4 p1

p4 p3 p2 {} p14 p5 p3

Figure2.3.70

ClearlyW4isalinguisticbinarytree.

Now we provide some examples of linguistic directed graphs(ofdecreasingrelation).

Let W5 be a linguistic directed graph given in Figure 2.3.71. W5 =

Figure2.3.71

Let W6 be a linguistic directed graph with entries from the linguistic power set P(S) of the linguistic set S given by the followingFigure2.3.72. W6=

p10 p2 p4 p1

p14 p3p2 p4 p1 p14 p5 p3

Figure2.3.72

Let W7 be the linguistic directed graph given by the followingFigure2.3.73.

W7 =

p1

p14 p2 p5 p10

p4 p9

p3

Figure2.3.73

Let W8 be the linguistic directed graph given by the followingFigure2.3.74.

p1

p6 p12  p5

W8 = Figure2.3.74

p3 p2 p11

p6

Clearly even W8 is not a complete linguistic directed graph.

p3

p1 p2 p14

p12 p5 p9 p4 p10

p11

Consider W9 the linguistic directed graph given by the followingFigure2.3.75. W9 = Figure2.3.75

We see W9 is also a linguistic directed graph which is not complete.

Now we provide linguistic subgraphs of linguistic graphs using partially ordered linguistic sets or subsets of linguistic powersetP(S)ofalinguisticsetS.

We deal both linguistic subgraphs as well as edge linguistic subgraphs of these graphs. We take basically the linguisticpowersetP(S)giveninexample2.3.5.

LetKbethelinguisticgraphgivenintheFigure2.3.76.

K= Figure2.3.76

 p3 p1 p7 p5 p11 p9 p14 p13 p1 p3

 p13 p9

We give some of the linguistic subgraphs of K in the following. B1 = Figure2.3.77

p14

ClearlyB1 isnotaconnectedlinguisticsubgraphofK.

B1 is a disconnected linguistic subgraph with two components.

p11

p1 p13

p9 p4

Let B2 be a linguistic subgraph of K given by the followingFigure2.3.78. B2 = Figure2.3.78

B2 isalsoadisconnectedlinguisticsubgraphofK.

The components of B1 are different from the components of B2 as B2 has a linguistic vertex but in case of B1 both are linguisticsubgraphs.

Let B3 be a linguistic subgraph of K given by Figure 2.3.79.

B3 =

p1 p4

p13

Figure2.3.79

p7 p7

NowwegiveafewedgelinguisticsubgraphsofK. Let C1 be a edge linguistic subgraph of K obtained by removingtheedges p3, p13, p1,p1p7,p3p7 andp7p11. C1 =

p1 p3 p5 p9 p11 p14 p13

Figure2.3.80

Let C2 be the edge linguistic subgraph of K given by the following Figure for which these edges p1, p1p7, p1p3, p1p5, p13, p7p11, p7p13, p7p3, p7p9, p9p13 and p14p13 and p14p13 are removed.

C2 =

Figure2.3.81

We see the edge linguistic subgraph is a line linguistic subgraphstartingfromp3 andendingatp9.Thatis C3 =

p3 p5 p11 p14 p9 p3 p5 p11 p14 p9

Figure2.3.82

WecanhaveseveralsuchlinguisticsubgraphsofK.

Now we find a few linguistic directed subgraphs of the linguistic directed graph M given by the following Figure 2.3.83.

p4 p2 p10

M= Figure2.3.83

Let N1 be the linguistic directed subgraph of M given by thefollowingFigure2.3.84.

N1 =

p6 p8 p12 p14 p8 p4 p10

p12 p14

Figure2.3.84

Let N2 be the linguistic directed subgraph of M given by theFigure2.3.85.

N2 = Figure2.3.85

p12 p8

p2 p6 p14

N2 is just a linguistic directed subgraph of M which is disconnected with one component just the node p6 and the other aproperlinguisticdirectedsubgraphofM.

Next we proceed onto give some edge linguistic directed subgraphR1 ofMgivenbythefollowingFigure2.3.86.

Let R1 be constructed removing the following edges fromM, viz. 1014pp

, 610pp

and 28pp

, 104pp

, 210pp

, 108pp

, 42pp

, 214pp

R1 =

p6

p4 p12

, 46pp

, 104pp

Figure2.3.86

Let R2 be a edge linguistic directed subgraph of M got by removing the following edges: 1412pp , 214pp , 128pp , 1014pp , 42pp

, 610pp

and 412pp

giveninFigure2.3.87.

p4 p2 p8

p8 p14 p10

Figure2.3.87

Now for the following linguistic directed graph Q (decreasingp14  p1).

We find its linguistic directed subgraph and edge linguisticdirectedsubgraphsofQ.

Q=

p1 p2 p4 p9

Figure2.3.88

Let D1 be the linguistic directed subgraph of Q given by thefollowingFigure2.3.89. D1 =

p3 p10 p14 p1 p2 p3 p10

p4 p9

Figure2.3.89

Clearly D1 is a linguistic directed subgraph of Q which is notconnected.

Let D2 be the linguistic directed subgraph of Q given in theFigure2.3.90.

D2 =

p14

Figure2.3.90

D2 is a directed linguistic subgraph which is connected andisatree.

Let E1 be the edge linguistic directed subgraph of Q got by removing the edges 141pp

and 149pp

given by the Figure 2.3.91. E1 =

p4 p3 p9 p10

p1 p2 p3 p4 p14

Figure2.3.91

E1 is a connected edge linguistic directed subgraph of Q. let E2 beaedge linguistic directed subgraphofQgotbymoving the edges 141pp

, 142pp

, 143pp

and 144pp

is given in the followingFigure2.3.92.

E= Figure2.3.92

p1 p3 p9 p10

p14

p4

E2 is a disconnected edge linguistic directed subgraph of Qandoneofthecomponentsisjustonevertexsetandotherisa properlinguisticdirectedsubgraphofQ.

Next, we consider the linguistic directed graph W where the edge / relation is of increasing order that is p1  p14 given by the following Figure 2.3.93 using the linguistic power set P(S)ofthelinguisticset S={good,bad,fair,verygood}

W=

p2  p2 p5 p13

p4 p7 p11

Figure2.3.93

p14 p9

WefindsomelinguisticdirectedsubgraphsofW.

Let F1 be alinguisticdirectedsubgraphofWgivenbythe followingFigure2.3.94.

F1 =

p4p2p14 p5 p7 p11

Figure2.3.94

Let F2 be alinguisticdirectedsubgraphofWgivenbythe followingFigure2.3.95.

F2 =

p9 p7 p4p2 p5 p9



Figure2.3.95

Let G1 be the edge linguistic directed subgraph of W got byremovingtheedges 2p

, 27pp

fromW.

, 49pp

, 25pp

213pp

, 11p

, 13p

and 14p

G1 isgiveninFigure2.3.96. G1 = Figure2.3.96

 p2 p5

p14

p4 p7 p9 p11 p13

We see none of the nodes of W are removed by the removaloftheseedgesfromW.

G1 is the edge linguistic directed subgraph of W with 10 edges removedfromW.

Now let G2 be a edge linguistic directed graph got by removingthefollowingedgesfromW;

that is edges 5p

, 14p

, 214pp

, 414pp

, 511p,p

, 514pp

, 714pp

, 914pp , 1114pp , 1314pp , 213pp , 413pp , 713pp and 25pp , 13p



V=

p2 p7

p4 p9 p11

Figure2.3.97

Now having seen linguistic directed subgraphs of both types we proceed describe define and develop the notion of linguistic adjacency matrices in case of the 3 types of matrices describedinsection2.1thischapter.

2.4 Linguisticedgeweightedgraphs

In this section we introduce the notion of linguistic edge weightedlinguisticgraphs.

The edge values also take only linguistic values relating thelinguisticedges.

Beforewedefinetheseconceptswedescribebysomeexamples.

Example2.4.1.Let

S = {good, bad, best, very bad, very good, fair, just bad, just good,worst}

be the linguistic set associated with the linguistic variable, the performanceofaworkerinafactory.

We give the linguistic edge set which evaluates the worker is improving or not improving or remaining good or remainingbadandsoon.

E = {i - improving, ni - not improving, vi - very much improved, ji - just improving, rg - remaining good, rbremainingbad,di-drasticimprovement}.

We just give a linguistic graph the values as labels. Let G be a linguistic graph with weighted linguistic edge values given bytheFigure2.4.1.

G=

rb

good

ni i ni

i rg rb ni ni rb ni

bad best very bad just good

justbad fair worst

Figure2.4.1

Let G1 be a line linguistic edge weighted graph given by thefollowingFigure2.4.2.

G2 =

best i very good Just good

ji fair

rb

ni bad

Figure2.4.2

Clearly G2 is a linguistic edge weight graph which is a linelinguisticgraph.

Let G3 be the complete linguistic edge weighted graph whichisgivenbythefollowingFigure2.4.3.

G3=

bestnibad ni good

i just good

yi fair

Figure2.4.3

bad

ji rb rb i i best rb

LetG4 bethestarlinguisticedgeweightedgraphgivenby thefollowingFigure2.4.4. G4 =

ni

ni ji ji rb

fair

just goodgood very good worst

Figure2.4.4

We now give wheel linguistic edge weighted graph inthe followingFigure2.4.5. G5 =

bad very bad

good rb

rbni

very good

ji ji i

fair very good best worst

rb rb ji ji i rb rb

Figure2.4.5

We wish to state though the direction is not specified in all these edge weighted linguistic graphs (dyads) we see that by seeingedgeweightsonecanforinstancesayifitis

rb besttovery igood from very good to best i orfrom

Figure2.4.6

so the direction is the worker has been assessed from the very good worker to a best worker which implies he/she is improving.Ifontheotherhandifthelinguisticedgeweight

besttovery igood from

Figure2.4.7 isniandcannotbei.

We can have several such examples of edge weighted linguistic graphs. We leave getting different undirected edge weightedlinguisticgraphsasexercisetothereader.

Now we give an example of edge weighted linguistic directedgraphs.

Example2.4.2. Let

S = {tall, verytall,just tall, tallest,medium short, verymedium, justmedium,short,veryshort,justshort,shortest}

bethegrowthoftheplantasobservedascientistassociatedwith thelinguisticvariableisgrowthoftheplantsingeneral.

LetusconsiderlinguisticedgeweightsofthisSgivenby

E = {good, bad, medium, very good, very bad, just good, just bad, worst (no proper growth), very very bad, very very good}.

Thatisthegrowthisgood,verygoodetcetc.

Let H be the edge weighted linguistic directed graph givenbythefollowingFigure2.4.8.

H=

tall good short

medium just short

Figure2.4.8

verytall bad good verybad goodtall justtall tallest

very good

good verytall

medium bad good

just good medium

Let B be the edge weighted linguistic directed graph with edge weights from the linguistic set E and nodes are taken from thelinguisticsetS.BisgivenbythefollowingFigure2.4.9. B= Figure2.4.9

We can have several such examples of such linguistic directed graphs which are neither of increasing nodes or of decreasingnodes.

Now we give some examples of directed edge weighted linguisticgraphswithincreasingnodesanddecreasingnodes.

Example 2.4.3. Let us consider the linguistic variable age of a person.

S = {old, oldest, just old, very young, adult young, youngest, justyoung,justmiddleage,middleage} bethelinguisticsetassociatedwiththelinguisticvariableage.

Let E be the linguistic edges which are weights which describesthefunctioning/abilitytodowork.

E = {very able - va, just able - ja, not that able - na, partially able-pa,able-a}

Let M be the linguistic directed edge weighted graph in increasingorderofage. M=

middle age va veryold na oldest young youngest justmiddle age

paja

Figure2.4.10

Let N be a edge weighted linguistic graph in the decreasingdirection(old  middleage)givenbythe following Figure2.4.11.

justold ja justmiddle age

na

niadult va middle age

oldest veryold

N= Figure2.4.11

va

Herewewishtokeeponrecordthefollowing:

1. The linguistic edge weights given for any linguisticsetSisinthehandsoftheexpert. The expertcangiveanysuitable/appropriatevalues forthesame.

2. Even the linguistic set associated with any linguistic variable is in the hands of expert to giveappropriately.

The flexibility at large helps the expert to choose with ease the linguistic sets which largely depends on the way he / shethinksisappropriate.

Example2.4.4.LetSandEbeasinexample2.4.2.

Let W be the linguistic edge weighted graphs given by thefollowingFigure2.4.12.

W=

tallgoodjust tall very good just medium good medium very medium good

Figure2.4.12

We give a few of its edge weighted linguistic subgraph H ofW. H1 =

just good shortestshortvery bad bad tallgoodjust tall shortestbadvery medium

Figure2.4.13

LetH2 betheedgeweightedlinguisticsubgraphH2 ofW.

H2 =

good just good

just medium goodvery medium very medium

Figure2.4.14

Let H3 be aedgeweightedlinguisticsubgraphofWgiven bythefollowingFigure2.4.15. H3 =

just good good

tallgoodjust mediumvery good medium very medium

Figure2.4.15

We see the linguistic edge weighted subgraph H1 is disconnectedit hastwocomponents both of whichare linguistic edgeweighteddyads.

The linguistic edge weighted subgraph H2 of W is a linguisticedgeweightedtriad.

The linguistic edge weighted subgraph H3 of W is a linguisticedgeweightedconnectedsubgraph.

Now having seen examples of linguistic edge weight subgraphs.

We now proceed onto give examples of linguistic edge weighteddirectedgraph.

We use the same linguistic set S and the linguistic edge set E given in example 2.4.2. and obtain a directed edge weightedlinguisticgraphP.

tall good tallest just tall

bad

P=

very bad

very tall

medium medium good mediumgoodshortbad very short just short bad

Figure2.4.16

Let Q1 be the linguistic edge weighted directed subgraph ofPgivenbythefollowingFigure2.4.17.

Q1 =

tall good just tall

medium

tallest medium just short bad very short

Figure2.4.17

Clearly Q1 is a disconnected edge weighted linguistic directedsubgraphofP.

The two components of Q1 is a linguistic edge weighted directedtriadandalinguisticedgeweighteddirecteddyad.

Let Q2 be a edge weighted linguistic directed subgraph of PgivenbythefollowingFigure2.4.18.

Q2 =

tall medium justtall medium

good short

good very short

Figure2.4.18

Nowcertainfactorsareimportant, whenweput

Q3 = Figure2.4.19

bad shortbadvery short

We mean that the plant was short after a time period it is not grown so for; its time of growth it is termed very short so weareforcedtoindicateitby“bad”.

Q4 is a connected edge weighted directed linguistic subgraphofP.

Consider Q4 a edge weighted linguistic subgraph of P givenbytheFigure2.4.20. Q4 =

tall medium just tall

tallest medium

Figure2.4.20

bad

good short very bad just short very short bad

Q4 is a disconnected edge weighted linguistic directed subgraph of P which are two components both are edge weightedlinguisticdirectedtriadsofP.

Now the interesting difference between the usual directed linguistic edge weighted graphs and the directed linguistic edge weightedgraphs

(increasingtall good verytall)areasfollows.

We can in case of the directed linguistic edge weighted graphswecanalsomark verytall bad  tall

this implies at one particular phase of time the plants was very tall for itsage(growthwasgood) butastheybecome older after a stipulated time they were only tall that is no growth so we cannotsaythegrowthisgoodsoonlywemarkitbybad.

This way of mapping is not possible in increasing or decreasing edge values. In fact, these are unique linguistic edge weighted directed graphs unlike the edge weighted linguistic directedgraphwhichcanbemanyandhasflexibility.

Now in most case we get only a complete edge weighted linguistic graphif the linguisticsetassociatedwith thelinguistic variablecanbetotallyorderable.

This is entirely different if the linguistic set is only a partially ordered set. We have surplus number of such partially ordered set by considering the linguistic power set P(S) of the linguisticsetS.

We provide example of linguistic edge weighted directed graph(increasingorderandfinditssubgraph).

For this also we use the linguistic set S and the linguistic edgesEgiveninexample2.4.2.

Let R be the linguistic edge weighted directed graph givenbythefollowingFigure2.4.21.

R=

shortest bad

good bad

tallgoodtallest justtall

short just bad

good

very good very short

Figure2.4.21

Let F1 be the linguistic edge weighted directed subgraph ofRgivenbythefollowingFigure2.4.22.

F1 =

tall good justtall

Figure2.4.22

F1 is a disconnected edge weighted directed subgraph of R which has two distinct components one is a edge weighted linguistic directed triad whereas the other one is a edge weightedlinguisticdirecteddyad.

Let F2 be a linguistic edge weighted directed subgraph of RgivenbythefollowingFigure2.4.23.

F2 =

tallest good very good shortest verytall bad tallest verygood justtallshort good shortest bad

Figure2.4.23

F2 is a connected edge weighted linguistic directed subgraphwhichisalinegraph.

Now using the same set of linguistic set S and the edge linguisticsetEgiveninexample2.4.2.

We give an example of a linguistic edge weighted directed graph (decreasing direction relation or edge) Z given by the followingFigure2.4.24.

short good shortest

bad very short

good justshort very bad bad justtall verytall bad tall good

Z= Figure2.4.24

good

Let X1 be the linguistic directed edge weighted subgraph ofEgiveninFigure2.4.25.

X1 =

short good very short bad

shortest good justshort

bad

bad tall verytall bad

Figure2.4.25

X1 is a disconnected edge weighted linguistic directed subgraphofE.

It has two components one is complete edge weight linguistic directed graph of order 4 and other component is just alinguisticdirectededgeweighteddyad.

Let X2 be the edge weighted directed linguistic subgraph ofEgivenbythefollowingFigure2.4.26.

good

short good just short bad justtall tall

Figure2.4.26

This is a line linguistic edge weighted directed subgraph ofE.

Now we find edge linguistic edge weighted directed subgraphs of linguistic edge weighted directed graph E in the following.

Let Y1 be a edge-linguistic edge weighted directed subgraphofEgotbyremovingtheedges.

(justtall)(justshort), (justshort)(veryshort), (short)(justshort)

and (justshort)(shortest)

isgiveninFigure2.4.27.

Y1 =

shortbad shortest veryshort

bad good justtallgoodverytall tall

bad good

Figure2.4.27

Y1 is a disconnected edge linguistic edge weighted directed subgraph of E. There are two components for Y1 both arelinguisticedgeweighteddirectedtriads.

Let Y2 be the edge linguistic edge weighted directed subgraphofEgotbyremovingtheedge (justshort)(veryshort)

, (veryshort)(short)

and (veryshort)(shortest)

giveninthefollowingFigure2.4.28.

shortbad shortest just short

good verybad bad justtall good verytall good tall

bad

Clearly Y2 is a connected edge linguistic edge weighted directedsubgraphofE. Figure2.4.28

Now we find the edge linguistic edge weighted directed subgraph of the linguistic edge weighted directed graph Rgiven inFigure2.4.21.

Let H1 be the edge linguistic edge weighted directed subgraphofRgotbyremovingtheedge

(short)(justtall)

givenbyFigure2.4.29.

H1 =

tall

tallest justtall

bad

short bad

Figure2.4.29

justbad

good verygood good shortest veryshort

Clearly H1 is a disconnected edge linguistic edge weighted directed subgraph of R and both the components are linguisticedgeweighteddirectedtriads.

Let H2 be a edge linguistic edge weighted directed subgraphofRgivenbythefollowingFigure2.4.30.

ThislinguisticsubgraphH2isgotbyremovingtheedges (tall)(good), (justtall)(tall), (shortest)(short) and (shortest)(veryshort)

H2 =

good justbad

tall verygood justtall short veryshort

Figure2.4.30

Clearly H2 is a connected edge-linguistic edge weighted directedsubgraphofR.

Now we find the edge linguistic edge weighted directed subgraphsofPgiveninFigure2.4.18.

Let N1 be a edge linguistic edge weighted directed subgraphofPobtainedbyremovingtheedges (medium)(justtall)

and (short)(medium)

fromPgivenbythefollowinginFigure2.4.31.

N1 =

tall good tallest

bad medium just tall

medium short very short

bad just short

verybad bad

very tall

Figure2.4.31

Let N2 be a edge linguistic edge weighted directed subgraph of P given by the following Figure 2.4.32 got by removingtheedges (tall)(tallest)

, (justtall)(tall)

givenbythefollowing

, (short)(veryshort)

N2 =

tallest badvery tall medium just tall short just shortvery short

Figure2.4.32

ClearlyN2 isaconnectedsubgraphofP.

2.5 SuggestedProblems

In this section we propose some problems which are mostly exercises for the reader to carefully attempt them in order to build a complete knowledge about this newly developinglinguisticgraphs.

This expertise will help the reader to build linguistic networksandlinguisticmodelsusingtheselinguisticgraphs.

2.5.1 Find all linguistic graphs using the linguistic set S where S = {tall, very tall, just tall, short, very short, medium, justmedium,veryveryshort}.

i) How many of these linguistic graphs using S arecompletelinguistictriads?

ii) How many linguistic graphs using S are completegraphsoforder5?

iii) FindthebiggestlinguisticcirclegraphusingS.

iv) FindthelargestlinguisticwheelgraphusingS.

iv) Give the largest linguistic star graph using the linguisticS.

2.5.2 Let G be a linguistic graph given in the following using thelinguisticsetSinproblem.

tall verytall short justtall medium just medium

very short

Figure2.5.1

i) Find all linguistic complete subgraphs of G of orderfour.

ii) FindalllinguisticsubgraphsofGoforderfour.

iii) FindallcompletelinguistictriadsofG.

iv) FindalllinguisticsubgraphsofGoforder5.

v) WhatisthebiggestlinguisticsubgraphofG? vi) How many such linguistic subgraphs of G exist mentionedinproblem5? vii) H1 = Figure2.5.2

tall justtall

verytall short tall justtall verytall short very short medium

Is H1 given in the above Figure a linguistic subgraphofG?Justifyyourclaim? viii) H2 = Figure2.5.3

Is H2 a linguistic subgraph of G or a linguistic edge subgraphofG?Justifyyourclaim.

2.5.3 Find some interesting properties enjoyed by linguistic graphsingeneral.

2.5.4 Let S = {fastest, fast, just fast, very fast, slow, just slow, slowest, very slow, medium speed, very very slow,  = (no speed just in static state which usually occurs in signals)}

be the linguistic set associated with linguistic variable movingvehicleonroad. =K

 fast very fast slow

Figure2.5.4 bethelinguisticwheel.

fastest fast very slow

i) Find all linguistic triads associated with the linguisticgraphK. ii) FindalllinguisticsubgraphsofKoforderfour. iii) FindalllinguisticsubgraphsofKoforder5. iv) How many linguistic circles can be got using thelinguisticgraphK?

v) Findalllinguisticsubgraphsoforder67.

vi) Obtain any other special property associated withK.

2.5.5 Let S be the linguistic set associated with the linguistic variable growth of paddy plants from the time it is sown tilltheyareharvested.

S = {average, not up to mark, no good yield good, stunted, very stunted, not good, good in h eight but was not green, infested by insects, very good, average, yield goodyield,juststunted,justgood,justaverage}.

Since it is over all growth S cannot be ordered for we cannotfindanyrelationwithcolour.

G=

average yield justgood growth stunted growth good growth verygood growth

good yield notagood yield very stunted

Figure2.5.5

LetGbethelinguisticgraph.

i) FindalllinguisticsubgraphsofG.

ii) FindallconnectedlinguisticsubgraphsofG.

iii) FindalledgelinguisticsubgraphsofG. iv) DoesGcontainalinguistictriad?

v) Obtain any other special features associated withthisG.

vi) Using the linguistic set S as the nodes obtain at least3linguisticgraphsoforder12,10and9.

vii) Study questions (i) to (iv) in case of these 3 linguisticgraphsgivenin(vi).

viii) CanSbeapartiallyorderedset?

ix) How many such type of partial orders can be definedonthelinguisticsetS?

x) Show linguistic sets can give many types of partialorderingonit.

xi) Can we separate S into two classes (say) S1 and S2 sothatbothS1 andS2 aretotallyordered?

xii) Can you supply a linguistic edge set for this linguisticsetS?

xiii) Can you give directed edges to the linguistic nodes used in the linguistic graph G given in Figure2.5.5?

xiv) Can you give directed increasing edges using the linguistic nodes of the linguistic graph G giveninFigure2.5.5?

xv) Draw the linguistic directed graph with decreasing edges using nodes of G given in the linguisticgraph2.5.5.

xvi) Now can we give edge weights to the linguistic graphGgiveninFigure2.5.5?

xvii) Obtain a linguistic edge weighted subgraphs of the linguistic edge weighted graph given in problem(xvi).

xviii) Using the nodes of the linguistic graph G given inFigure2.5.5draw.

a) The linguistic edge weighted directedgraph (increasingorder).

b) The linguistic edge weight directed graph (decreasingorder).

xix) Find all linguistic edge weighted directed subgraphsinxviii(a)and(b).

xx) Find all edge linguistic edge weighted subgraphsinxviiiof(a)and(b).

xxi) Does these linguistic edge directed weighted subgraphs in xviii (a) have linguistic directed edgeweightedtriads?

xxii) How many edge linguistic edge weighted directed subgraph in xviii(b) are linguistic directededgeweightedtriads?

2.5.6 Let S = {good, bad, very good, best, lazy, very fair, medium, very bad, fair, just fair, just good, just bad, very medium}

be the linguistic set associated with the linguistic variable performanceoftheworkeratafactory.

i) IsthelinguisticsetStotallyorderable?

ii) Give alinguisticedgesetEfor thelinguisticset S.

iii) Let H be the linguistic graph with linguistic set SgivenbythefollowingFigure2.5.6.

H=

just good

bad very bad

good fair lazy very medium very good best

Figure2.5.6

iv) Study all the problems (i) to (xxii) of 2.5.5 for thisHgiveninlinguisticgraphinfigure2.5.6.

2.5.7 Let S = {good, bad, best, very good, just bad, fair, worst, verybad}

bethelinguisticset.P(S)bethelinguisticpowersetofS.

{bad,best, fair} {justbad,very bad,bad,best, fair}

LetK= Figure2.5.7

{good bad}

{bad}

{fair,very bad} {best,bad, verygood,fair, worst}

a) Study all the problem 2.5.5 (i) to (xxii) using the linguistic graph K given in Figure 2.5.7.

b) What is the difference between the linguistic set associated with the graph K (Figure 2.5.7) and graphs G (Figure 2.5.5) andH(Figure2.5.6)?

ADJACENCY LINGUISTIC MATRICES OF LINGUISTIC

GRAPHS AND THEIR APPLICATIONS

3.1 Introduction

In this chapter, we introduce adjacency matrices, which arelinguisticinnatureandrelatedtolinguisticgraphs.

Here we also introduce the notion of linguistic bigraphs, and finally, we describe and discuss their applications. This chapter has four sections. Section 1 is introductory in nature; section two introduces the adjacency matrices in the case of linguistic graphs and linguistic graphs, which are directed generallydirectedinincreasingorderanddecreasingorder.

The third section describes the adjacency matrices of edge-weighted linguistic matrices. Section four suggests problemsbasedonthesestudies.

3.2 Adjacencylinguisticmatricesoflinguisticgraphs

Here we obtain adjacency matrices of four types of graphsandobtainsomeinterestingpropertiesaboutthem. Wewillfirstillustratethisbysomeexamples.

Example3.2.1.Let

S = {good, bad, best, just good, fair, just bad, very good, very bad}

be a linguistic set associated with the linguistic variable performanceaspectsofaschoolstudentintheclassroom.

Let G be the linguistic graph given by the following Figure3.2.1.

G=

verybad

good fair justgood

bad best

Figure3.2.1

Recall we have given in Chapter two of this book that if we have an edge between two linguistic terms, it is denoted by e, which means edge exists, and if the linguistic terms say good and fair, we say good and fair are adjacent and denote it by, goodfair=e.

If they have no edge connecting two linguistic terms, we say they are not adjacent and denoteit by f, whichis no edge or empty edge. This is similar to the classical graphs where 1 denotes the edge between any two vertices and 0 no edge betweentwovertices.

Further, we assume that there is no edge between a vertex withitself,thatis,thereisnoloop.

Now with this similarity in mind, we can get the linguistic adjacency matricesin the case of linguisticgraphs. The edgeset in this case would be {f, e} analogous to {0, 1}, 0 replaced by emptytermfand1replacedbye.

The linguistic adjacency matrix associated with the linguisticgraphGgiveninFigure3.2.1isasfollows.

LetMbethelinguisticadjacencyofthelinguisticgraphG.

Thefollowingobservationsaremandatory.

i) Always the adjacency matrix of a linguistic graph (whichis nota bipartitegraph) is asquare linguistic matrix(we callMthelinguisticadjacencymatrix of thelinguisticgraphG).

ii) They are always symmetric linguistic matrices unless the given linguistic graph is a directed linguisticgraph.

iii) Alwaysthe valuesor entries ina linguistic matrixis from the set {, e} unless we take edge-weighted linguisticgraphsandthemaindiagonalisalways 

With these concepts in mind, we now define the linguistic adjacencymatrixofalinguisticgraphG.

Definition 3.2.1 Let S = {a1, …, an} be the linguistic set associated with the linguistic variable L. Let G = (V, E) be a linguistic graph with V a linguistic vertex subset of S where edgessetE= {,e}andV= {a1,a2,…,am}.

The linguisticadjacencymatrixM ofGisgivenby

Clearly, M is an m  m linguistic square matrix which is symmetricaboutthemaindiagonal.

We provide more examples of adjacency linguistic matrices.

Example3.2.2. Let S = {tall, just tall, very tall, tallest, short, very short, medium, justshort,shortest,justmedium} be the linguistic variables associated with the linguistic variable height.

tall verytall medium

tallest short

LetHbethelinguisticgraphgivenbyFigure3.2.2. H = Figure3.2.2

justshort veryshort

LetWbethelinguisticadjacencymatrixassociatedwithH.

     

W= tallmediumshorttallestveryjustvery tallshortshort talleee very ee tall mediumee shortee tallesteee just ee short very ee short

Wisasymmetric7  7squarelinguisticmatrix.

Herewemakethefollowingobservationthenodes {tall,verytall,…,veryshort}isnotorderedbyus.

When getting the linguistic adjacency matrices for a totally ordered set, it is not mandatory, we need not mention the naming of rows or columns in the increasing or in decreasing order they can be anyway. It need not be in a form given by the adjacency matrix X, the adjacency matrix in W is also equally allowed.

Now if we order this linguistic subset, that is the vertex subsetofHwesee very short < short < just short < medium < tall < very tall < tallest

SowecandenotethevertexsetVofHby

V = {very short, short, just short, medium, tall, very tall, tallest}.

LetXdenotethelinguisticmatrixofHassociatedwithV. X= veryjustveryshortmediumtalltallest shortshorttall very ee short shorteee just ee short mediumee talleee very ee tall tallestee

      

.

Clearly W and X look different however one can be got fromtheotherbyshiftofrows.

In W if we shift last row to first row, 4th row to 2nd row, 6th rowto3rd rowandsoonwewillgetthelinguisticmatrixX.

Now having seen example of adjacency linguistic matrix of a linguistic graph we now give an example of the adjacency linguisticmatrixoflinguisticdirectedgraph.

Example3.2.3. LetSthelinguisticsetbeasinexample3.2.1.

Let P = {V, E}; V  S be the linguistic directed graph withentriesfromSgivenbythefollowingFigure3.2.3.

good justgood

P= Figure3.2.3

bad best verybad justbad verygood

fair

Let K be the linguistic adjacency matrix of the linguistic directedgraphPgiveninthefollowing.

       

K= goodbadbestfairveryjustveryjust badbadgoodgood goode bade very ee bad bestee just bad very good fairee just e good

ClearlyKisalsoasquarelinguisticmatrixwiththemain diagonalentriestobe 

ClearlyKisnotalinguisticsymmetricmatrix.

NowiftotallyorderthevertexsetofKas

verybad<bad<justbad<fair<justgood <good<verygood<best

We get the following adjacency matrix with the order set ofverticesforP; V = {very bad, bad, just bad, fair, just good, good, very good, best}  S.

Let R be the linguistic adjacency matrix of K using the totally orderedsetV.

R=

veryjustjustverybadfairgoodbest badbadgoodgood very ee bad bade just bad faire just e good goode very good beste

  

We see the linguistic matrix R can be got from K by row echelonmethodbyshiftofrows.

Now having seen examples of adjacency linguistic matrices related with linguistic directed graphs we now proceed onto find linguistic adjacency matrices of linguistic directed graphs (increasing order of nodes) by some examples. The above linguistic graph 3.2.3 is not in the increasing order of nodesbutonlyarbitrary.

Example 3.2.4. Let the linguistic set S be as in example 3.2.2 associatedwiththelinguisticvariableheight.

Let B = (V, E), V  S; be the linguistic directed graph (increasingorderofnodes)giveninFigure3.2.4.

tallest just medium

just tall

verytall tall veryshortshort

Figure3.2.4

medium

Clearly veryshort<short<justmedium<medium <justtall<tall<verytall<tallest isatotallyorderedlinguisticset

V = {very short, short,just medium, medium, just tall, tall, very tall,tallest}  S be thetotally orderedlinguistic set associatedwiththelinguistic nodesofthelinguisticdirectedgraphBwherethedirectionasin the case of performance of students, cannot be changed as it is heightwecanonlygetthelineinincreasingorder.

TheadjacencylinguisticmatrixQofBisasfollows.

Q=

veryjustjust shortmedium shortmediumtall very eeee short shorteee just ee medium mediume just tall tall very tall tallest

        very talltallest tall eee eee eee eee eee ee e   

Clearly, Q is a square linguistic matrix but it is not symmetric.

Infactthemaindiagonalelementsare .

Q is upper triangular linguistic matrix with empty main diagonalentries.

For anylinguistic directedgraphwith(increasingorderof nodes (or direction)) we see the resultant adjacency linguistic matrix is not symmetric butupper triangular matrix whose main diagonalisalsoempty.

This will be the case only in case of graphs where totally ordered linguistic sets are used. We give an example of a linguistic directed graph (decreasing order of nodes) using the samesetS.

Let C = {V, E}; V  S be the linguistic subset of S for thelinguisticdirectedgraph.

WegiveVinthedecreasingorder,V={tallest  verytall  tall  just tall  medium  short  just short  shortest} andE={0,e}

tallest tall verytall medium

Justtall

shortestjustshortshort

Figure3.2.5

Now we give the linguistic adjacency matrix Tassociated with the linguistic directed graph C (decreasing order) is given byT. T= justjust shortestshortmediumtall shorttall shortest just e short shortee mediumeee just eeee tall talleeeee very eeeeee tall tallesteeeeee

T is a 8  8 linguistic square matrix whose main diagonal is empty. But it is only a linguistic lower triangular matrix whose maindiagonalentriesareempty.

Now we proceed on to get linguistic directed graphs usingpartiallyorderedsets.

Example3.2.5. Let S={good,bad,best,verygood,verybad,fair} be a linguistic set associated with the performance of teacher in aclassroom.ThelinguisticpowersetP(S)ofSisasfollows:

P(S) ={,t1 = {good},t2 ={bad}, t3 = {best},t4 = {verygood}, t5 = {very bad}, t6 = {fair}, t7 = {good, bad}, t8 = {good, best}, t9 = {good, very good}, t10 = {good, very bad}, t11 = {good, fair}, t12 = {bad, best}, t13 = {bad, very good}, t14 = {bad, very bad},…,t31 ={S}}.

Let L= {V, e}bea linguistic directed graphwhere V is a subsetofP(S)andE={.E}.

V={,t2,t4,t7,t12,t9,t18 ={good,bad,best}, t20 ={verybad,fair,best}, t26 ={verybad,good,best,fair,verygood}, t27 ={bad,good,verygood}, t31 ={S}}

t2t4

t12

 t31

L= Figure3.2.6 WehaveZtobetheadjacencylinguisticmatrix. Z=

t18 t27

t7 t9 t20 t26

           

2479121820262731 2 4 7 9 12 18 20 26 27 31

tttttttttt eeeeeeeeee teeee teeee teee teee te te te te te te

Clearly Z is not a symmetric linguistic matrix however it a 11  11 square matrix whose diagonal entries are ; empty linguisticterm.

Let W = {V, E} be the linguistic directed graph (decreasingorder)with V={,t3,t5,t6,t8,t9,t11 ={fair,good}, t10 = {fair, good, bad, best}, t31, t22 = {fair, good, fair, bad, best}} andE={,e}giveninthefollowingFigure3.2.7. X=

         

3568911192231 3 5 6 8 9 11 19 22 31

ttttttttt te te te tee te tee teeee teeeee teeeeeeeeee

X is not symmetric linguistic matrix only a lower triangular matrix filled with linguistic empty words / terms alongthemaindiagonal.

W=

t5 t19

 t3 t6 t9 t8

t11 t31

Figure3.2.7

t22

Next, we proceed on to describe and develop the notion of adjacency linguistic matrix, which are edge weighted by linguisticterms/wordsbysomeexamples.

Example3.2.6. Let

S = {bad, very bad, very good, good, just good, just bad, fair, best,worst} be the linguistic set associated with the linguistic performance ofworkersinafactory.

Letusdefinethelinguisticedgeset

E = {sincere, very sincere, careless, careful, loyal, trust worthy, not lazy, lazy, active, medium, industries, indifferent, efficient,inefficient} forthelinguisticsetS.

indifferent careless justbad

bad very bad good

lazy verylazy best very good fair

medium sincere

Now we construct the edge weightage linguistic graph G = {V, G} using the subset of S for its nodes and edge set E fortheedges.GisgivenbythefollowingFigure 3.2.8. G= Figure3.2.8

Let B denote the linguistic adjacency matrix associated withthelinguisticgraphG.

B=

badverybadgood badindifferentcareless verybadindifferentinefficient goodcarelessinefficient justbadmediumlazyverylazy fairmedium bestsincere verygoodefficient

      justbadfairbestverygood medium lazy sincereefficient activesincere activeloyal sincereloyal

      

We see B is a linguistic edge-weighted matrix, which is a square matrix. It is symmetric. However, we have not ordered thelinguisticsetS.

Next, we proceed onto give the linguistic matrix for the samelinguisticgraphGbutnowweorderthevertexset verybad  bad  justbad  fair  good  verygood  best

V={verybad,bad,…,best}.

LetWbethelinguistic(adjacency)matrix.

W=

verybadbadjustbadfair verybadindifferentlazy badindifferentmedium justbadlazymedium fair goodinefficientcarelessverylazymedium verygood best

   

  goodverygoodbest inefficient careless verylazy medium efficientsincere efficientloyal sincereloyal

      

Thisisalsoasymmetriclinguisticmatrix.

One can easily get from the linguistic matrix B to W and viceversabyinterchangingtherows.

Next, we proceed onto describe directed edge weighted linguisticgraphsbyanexample.

Example 3.2.7. Let S be a linguistic set associated with the linguisticvariable,natureorcharacterofpeople,

S = {loyal, disloyal, active, inactive, sincere, not sincere, efficient, not efficient, very efficient, just efficient, industrious,

traitor, lazy, careless, very lazy, trustworthy, not trustworthy, verycareful}

andletEbethelinguisticedgesetassociatedwithS.

E = {good, bad, very good, best, just good, just bad, very bad,worst,veryfair,fair} bethelinguisticedgesetassociatedwithS.

Let H be the linguistic edge weighted directed graph givenbythefollowingFigure3.2.9. H=

good very lazy not carelesssincere

active sincere trust worthy

verybad

verygood efficient good

verygood loyal

Figure3.2.9

Let M be the linguistic adjacency matrix associated with the edge weighted linguistic directed graph H given in Figure 3.2.9.

M=

verylazycarelessnotsincereactive verylazybadverybad carelessworst notsincere active sincere trustworthy loyal efficient

       

sinceretrustworthyloyalefficient goodbest verygood verygoodgood

       

We see the linguistic directed graphs adjacency matrix is a 8  8 square matrix with all the main diagonal elements are emptyterm  andisnotsymmetric.

It is pertinent to keep on record that the linguistic set S is only a partially ordered set so if we try to label the edge weight only which falls under order  (or ) for otherwise the two linguistic nodes are not adjacent they only yield a empty linguistictermedge .

To give a concrete example of both edge-weighted linguistic directed graph (in increasing order of nodes) and linguistic edge-weighted directed graphs (with decreasing order of linguistic nodes) we have to have basically a totally ordered linguistic set for its nodes. So, we take a totally ordered linguisticsetandprovideanexampleofthesame.

Example 3.2.8. Let us consider linguistic set S associated with age of people who have specific all worked in a factory and theirproductivecapacityinthoseagegroups.

S = {just old, middle aged, just middle age, youth close to middle age, just youth (just very away from middle age)youth(attheprimeage),old}.

Let us define the linguistic set E associated with their performance related to their age which is given as weight of the edges.

E={good,bad,verygood,justgood,verybad,veryfair,best}.

Nowwesee the linguistic setSis atotally orderablesetS hasthefollowinglinguisticchain

justyouth  youth(prime)  youth

 justmiddleage  middleage  justold  old.

Let F = (V, E) be the directed linguistic edge weighted graph (increasing order) where V  S and E1  E given by the followingFigure3.2.10.

veryfair good

F= Figure3.2.10

youth prime Just middleage

justgood best verygood verygood middleage

 



youthjustyouthyouthprime

Infact the linguistic adjacency matrix has only the values just below the main diagonal and all other values are linguistic emptyword 

Further it is to be noted the rows or columns are formed by decreasing order only the edge weights are given in the increasingorder.

How does the adjacency linguistic matrix look like if the row elements and column element are marked of the decreasing order.

Let Q be the linguistic adjacency matrix of the linguistic graph F where the row and column nodes are given in increasingorderQisasfollows.

Q=

youthjustjust youth primeyouthmiddleage youth verygood prime just best youth youthverygood just middleage middleage justoldage oldage

      

middleagejustoldageoldage good justgood justgood

      

.

We see Q is a linguistic square matrix which has only nonempty linguistic just above the diagonal when the linguistic nodesareorderedinincreasingorder.

Let K denote the linguistic edge weighted directed graph with (decreasing order of nodes) given by the following Figure 3.2.11. Decreasingorder of nodes can never exist inthe case of thelinguisticvariableage.SothelinguisticgraphKisempty.

K=

youth

justold middle age just middle age youth prime just youth

Figure3.2.11

We see we cannot measure practically from old age to justoldageforageisalwaysanincreasingfactor.

So the linguistic directed graph is only an empty directed graph.

We just proceed onto describe an example wherethissort oflinguisticgraphispossible.

Example 3.2.9 Let S = {good, bad, best, very good, just good, verybad,fair,justfair,justbad}

be the linguistic variable associated with the performance of a studentinaclassroom.

verybad  bad  justbad  fair  justfair  justgood  good  verygood  best.

NowletEbethelinguisticsetgivingthe edgeweightsset forlinguisticgraphsusingthelinguisticsetS.

E = {hard working, lazy, just work, very hard worker, indifferent, does not work, does work, not alternative,verylazy}

bethelinguisticsetassociatedwithS.

bad doesnotwork good very good

best just good doesnotwork hard

doesworks justhard worker nothard worker

lazy lazy

lazy

LetGbethelinguisticedgeweighteddirectedgraph(decreasing orderofnodes)givenbythefollowingFigure3.2.12. G= Figure3.2.12

indifferent very bad fair

works just fairdoeswork

lazy

We provide the linguistic adjacency matrix M related withG. M= verybadbadjustfairfair verybad badindifferent justfairnotattentivelazy fairnotattentivedoeswork justgoodworks gooddoesnotwork verygoodlazylazylazy bestinefficientindifferent

        justgoodgoodverygoodbest doesnotwork notahardworker notahardworkerjustworkjusthardworker

       

We see M is a linguistic lower diagonal 8  8 square matrix which is lower diagonal and the main diagonal elements arejusttheemptyterm.

Now having seen examples of totally ordered linguistic sets we proceed onto work with partially ordered set; in particularthelinguisticpowersetP(S)ofS.

Example 3.2.10. Let S = {bad, good, fair, very fair, best, very good,verybad,justbad,justgood}

be a linguistic set associated with the linguistic variable performanceaspectsofateacher.

LetP(S)bethepowersetofS; P(S) = {, t1 = {bad}, {good} = t2 {fair}, = t3 {very fair} = t4, t5 = {best}, 6 = {very good}, t7 = {very bad}, t8 = {just bad}, t9 = {just good}, t10 = {bad, good}, t11 = {bad, fair}, t12 = {bad, very fair}, t13 = {bad, best}, t14 = {bad, very good}, t15 = {bad, very bad},t16 ={bad,justbad},t17 ={bad,verybad},…,t51 ={S}}, whichisapartiallyorderedset.

Let G = {V, E} be a linguistic directed graph with linguisticedgesgivenby E={veryweak,weak,justweak,medium, just medium, strong, verystrong,juststrong}.

V = {, t1, t2, t3, t6, t9, t12, t16, t26 = {just bad, just good}, t28 = {just good, very good}, t33 = {bad, good, very good}}, t51 ={S}}.

WeseeV  P(S)ando(V)=11.

Wesayveryweakwhentherelationiswith  just weak if we have a singleton set related with higher order sets

weakif wehave apairof elements inScontainedinanyset and soon}

verystrongifthesetti  tj with|ti|=28and|tj|>28

ThegraphofGisgiveninFigure3.2.13.

t1

very weak j.weak j.weak

 t3 t2 t12 t26 t16 t33

t9

G= Figure3.2.13

j.weak j.weakj.weakj.weakj.weak

t6 t28

v.weak

Gisjustalinguisticgraphwithedgeweights. LetDbethelinguisticadjacencymatrixassociatedwithG.

D= 12369 1 2 3 6 9 12 16 26 28 33

ttttt

v.weakv.weakv.weakv.weakv.weak tv.weak tv.weak tv.weak tv.weak tv.weak tv.weakj.weak tv.weak tv.weakj.weak tv.weakj.weakj.weak tv.weakj.weakj

           .weakj.weak 1216262833 ttttt

v.weakv.weakv.weakv.weakv.weak j.weakj.weak j.weak j.weak j.weakj.weak

          v.weak–veryweak;j.weak–justweak. We see D the linguistic adjacency matrix of G. D is a 11  11squaresymmetriclinguisticmatrix.

Let J = (W, E2) be a linguistic edge weighted directed graphwhere W  P(S) andE2  Egiveninthe followingFigure 3.2.14.

HereW={t1,t3,t5,t7,t9,t11,t15,t13,t18 ={bad,justgood}, t19 ={bad,fair,verybad},t20 ={bad,best,justgood} t21 ={bad,fairbest,verybad,justgood,verygood}}.

LetJ bethelinguisticedge-weighteddirectedgraphgiven bythefollowingfigure. J=

t3 t5 t7 t11 t15 t18 t19 t21

t1 t9 t20

j.weak medium weak weak j.weak mediumweak

t13

Figure3.2.14

Let T be the linguistic adjacency matrix related with linguisticedgeweighteddirectedgraphJ:

T=

ttttt t t t t t t t t t t t t

           

13579 1 3 5 7 9 11 13 15 18 19 20 21

11131518 tttt j.weakj.weakj.weakj.weak j.weak

          

j.weakj.weak j.weak j.weak j.weak

j.weak–justweak

192021 ttt j.weakj.wekj.weak j.weak j.weakj.weak j.weak j.weakj.weakj.weak j.weakj.weakj.weak weak weakweak weak weakweak medium medium

        

Now we just show how the linguistic subgraph H of a linguisticgraphGrelatestheadjacencylinguisticmatrix.

If K is the linguistic adjacency matrix of the linguistic graph G then the linguistic adjacency matrix of H is a linguistic matrix K. This linguistic matrix K of the subgraph H is defined as linguisticsubmatrixofthelinguisticmatrixM.

Wewillillustratethissituationbyanexampleortwo.

Example 3.2.11. Let Q bethe linguistic subgraph of G givenin Figure3.2.1.

bad very bad

A= Figure3.2.15

goodbadverybadfair

goode bade verybadee fairee

ClearlyXisa linguisticsubmatrixof the linguisticmatrix M.Thus,thesubgraphofGcorrespondstoasubmatrixofM.

bad good

just v.lazybad

very bad indifferent careless sincere best

lazy

Example 3.2.12. Consider the linguistic subgraph K of the linguisticgraphPgiveninFigure3.2.8. K = Figure3.2.16

The adjacency linguistic matrix Y associated with K is as follows. Y=

badverybadgood badindifferentcareless verybadindifferentinefficient goodcarelessinefficient justbadmediumlazyv.lazy bestsincere

   

justbadbest medium lazy v.lazysincere

   

ClearlyYisa linguisticsubmatrixof the linguisticmatrix B of the linguistic edge weighted graph G for which K is a linguisticsubgraphofG.

Example 3.2.13. Let us consider a linguistic edge-weighted directed subgraph L of the linguistic edge weighted linguistic directedgraphHgiveninFigure3.2.9. L=

activegood efficient

best loyal

sincere verygood verygood good

Figure3.2.17

trust worthy

Let N be the adjacent linguistic matrix associated with linguisticedgeweighteddirectedsubgraphLofH.

N=

trustefficient activesincereloyalworthybest activegood very sincere good trust worthy very loyal good very efficient good

 

Clearly N is a linguistic submatrix of M got by moving thefirstthreerowsandfirstthreecolumns.

Example 3.2.14. Considerthe linguistic edgeweighteddirected subgraph Q of the linguistic edge weighted directed graph F giveninFigure3.2.10.

Q=

youth just youth

Figure3.2.18

youth prime justmiddle age

U=

LetUbethelinguisticadjacencymatrixofQ

justmiddlejustyouth youth ageyouthprime justmiddle age youthverygood justyouthbest youthprimeverygood

   

Clearly U is a submatrix of the linguistic matrix R got by removingthefirstthreerowsandfirstthreecolumns.

Example 3.2.15: Let W be the linguistic edge weighted directedsubgraphofthelinguistic edge weighteddirectedgraph GgiveninFigure3.2.12.

W=

verybad

verygood

Figure3.2.19

just good best doesnotworkhard justhardworker inefficient notahard worker lazy justfair lazy lazy notattentive

works lazy doeswork

fair

verybadjustfairfair verybad justfairlazy fairnotattentivedoeswork justgoodworks verygoodlazylazylazy bestinefficient

     justgoodverygoodbest nothardworker doesnotworkhardjusthardworker

Thus we have a correspondence from subgraph to the submatrixoftheadjacencymatrixandviceversa.

ForinstancetakethesubmatrixTofMwhere

T=

badjustfairfairgoodbest bad justfairlazy does fair work does good notwork doesnotjust bestindifferent workhardwork

    

The linguistic directed edge weighted subgraph A associated with the submatrix T of the linguistic adjacency matrixMisasfollows.

bad doesnotwork good justworks best

fair

A= Figure3.2.20

lazy doesnotworkhard

justfair indifferent

doeswork

Now in view of all these notions we have the following theoremwhichweleaveitisanexercisetothereader.

Theorem 3.2.1. Let S be a linguistic set associated with some linguistic variable. E be the edge set either { , e} or {linguistic

weighted edges connecting nodes of S}. Let G be a linguistic edgeweighteddirectedgraphorotherwise.

Let M be the linguistic adjacency matrix of Gwith entries fromE.

Any linguistic submatrix ofM has anassociatedlinguistic subgraphwith itandvice versa.

Next we proceed onto describe linguistic bipartite graphs inthefollowingsection.

3.3 LinguisticBipartiteGraphsandtheirProperties

In this section we for the first time define the concept of linguistic bipartite graphs. The main criteria for building these graphs is we must have two linguistic sets D and R they should be such that D is disjoint in relations with itself and similarly R should also be disjoint with itself, so that we have linguistic relations are edges only connecting linguistic nodes from D to linguisticnodesfromR.

Before we define it we illustrate this situation by some examples.

Example 3.3.1. Suppose we want to relate the industry's profit or loss and the employee's work with the type of payment he gets.Werelatethemusinglinguisticbipartitegraphs.

Let D be the linguistic variables related to the employee working in an industry is taken as the linguistic domain space, andRisthelinguisticrange space whichdescribestheattributes relatedtotheindustry(likelossorgain).

D={W1 =Paywithbonusandallowances,

W2 =Onlypaynobonusorallowances,

W3 =Paywitheitherbonus(orallowance)

W4 =Employeeperformanceisatbest

W5 =Averageperformancebytheemployee

W6 =Employeeperformancepoorly

W7 =Employeeworksmorehoursand

W8=Employeeworksforonlylessnumberofhours}

Let the range space R be the linguistic attributes/terms associatedwiththeindustry.

The linguistically associated variable with industries are loss, gain, profit, no loss, less gain, good profit, no loss or gain andsoon.

R={ I1 =Maximumprofitmadebytheindustry, I2 =onlyjustprofittotheindustry, I3 =Neitherprofitnorlosstotheindustry, I4 =losstotheindustry, I5 =heavylosstotheindustry}.

Now we get the linguistic directed bigraph B1 given in Figure3.3.1.

W1 W2 W3 W4 W5 W6 W7 W8

B1 = Figure3.3.1

I1 I2 I3 I4 I5

Now we provide the related linguistic adjacency matrix N1 ofthelinguisticbipartitegraphB1 isasfollows. N1 =

12345 1 2 3 4 5 6 7 8

IIIII We We We We We We We We

       

Thusthelinguistic adjacencymatrixofB1 isarectangular 8  5matrixwithedgesetfromE={,e}.

Suppose B1 is the linguistic bipartite graph given by the firstexpert.

W1

W2 W3 W5 W6 W7 W8

W4

Figure3.3.2

I2

We can take the second experts opinion and get the linguistic bipartite graph B2 and the related linguistic adjacency matrixN2 inthefollowingFigure3.3.2. B2 =

I3

I4

I1 I5

N2 isthelinguistic adjacencymatrixassociatedwith thelinguisticbipartitegraphB2.

N2 =

IIIII Wee We We Wee We We We Wee

       

12345 1 2 3 4 5 6 7 8

Thus depending on the expert we can have different linguisticbipartitegraphs.

These are related with the real world of problem of analyzing the employees pay and work and its impact on the gainorprofitoreitheroftheindustry.

Now we give some properties of linguistic directed bipartitegraphsingeneral.

Weprovidesomeexampleslinguisticbipartitegraphs.

Example 3.3.2. Let G be a linguistic bipartite graph with D={d1,d2,d3,d4,d5,d6}aset oflinguisticsets whichisneither totally ordered or partially ordered. That is no relation exists amongthem.

Let R = {R1, R2, R3, R4, R5, R6, R7} be a set of linguistic terms which do not hold any relation among themselves which will serve as the linguistic set for the range space the nodes are RandDarerelatedlinguistically.

d1 d2 d3 d4 d5 d6

R2 R3 R4 R5 R6

R1 R7

G= Figure3.3.3 Let W be the linguistic adjacency matrix W associated withthebipartiteGisgiveninthefollowing. W=

1234567 1 2 3 4 5 6

RRRRRRR dee dee deee deee deee dee

      Wisa6  7linguisticmatrixwithentriesfromE={,e}.

On these we define some operations which are of a specialtype.

SupposeifX={(a1,…,a6)whereai  {0,e}; 1  i  6thenwecandefine

max{min{X,W}}=max{min{(e, , ,e, ,e),W}} (whereX=(e, , ,e, ,e}andmin{e, }=  andmax {e, }=e). max{min{X,W}}=max{minX,

e ee eee eee eee ee

               

} =(e, ,e, ,e,e,e}=Y. max{min{Y,G}}=max{min{(e, ,e, ,e,e,e), eee e eee e ee ee eee

                 

}}=(e,e,e,e,e,e)=X1 max{min{X,G}}=(e,e,e,e,e,e,e) =Y1 max{min{Y1,Gt}}=(e,e,e,e,e,e)=X2 =X1.

Thuswegetasamevaluehowmanytimewerepeat.

We call this as a linguistic dynamical fixed point of the linguisticsystem.

Next we proceed onto define linguistic graphs (bipartite) with linguistic edges different from using the set E = {, e} to {0, 1} where 0 corresponds to the empty linguistic term / word and 1 corresponds to the linguistic term e, whichmeans an edge eexists.

Example 3.3.3. Now we study the linguistic variables ‘economic’ states and ‘female infanticide’ as a huge social problem in India for people to give birth to many female children. It is also considered a social stigma. It is an enormous burden on the parents who give birth to many female children, for the parents have to pay much dowry in marriage for the femalechildren.

Unlike male children, female children are viewed more as aliabilitythananasset.

Here westudythisbyfirst definingthe linguisticvariables associated with riches and the problems faced by families with morefemalechildren.

Wetakethedomainspaceofthe linguistic bipartitegroup tobetheeconomicstatusofthepeopledenotedbyD e1 –veryrich e2 –rich e3 –uppermiddleclass

e4 –middleclass

e5 –lowermiddleclass

e6 –poor

e7 –verypoor

D = {e1, e2, e3, e4, e5, e6, e7} is taken as the linguistic nodes of thedomainspaceD.

Let the nodes of the range space R tests out the social / economic problems due to many female children in then families. The linguistic terms associated with the linguistic variablesocialoreconomicproblemislistedinthefollowing.

s1 - numberoffemalechildren

s2 - socialstigmaofhavingmanyfemalechildren

s3 - torture by in-laws for having many / only female children

s4 - economicloss/burdenduetofemalechildren

s5 - insecurityduetohavingonlyfemalechildren(forin days whenparents are old they have nomale child/ sontotakecareofthem)

R={s1,s2,s3,s4,s5}

Now we give the linguistic edges connecting the possible domainandrangelinguisticnodes

E={t1 =familyproblems

t2 =societalproblems

t3 =economicloss

t4 =noonetotakecareofparentsintheirolderage

t5 =feedingthemalosstofamily

t6 =socialburdenandresponsibility}.

We have mentioned a few probable edges. We have explainedeachofthelinguisticedgesti;1  i  5.

t1 –familyproblems:

In the existing scenario, in India, male children are preferred. Family problems refer to the problems faced by a young woman married into another family. It is customary to marry a female child at a young age and sometimes even to a man who is her parents' choice, not that girl's choice. It is a tradition in India that the girl must live with her husband's family and take care of her husband and his parents. Most household work and child care are done by the woman, whether she is educated or otherwise. In this state of affairs, if she gives birth to only female children, sheis scorned by her husband and hisparents.

They will torture her and ill-treat her. Some families will send her back to her parents or kill the female babies. In some cases, they may remarry her husband with the hope that the secondmarriagewillprovidemalechildren.

These women face a lot of social insults and humiliations andtorturebyothers.

t2 :Societalproblems:

When we say societal problems, we mean they are illtreated in all family functions and sometimes are not allowed to stepoutofthehouse.

Society in India looks down upon a woman with no male children.

t3:Economicloss:

The very unusual tradition in India of marrying a female child/adult in India to a stranger, mostly a man chosen by her parents, with a huge amount of dowry. To get her married, the parents of the female child have to spend money on performing the marriage, give cash as dowry, and jewellery in gold for the daughter, which will be demanded from the husband's side as social status and so forth. So ultimately, it entitles in a considerablelosstothefamilywhentheyarenotveryrich.

So they consider this only as a tremendous economic loss tothem.

t4 :Noonetotakecareofparentsintheirolderage

It is essential to mention that it is a custom in India that parents only will live with their sons in their old age. They feel it is less prestigious to live in their daughter's house. So if a couple has no male child,they have to spendtheir oldage alone with no help or live with their daughter. So they curse a daughter'sbirthandrejoiceinason'sbirth.

t5:Feedingthemalosstothefamily:

Now because a female child will incur that significant economic loss, some families in India do not feed them with good or nutritious food. In fact, many families feed the female children whatisleft over afterothers eat. Theyconstantlybrood overthattheyareaburdentothem.

Some uneducated families never send female children to school or give them a college education because they think it is a waste to invest in them. Only male children are educated in theirfamilies.

In most families, the female children are trained from the age of four or five to take care of their younger siblings and do allhouseholdwork.

t6:Socialburdenandresponsibility:

In most families, they always feel female children are a social burden and their responsibility, hence a liability to them. Thetermssocialburdenandresponsibilitymeanthefollowing.

When they say responsibility, they have to guard her from ayoungagesothatsheiskeptundertheireyes,fortheyfearshe may marry someone of another caste or religion and thereby bringingdishonourtotheirfamily.

Most honour killings are committed to harm younger women who marry into other castes and whose parents disagree withit.

That is why the term social burden and responsibility is used.

After explaining all the terms, we develop a linguistic bipartite graph using D as the linguistic domain space, R as the linguistic range space and E as its linguistic edges connectingD and R. In the existing scenario, in India, male children are preferred.

Let Bbe the linguistic bipartitegraph given bythe Figure 3.3.4.

B= Figure3.3.4

t4 t6t4 t2

t1

s2

s1 e2 s3 s4

t1t3 t5t4

e1 e3 e4 e5 e6 e7 s5

t2 t1t2 t1 t3 t3 t4 tt3 4 t3

Let S be the linguistic adjacency matrix of the linguistic bipartitegraphB.

S=

sssss ettt ettt ettt ettt ettttt etttt ett

     

12345 1214 2134 3621 4345 512345 61245 713

Theti’sarealreadydescribed.

We give a simple social problem and show how to work withthembyanexample.

Example 3.3.4. Let us now study social problems of child labourinIndia.

The linguistic terms / attributes associated with governmentfortacklingthechildlabourproblemare

g1 = childrendonotcontributetovotebank

g2 = the main source of vote politics are industrialists andbusinessmen

g3 =government should provide free and compulsory educationforchildren

g4 =government should put stringent laws against those whopracticechildlabour

R = {g1, g2, g3, g4} is taken as the range space of our linguisticbipartitegraph.

Nowforthedomainspaceof thelinguistic bipartitegraph as the linguistic attributes associated with children who are forcedtodochildlabour.

c1 - abolitionofchildlabornotputaslaw

c2 - parentsarenoteducated

c3 - school dropout or childrenwho havenever attended anyschool

c4 - socialstatusofchildlabourers

c5 - povertyorchildrenasourceoflivelihood

c6 - child labourers are runaways from home, orphans, parentsinprison/parentsarebeggars.

c7 - habitslikecinema,smoking,drinkingetc.

The linguistic bipartite graph associated with child labour takingthedomainlinguisticnodesas D={c1,c2,c3,c4,c5,c6,c7} and range of the linguistic nodes R = {g1, g2, g3, g4} the given bythefollowingFigure 3.3.5.

c1 c2 c3 c4 c5 c6 c7

Figure3.3.5

g1 g2 g4

g3

Let M be the linguistic adjacency matrix associated with thelinguisticbipartitegraphG.

M=

gggg cee cee cee ce cee cee cee

      

1234 1 2 3 4 5 6 7

SupposewetakeX=(e, , , , , , )andfind

max{min{X,M}}=(, ,e, )=Y

max{min{Y,Mt}=(e,e, , , ,e,e)=X1 max{min{X1 M}} =(e,e,e,e)=Y1

max{min{Y,Mt}} =(e,e,e,e,e,e,e)=X2

max{max{X2,M}} =(e,e,e,e)=Y2 (=Y1)

and so we see if “Abolition of child labour law not put as law”,wesee allthelinguisticnodes come toonstate inboththe linguisticnodes.

Now having seen how these linguistic bipartite graphs can be used as linguistic models we now proceed onto define this concept for the sake of completeness and for researchers to usetheminrealworldproblemsrelatedtosociety.

Definition 3.3.1, Let D = {d1, …, dn} the domain space where di  D are linguistic terms / attributes which serve as linguistic nodes(1 i n) associatedwithsomeproblem.

Let R = {r1, r2, …, rn} be the range space where rj in R are linguistic nodes / terms / attributes associated with the relatedproblemwith D;1 j  m.

LetE={,e} bethelinguisticstate (-emptylinguisticset,ethe existence ofedge).

LetX={(x1,..,xn)/xi E, 1 i n} be the linguisticstatevector space associatedwithDand Y = {(y1, …,ym) / yi E; 1 i m} be thelinguisticstate vector spaceassociatedwithR.

So a state vector (x1, …, xn) denotes the on-off state positionofthelinguisticnodeataninstant

xi = off state = eonstate.

Wehavethe bipartitelinguisticn mgraphM givenby M =

      

mm mm mm

111m 212m n1nm

where mij E={,e};1 i nand1 j m}.

We find max{min{X,M}} where X is an instantaneous linguistic vector, let max min{X, M} = Y where Y is a 1  m resultantstatevector.

Wefindmax {min{Y,Mt}}= X1 (say) thenwefind max{min{X,M}}=Y1 andsoon.

Wewillarriveatafixedlinguistic pairoralinguistic limitcycle whichwecallastheequilibriumstate.

We for the present call this as Linguistic Relational Maps (LRM) is a linguistic directed graph or a map from D to R as linguistic nodes and causalities as linguistic edges.It represents a linguistic causal relations between D and R with edges weightedfromE={,e};linguistictermsorlinguisticset.

We assume the linguistic edges form a directed cycle. ThatisLRMissaidtobeacycleifitpossessesadirectedcycle.

LRM with cycles is said to be a LRM with feedback when there is feedback in the LRM that is when the causal linguistic relations flow through a cycle in a revolutionary manner;theLRMiscalledalinguisticdynamicalsystem.

LRM functions analogous to Fuzzy Relational Maps (FRM) where the values of edges or edge weights are linguistic and the nodes of the linguistic bipartite graph is also linguistic nodes. These are very helpful in applications of real word problemsespeciallysocialproblems.

Next on similar lines we define and develop the new notionofLinguisticCognitiveMaps(LCMs)inthefollowing.

Definition 3.3.2. A Linguistic Cognitive Map (LCM) is a linguistic directed graph with linguistic attributes / nodes / events etc. as linguistic nodes and linguistic causalities as edges. This represents linguistic causal relationship between concepts.

Wewillprovideanexampleofthesameinthefollowing.

Example 3.3.5. In India there are several graduates and post graduates in last one decades. Several of them are unemployed or under employed. A expert who is a socioscientist spells out the five major concepts relating to unemployed graduates and postgraduates.

e1 - frustration

e2 - unemployment

e3 - increaseofeducatedcriminals

e4 - underemployment

e5 - takinguptodrugs,drinksetc.

The linguistic directedgraphwheree1,e2,e3,e4 ande5 are taken as the linguistic nodes and causalities as linguistic edges takingvaluesfromthelinguisticsetE={,e}.

TheexpertgivesthefollowinglinguisticdirectedgraphG.

G=

e1 e4 e3 e5

Figure3.3.6

e2

This is just an illustration of Linguistic Cognitive Maps (LCMs).

Definition 3.3.3 When the nodes of the FCMs are linguistic setsthentheyare calledasLinguisticCognitiveMaps(LCMs).

Definition 3.3.4. LCMs with edge weights from E = { , e} are calledassimple LCMs.

Definition 3.3.5: Consider the linguistic nodes or concepts c1, c2,…,cm oftheLCM.

Suppose the linguistic directed graph is drawn using the linguistic edge weight from E = { , e}. The matrix M is defined by M = (mij) where mij  { , e}; 1  i, j  n is the linguistic weight of the directed edge cicj. M is called the linguistic adjacency matrix of the LCM also known as the linguistic connectionmatrix oftheLCM.

Definition 3.3.6. Let c1, c2, …, cn be the linguistic nodes of a LCM.

X = (x1, x2, …, xn) where xi  { , e} = E. X is called as the linguistic instantaneous state vector and it denotes the on-offpositionofthe linguisticnode ataninstant.

if ai isoffand xi = e ifai ison fori= 1,2,…,n

Definition 3.3.7. Let c1, c2, …, cn be the linguistic nodes of a LCM. Let 12 ,

cc , 23 ,

cc …, cicj be the linguistic edges of the LCM (i j).Thenthe linguisticedgesformadirectedcycle.ALCM is said to be cyclic if it possesses a linguistic directed cycle. A LCM is said to be acyclic if it does not possesses a linguistic directedcycle.

We define a LCM with cycles is said to have a feedback. When there is a feedback in an LCM, i.e., when the causal relations flow through a cycle in a revolutionary way, the LCM is calledlinguistic dynamicalsystem.

Let

12c,c , 12c,c

…, n-1nc,c

be a linguistic cycle. When ci is switched on and if the causality flows through the linguistic edges of a cycle and if it again causes ci, we say that the linguistic dynamical system goes round and round. This is true for anynodeci,fori =1,2,…,n.

The equilibrium state for this linguistic dynamical system is calledthelinguistic hiddenpattern.

If the equilibrium state of the linguistic dynamical system is a unique linguistic state vector then it is called a linguistic fixedpoint.

If the LCM settles down with a linguistic state vector repeatinginthe form X1 X2 …Xi X1 Thenthisequilibriumis calledalinguisticlimitcycle.

Now for the socio economic problem of the unemployed graduates and post graduates in India let the linguistic causal connection matrix M of the linguistic directed graph in Figure 3.3.6isasfollows.

Now we findusing the linguisticdynamical systemM we workwithdifferentlinguisticinstantaneousstatevectors.

Letusconsiderthelinguisticstatevector

X = (e, , , , ) where only the linguistic node frustrationisintheonstateallothernodesareintheoffstate. Wefindmax{min{X,M}}

=max{min{(e, , , , ),

e eee ee e

            

}}=(, , , ,e).

Nowweupdatetheresultantlinguisticvector (, , , ,e) ↪ (e, , , ,e)=X1 (say) (↪ symboldenotesthelinguisticvectorhasbeupdated).

max{min{X1,M}}=(, ,e, ,e) ↪ (e, ,e, ,e) =X2 (say)

max{min{X2,M}}=(, ,e, ,e) ↪ (e, ,e, ,e) =X3 (say)=X2.

Thustheresultantisalinguisticfixedpointgivenby (e, ,e, ,e).

Weseewhenthelinguisticnodefrustrationaloneisinthe on state, then it leads a fixed linguistic point which makes the linguistic nodes e3 which is “increase of educated criminals” ande5 whichis “taking uptodrug,drinksetc”comestoonstate their by making one understand the frustration due to unemployment or underemployment of the educated (graduate and post graduates) leads to increase in educated criminals and theytakingdrugsanddrinkstocoveruptheirfrustration.

Now let us consider the on state of the linguistic node unemployment of the postgraduate and undergraduates, that is Y=(,e, , , )bethelinguisticstatevector.

WestudytheeffectofYonM;

max{min{Y, M}} = (e, , e, , e) ↪ (e, e, e, , e) = Y1 (say) max{min{Y1,M}}=(e, ,e, ,e) ↪ (e,e,e, ,e)=Y2 (say).

Nowwefindout

max{min{Y2,M}}=(e, ,e, ,e) ↪ (e,e,e, ,e) =Y3 (=Y2).

We see the on state of the linguistic node unemployment makes the linguistic nodes frustration, increased educated criminals and taking up drugs as the resultant linguistic state vector.

Now let us consider the linguistic state vector in which only the node increase of educated criminals, e3 is in the on state.

We find the effect of Z = (, , e, , ) on the linguistic dynamicalsystemM.

max{min{Z,M}}=(, , , , ) ↪ (, ,e, , )=Z1 (=Z).

We see according to this socio scientist increase of educatedcriminalshavenoeffectonthedynamicalsystemM.

Now consider the on state of the linguistic node under employmente4.

Let A = (, , , , e, ) be the linguistic state vector, we study theeffectofAonM. max{min{a,M}}=(e, ,e, , ) ↪ (e, ,e,e, )=A1 (say)

NowwefindtheeffectofA1 onthedynamicalsystemM. max{min{A,M}}=(e, ,e, ,e) ↪ (e, ,e,e,e)=A2 (say).

NowwefindtheeffectofA2 onthedynamicalsystemM. max{min{A2,M}}=(e, ,e, ,e)  (e, ,e,e,e) = A3 (=A2).

Thus the linguistic resultant of A on M gives a linguistic fixedpointgivenby (e, ,e,e,e).

That is on state of the linguistic node under employment leads to the on state of the linguistic nodes, frustration, increased educatedcriminalsandtotakinguptodrugsetc.

The onlylinguisticnodeswhichremainsintheoff state is e2 =unemployment.

Finally we study the effect of the linguistic node taking uptodrugsetc.thatise5 intheonstate.

Let B = (, , , , e) be the linguistic state vector in which only the linguistic node e5, taking up to drugs etc., is in the on state and all other nodes variable, taking upto drugs, drinks etc. leads to the on state of the increase of educated criminals.

Thuswehaveprovidedasexampleofhowwecanusethe conceptofLCMtomodelsocialproblems/issues.

Now we have given some simple basic applications of linguisticgraphstolinguisticmodelsLCMandLRM.

3.4SuggestedProblems

In the following section we provide some problems for the reader to solve so that by working out these problems they become well versed in linguistic graphs and their properties and applicationstosomesimplelinguisticmodels.

The problems with * or starred problems are different to besolved.

3.4.1. Let G be a linguistic graph associated with the linguistic set

S = {tall, very tall, short, very short, just short, just tall, verymedium,tallestmediumandjustmedium} givenbythefollowingFigure3.4.1.

G=

very tall tall just short

very short

short

medium

Figure3.4.1

i) Give the adjacency linguistic matrix M associated withG.

ii) IsMasquaresymmetricmatrix?

iii) Find all the linguistic subgraphs of G of order four andfindtheirlinguisticadjacencymatrices.

iv) Is the linguistic adjacency matrices in problem (iii) linguisticsubmatricesofM?Justifyyourclaim!

3.4.2 LetSbeasinproblem(1)

Let H be a linguistic directed graph with linguistic nodes fromSgivenbythefollowingFigure 3.4.2

H=

tallest tall very tall just tall just medium short

mediu m tallest tall just medium short

Figure3.4.2 i) Studyquestions(i)to(iv)ofproblem(1)forthisH. ii) LetP=

Figure3.4.3

bethelinguisticdirectedsubgraphofH.

a) Find the linguistic adjacency matrix T associated withP.

b) Is T a linguistic submatrix of the linguistic matrix associatedwithH?

c) What are the special features enjoyed by this linguistic graph P and its adjacency linguistic matrix?

3.4.3 Enumerate some special features enjoyed by linguistic directedgraphsingeneral.

3.4.4 Let S be a linguistic set associated with the linguistic variablespeedofavehicleontheroad;

S = {fastest, just fast, very fast, fast, moderate speed, just moderatespeed,slow,veryslow, ,justslow}.

Let G bethe directedlinguistic graph (increasing orderof linguisticnodes)givenbythefollowingFigure3.4.4.

just fast

G=



fast fastestmoderate justslowvery slow slow

Figure3.4.4

i) IsthelinguisticsetSortotallyorderedset?

ii) Find the adjacency linguistic matrix M associated withG.

iii) IsMasymmetriclinguisticmatrix?

iv) Is M a upper triangular linguistic matrix or a lower triangularlinguisticmatrix?

v) How many linguistic directed subgraphs in G are completelinguisticsubgraphs?

vi) Find at least four linguistic submatrices of M and find the corresponding linguistic directed subgraphs associatedwiththem.

vii) Find five distinct linguistic direct subgraphs of orders 3, 4, 5, 6 and 7 and find the corresponding linguistic matrices and show they are also linguistic submatricesofM.

viii) Let K be an edge directed linguistic subgraph of G giveninFigure3.4.5.

fast slow very slow 

K= Figure3.4.5

just fast

Find the related linguistic adjacency matrix of K. IsKalinguisticsubmatrixofM?

ix) Can we prove every edge linguistic directed submatrix of G contribute to a linguistic submatrix ofM?Proveordisprovethesame!

3.4.5 Let G be a linguistic bipartite graph associated with some linguistic domain and range sets D and R given by the Figure3.4.6

(D={x1,x2,x3,x4,x5}andR={y1,y2,y3,y4}) G= Figure3.4.6

x1 x2 x3 x4 x5 y4

y1 y2 y3

i) FindthelinguisticadjacencymatrixNofG.

ii) Prove all linguistic bipartite subgraphs of G is associated with the linguistic submatrix of N and viceversa.

iii) Is(ii)trueiflinguistic bipartitesubgraphisreplaced byedgelinguisticsubgraph?Justifyyourclaim.

3.4.6* Let S = {dense, very dense, not that dense, scattered, moderate, very moderate, very scattered, just dense, justscattered,sparse,justsparse,verysparse}

bethelinguisticterms/setassociatedwiththelinguistic variablegrowthofpaddysowninhandinafield.

Let the linguistic edge set E which measures the yield usingthelinguisticsetS.

E = {good yield, moderate yield, poor yield, very poor yield,justgoodyield,justmoderate},

i) Construct linguistic edge weighted graph using the linguisticsetSandedgeweightfromE.

ii) Construct the linguistic adjacency matrix for the graphconstructedin(i)

iii) Obtain a linguistic edge weighted directed graph H using the linguistic nodes S and the linguistic edges fromE.

iv) Is the linguistic adjacency matrix L associated with graph H given in (iii) a upper triangular matrix or a lowertriangularmatrixorneither?

v) IsSatotallyorderedlinguisticset?

vi) Is the linguistic edge weighted directed graph W(in the increasing order of nodes) yield a upper triangular linguistic adjacency matrix? Justify your claim.

3.4.7 Obtain any interesting property associated with linguistic edgeweighteddirectedgraphs.

3.4.8*Construct a LCMs which involves indeterminacy as linguisticnodes.

3.4.9 Apply linguistic directed graph with edge weights from E = {, e} using LCM for any social problem (real world) anddrawconclusionsbasedonit. CompareitwithclassicalFCM.

3.4.10Apply LCM model to the problem of symptom-disease model in children. The 8 linguistic nodes are supplied in thefollowing

d1 - coldwithfever d2 - vomiting,loosemotionwithfever d3 - feverwithlossofappetite d4 - coughandfever d5 - respiratorydiseases d6 - gastroenteritis d7 jaundice d8 - tuberculosis

The linguistic directed graph G given by an expert who is adoctorisasfollows

G = Figure3.4.7

d1 d5 d2 d7 d6 d3 d8

d4

i) Obtain the adjacency linguistic matrix/connection linguistic matrix M of the linguistic graph G with edgeweightfromE={,e}.

ii) Find the linguistic fixed point / limit cycle for the linguistic point / limit cycle for the linguistic state vectorX=(e, , , ,e, , , )

iii) Suppose Y = (, , , e, e, , , ) be a linguistic state vector using linguistic dynamical system M of LCMfindthelinguistichiddenpatternofY.

(The equilibrium state of the linguistic dynamical system of theLCMis calledasthelinguistic hidden pattern).

iv) Suppose A = (, , , , , e, e, e) is the linguistic state vector find the linguistic hidden pattern of A oftheLCM.

v) Find the maximum number of linguistic fixed pointsgotusingLCMofall28 statevectors.

vi) How many of the linguistic hidden patterns are linguisticfixedpoints?

vii) How many of the linguistic hidden patterns are linguisticlimitcycles?

viii) Obtain any other special feature associated with the problem.

3.4.11 Let us stud the relation between the teacher and the student. Let D denote the linguistic nodes associated withteacher

D={(t1,t2,t3,t4,t5,t6 t7}

Where t1 =goodteacher t2 =poorteacher t3=teacheriskind t4=teacherisharsh(rude) t5 =teacherisindifferent t6=teacherismediocre t7 =teacherisdevoted

Let R be the linguistic nodes associated with the student R={s1,s2,s3,s4,s5}where

s1 =goodstudent

s2 =badstudent

s3 =averagestudent

s4 =studentindifferenttostudies

s5 =lazystudentinclass

i) Using the linguistic nodes D and R construct a LRM;thatisalinguisticbipartitegraphB.

ii) If P1 is the 7  5 linguistic connection matrix of B with edge weights from E = {, e}; determine the following.

a) The linguistic hidden pattern of X = (e, , , , , , )usingtheLRM.

b) Using the LRM the dynamical system P1 t find the linguistic hidden pattern of Y = (, e, , , ).

Will Y leadto a linguistic fixed point pair or a limit cyclepair?

c) Find outatleast 2 linguistic state vector from X = {(a1, a2, a3, a4, a5, a6, a7) / ai  E = {, e}; 1  i  7} whose linguistic hidden pattern is a linguisticlimitcyclepair.

d) Find out all linguistic hidden patterns whichare linguisticfixedpointspairoftheLRM,P1.

iii) Can E = {, e} be replaced by linguistic edge sets forthisDandR?

FURTHER READING

1. Broumi S, Ye J, Smarandache F. An extended TOPSIS method for multiple attribute decision making based on intervalneutrosophicuncertainlinguisticvariables.Infinite Study;2015Jul1.

2. DatLQ,ThongNT,AliM,SmarandacheF,Abdel-Basset M, Long HV. Linguistic approaches to interval complex neutrosophic sets in decision making. IEEE access. 2019 Mar4;7:38902-17.

3. Fan C, Feng S, Hu K. Linguistic neutrosophic numbers einstein operator and its application in decision making. Mathematics.2019Apr28;7(5):389.

4. Fan C, Ye J, Hu K, Fan E. Bonferroni mean operators of linguisticneutrosophicnumbersandtheirmultipleattribute group decision-making methods. Information. 2017 Sep 1;8(3):107.

5. Fang Z, Ye J. Multiple attribute group decision-making method based on linguistic neutrosophic numbers. Symmetry.2017Jul7;9(7):111.

6. Herstein., I.N., Topics inAlgebra,WileyEastern Limited, (1975).

7. Kamacı H. Linguistic single‐valued neutrosophic soft sets with applications in game theory. International Journal of IntelligentSystems.2021Aug;36(8):3917-60.

8. KandasamyWV,IlanthenralK,SmarandacheF.Algebraic StructuresonMODPlanes.InfiniteStudy;2015.

9. Lang,S.,Algebra,AddisonWesley,(1967).

10.Li YY, Zhang H, Wang JQ. Linguistic neutrosophic sets and their application in multicriteria decision-making problems. International Journal for Uncertainty Quantification.2017;7(2).

11.Meyer A, Zimmermann HJ. Applications of fuzzy technologyinbusinessintelligence.InternationalJournalof Computers Communications & Control. 2011 Sep 1;6(3):428-41.

12.PengX,DaiJ,SmarandacheF.Researchontheassessment of project-driven immersion teaching in extreme programming with neutrosophic linguistic information. InternationalJournalofMachineLearningandCybernetics. 2022Oct10:1-6.

13.Peng, X., Dai, J. & Smarandache, F. Research on the assessmentofproject-drivenimmersionteachinginextreme programmingwithneutrosophiclinguisticinformation.Int. J. Mach. Learn. & Cyber. (2022). https://doi.org/10.1007/s13042-022-01669-6.

14.Smarandache F, Broumi S, Singh PK, Liu CF, Rao VV, Yang HL, Patrascu I, Elhassouny A. Introduction to neutrosophyandneutrosophicenvironment.InNeutrosophic Set in Medical Image Analysis 2019 Jan 1 (pp. 3-29). AcademicPress.

15.Smarandache F. Linguistic paradoxes and tautologies. Collect.Pap.Vol.III.2000;3:122.

16.Smarandache, Florentin, Neutrosophic Logic— Generalization of the Intuitionistic Fuzzy Logic, Special SessiononIntuitionisticFuzzySetsandRelatedConcepts, International EUSFLAT Conference, Zittau,Germany, 1012September2003.

17.Tian ZP, Wang J, Zhang HY, Chen XH, Wang JQ. Simplified neutrosophic linguistic normalized weighted Bonferroni mean operator and its application to multicriteria decision-making problems. Filomat. 2016 Jan 1;30(12):3339-60.

18.VasanthaKandasamyWB,SmarandacheF,AmalK.Fuzzy Linguistic Topological Spaces. arXiv e-prints. 2012 Mar:arXiv-1203.

19.Vasantha Kandasamy, W.B., Ilanthenral, K., and Smarandache,F.,MODplanes,EuropaNova,(2015).

20.Vasantha Kandasamy, W.B., Ilanthenral, K., and Smarandache,F.,MODFunctions,EuropaNova,(2015).

21.Vasantha Kandasamy, W.B., Ilanthenral, K., and Smarandache, F., Multidimensional MOD planes, EuropaNova,(2015).

22.Vasantha Kandasamy, W.B., Ilanthenral, K., and Smarandache,F.,NaturalNeutrosophicnumbersandMOD Neutrosophicnumbers,EuropaNova,(2015).

23.Vasantha Kandasamy, W.B., Ilanthenral, K., and Smarandache, F., Linguistic functions, Editorial Global knowledge,(2022).

24.Vasantha Kandasamy, W.B., Ilanthenral, K., and Smarandache, F., Linguistic matrices, Editorial Global knowledge,(2022).

25.YeJ.Someaggregationoperatorsofintervalneutrosophic linguistic numbers for multiple attribute decision making. Journal of Intelligent & Fuzzy Systems. 2014 Jan 1;27(5):2231-41.

26.Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning—I. Information sciences.1975Jan1;8(3):199-249.

27.Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning—II. Information sciences.1975Jan1;8(4):301-57.

28.Zadeh LA. The concept of a linguistic variable and its application to approximate reasoning-III. Information sciences.1975Jan1;9(1):43-80.

29.Zimmermann HJ. Description and optimization of fuzzy systems.InternationaljournalofgeneralSystem.1975Jan 1;2(1):209-15.

30.ZimmermannHJ.Fuzzysettheory.Wileyinterdisciplinary reviews:computationalstatistics.2010May;2(3):317-32.

31.Zimmermann HJ. Fuzzy set theory—and its applications. SpringerScience&BusinessMedia;2011Jun27.

NDEX

A

Adjacencymatricesofedge-weightedlinguisticgraphs, 150-8

C

Circlelinguisticgraph, 53-62

Completeedgeweightedlinguisticdirectedgraph, 134-140

Completeedgeweightedlinguisticgraph, 118-124

Completelinguisticdirectedgraphs, 79-85

Completelinguisticdirectedsubgraphs, 79-85

Completelinguisticgraphs, 38-45

Completelinguistictriad, 35-40

Connectededgelinguisticedgeweightedsubgraph, 136-145

Connectededgeweightedlinguisticgraphs, 128-133

Connectedlinguisticedgeweightedgraphs, 128-133

Connectedlinguisticgraph, 59-66

D

Directededgeweightedlinguisticsubgraph,134-140

Directedlinguisticemptygraphs, 99-105

Directedlinguisticgraphs, 30-40

Directedlinguisticsubgraphs, 79-85

Disconnectededgelinguisticedgeweighted subgraphs 136-145

Disconnectedlinguisticedgeweighted graphs, 128-130

Disconnectedlinguisticgraph, 35-40

E

Edge-directedlinguisticsubgraphs 90-97

Edgelinguisticdirectedsubgraphs, 90-97

Edgelinguisticedge-weighteddirected subgraph, 135-40

Edgelinguisticsubgraph, 57-65

Edge-weightedcompletelinguisticgraph, 118-124

Edge-weightedlinguisticgraphs, 116-120 Edgeweightedlinguisticsubgraph, 133-140 Emptylinguisticgraphs, 33-40

F

FuzzyRelationalMaps(FRMs) 210-8

I

Identicallinguisticdirectedgraphs, 87-93 Incompletelinguistictriad, 35-40

Increasingordecreasingrelationof linguisticnodes, 69-76

L

Linelinguisticedgeweighteddirectedgraph,135-140

Linelinguisticedgeweightedgraph, 118-124

Linguisticadjacencymatrices, 116–120,160-9

Linguisticadjacencymatrixofa linguisticbipartitegraph, 196-200

Linguisticbipartitegraphs, 193-200

Linguisticcirclegraph, 53-62

LinguisticCognitiveMaps(LCMs), 210-8

Linguisticcompletegraphs, 38-45

Linguisticcompletetriad, 35-40

Linguisticconnectedgraph, 59-66

Linguisticconnectionmatrix, 212-9 Linguisticcycles, 212-9

Linguisticdirectedcompletegraphs, 79-85

Linguisticdirectedcompletesubgraphs, 79-85

Linguisticdirectedcycle, 212-9

Linguisticdirectededgeweightedsubgraph,134-140

Linguisticdirectedemptygraphs, 99-105

Linguisticdirectedgraphs, 30-40,59-65

Linguisticdirectedsubgraphs, 79-85

Linguisticdisconnectedgraph, 59-67

Linguisticdyads, 32-40

Linguisticdynamicalsystem 212-9

Linguisticedgeweightedconnectedgraphs,128-133

Linguisticedgeweighteddirectedtriads, 138-142

Linguisticedgeweighteddisconnected graphs, 128-130

Linguisticedgeweighteddyads, 120-125

Linguisticedgeweightedgraphs, 116-120

Linguisticedgeweightedlinegraph, 118-124

Linguisticedgeweightedmatrix, 169-76 Linguisticedgeweightedstargraph, 119-128

Linguisticedgeweightedsubgraph, 133-140

Linguisticedgeweightedwheelgraph, 119-128

Linguisticedge, 30-40

Linguisticemptygraphs, 33-40

Linguisticfixedpoint, 214-9

Linguisticgraph, 7-25

Linguisticgraphswhicharenotdirected, 30-40

Linguistichiddenpattern, 213-9 Linguisticincompletetriad, 35-40 Linguisticlimitcycle, 213-9 Linguisticlowertriangularmatrix, 164-175

Linguisticnode, 31-40 Linguisticpowerset, 20-30 Linguisticrelation, 30-40

LinguisticRelationalMaps(LRM), 209-12

Linguisticsentences, 7-9

Linguisticset, 7-20

Linguisticstargraph, 51-9 Linguisticstatevector, 213-9

Linguisticsubgraphs, 30-40 Linguisticsubmatrix, 190-6 Linguistictriads, 35-40 Linguisticvalue, 7-25 Linguisticvariable, 7-30

Linguisticweightedgraphs, 30-40 Linguisticwheelgraph, 52-61 Linguisticwords, 7-25 Linguisticallycomparable, 18-30 N

Notorderable, 13-30

Orderableset, 14-30 Orderingoflinguisticsets, 13-30

P

Partialorderingoflinguisticsets, 13-31 Partiallyorderableset, 14-30

S

Squarelinguisticmatrices, 159 Starlinguisticedgeweightedgraph, 119-128 Starlinguisticgraph, 51-9

T

Totallyorderablelinguisticset, 14-25 Totallyorderableset, 14-30 Totallyorderedlinguisticgraph, 69-75

U

Undirectedlinguisticgraphs, 30-40 Uppertrianglelinguisticmatrix, 162-9

W

Wheellinguisticedgeweightedgraph, 119-128 Wheellinguisticgraphs, 52-61

ABOUT THE AUTHORS

Dr.W.B.Vasantha Kandasamy is a Professor in the School of ComputerScienceandEngg,VITUniversity,India.Inthepastdecade she has guided 13 Ph.D. scholars in the different fields of nonassociative algebras, algebraic coding theory, transportation theory, fuzzy groups, and applications of fuzzy theory of the problems faced in chemical industries and cement industries. She has to her credit 694 research papers. She has guided over 100 M.Sc. and M.Tech. projects. She has worked in collaboration projects with the Indian Space Research Organization and with the Tamil Nadu State AIDS Control Society. She is presently working on a research project funded by the Board of Research in Nuclear Sciences, Government of India. This is her 134th book. On India's 60th Independence Day, Dr.Vasantha was conferred the Kalpana Chawla Award for Courage and Daring Enterprise by the State Government of Tamil Nadu in recognition of her sustained fight for social justice in the Indian Institute of Technology (IIT) Madras and for her contribution to mathematics. The award, instituted in the memory of IndianAmerican astronaut Kalpana Chawla who died aboard Space Shuttle Columbia, carried a cash prize of five lakh rupees (the highest prizemoneyforanyIndianaward)andagoldmedal. Shecanbecontactedatvasanthakandasamy@gmail.com WebSite:http://www.wbvasantha.in

Dr. K. Ilanthenral is Associate Professor in the School of Computer Science and Engg, VIT University, India. She can be contacted at ilanthenral@gmail.com

Dr. Florentin Smarandache is a Professor of mathematics at the University of New Mexico, USA. He got his MSc in Mathematics and Computer Science from the University of Craiova, Romania, PhD in Mathematics from the State University of Kishinev, and Postdoctoral in Applied Mathematics from Okayama University of Sciences, Japan. He is the founder of neutrosophy (generalization of dialectics), neutrosophic set, logic, probability and statistics since 1995 and has published hundreds of papers on neutrosophic physics, superluminal and instantaneous physics, unmatter, quantum paradoxes, absolute theoryofrelativity,redshiftandblueshiftduetothemediumgradient andrefractionindexbesidestheDopplereffect,paradoxism,outerart, neutrosophyasanewbranchofphilosophy,LawofIncludedMultipleMiddle, multispace and multistructure, degree of dependence and independence between neutrosophic components, refined neutrosophic set, neutrosophic over-under-offset, plithogenic set, neutrosophic triplet and duplet structures, quadruple neutrosophic structures, DSmT and so on to many peerreviewed international journals and many books and he presented papers and plenary lectures to many international conferences around the world. http://fs.unm.edu/FlorentinSmarandache.htm Hecanbecontactedatsmarand@unm.edu