Study On Neutrosophic Graph with Application in Wireless Network

CAAITransactionsonIntelligenceTechnology

ResearchArticle

Studyonneutrosophicgraphwithapplication inwirelessnetwork

YaoFei

ISSN2468-2322

Receivedon16thMay2020 Revisedon24thJune2020 Acceptedon29thJune2020 doi:10.1049/trit.2020.0093 www.ietdl.org

CollegeofInformationScienceandTechnology,GansuAgriculturalUniversity,Lanzhou730070,People’sRepublicofChina ✉ E-mail:yaofei111111@sina.com

Abstract: Aneutrosophicnetworkisanextensionofanintuitio nisticfuzzynetworkthat providesmoreprecision compatibilityandflexibilitythananintuitionisticfuzzygraphinstructuringandmodellingmanyreal-lifeproblems.The authorshaveexploredtheuseofaneutrosophicnetwo rkformodellingthepassiveopticalnetwork,mobile adhoc network(MANET),andwirelesssensorgraph.Theyhavepresentedtheideaofstrongarc,weakarcstrongdomination numbers,andstrongperfectdominationofneutrosophicnetwork.Theyhavedescribedthemethodtofindthevalues ofstrongandstrongperfectdominationofneutrosophicn etwork.Finally,theyuseth eideaofastrongarcstrong dominationnumberinMANETandwirelesssensorgraphs.

1Introduction

Graphtheories[1–3]haveseveralreal-lifeapplicationsintheareaof computerscience,transportation,physics,systemsanalysis,biology, economics,astronomy,andoperationsresearch.Inthoseproblems, thegraphemergesasamathematicalgraphicalmodelofthe observedreal-lifeproblems.Let G= (V , E )beagraph,where V representsthenon-emptysetofallnodes/verticesand E represents thesetofallarcs/edgesofthegraph G.Twonodes k and l ina graph G aredefinedasanadjacentnodein G ifandonlyif(k , l )is anarcof G.Thegraph G isasimplegraphif G hasnomultiple arcsandloops.Asimplegraph G issaidtobeacompletegraphif andonlyifeverypairofdistinctnodesof G isjoinedbyanarc. Ifthecompletegraph G has n numberofnodesthenithasalways n(n 1)/2arcs.Inthisstudy,wehaveworkedondifferenttypes ofneutrosophicnetworks.Manydecision-makingproblems, e.g.trafficplanning,texturemapping,shortestpathproblem, spanningtreeproblemetc.canbemodelledasagraph,which consistsofobjects/items/elementsandrelationships.Expertsuse thenodesofthegraphtorepresenttheobjects/items/elementsand edgestorepresenttherelationshipbetweenobjectsoftheproblem. Inalmosteverydecision-makingproblem,theusableinformation isgenerallyapproximateorimpreciseduetomanydistinctreasons suchaslackofevidence,imperfectstatisticalinformation,lossof information,inaccuratedata,andinsufficientinformation.

Thebasicideaofthecrispsettheoryhasbeenintroducedby Cantor,whichisusedinclassicalgraphtheory.Inagraph,forany nodeorarc,thereexisttwodistinctpossibilities:the firstone eitherpresentinthegraphorisnotpresentinthegraph.Owingto thisreason,thesimplegraphtheoryisunabletomodeluncertain real-lifeproblemsproperly.Fuzzyset[4]isanupperversionofa crispset,whereeachandeveryitem/elementshavevarying membershipgrade.Itcanshowthatitselementshavedistinct membershipgradesbetweenintervals1and0.Theideaof membershipdegreesisdifferentfromprobability.

TheideaofafuzzygraphhasbeenpresentedbyKaufmann[5] usingthefuzzyrelation.In[6],Rosenfeldhasdescribedthe variousideasoffuzzybridges,fuzzycycles,fuzzypaths,fuzzy trees,andfuzzyconnectednesstoafuzzygraphandpresented someofthepropertiesofthem.Severalmathematicians,suchas RashmanlouandPal[7],SamantaandPal[8],GhoraiandPal[9], Paramik etal.[10],Rashmanlou etal. [11, 12],Nandhiniand Nandhini[13],andBorzooei etal.[14]haveresearchedalotin theareaofthefuzzygraphanditsapplicationsinreal-life

problems.Atanassov[15–17]haspresentedthenewtypeoffuzzy set,i.e.intuitionisticfuzzyset,asamodificationoftype1fuzzy set.Thetype1fuzzysetshaveonlyasinglemembershipgrade; however,theintuitionisticfuzzysetalwaysconsiderstwo independentmembershipgrades:membershipgradeand non-membershipgradeforeachandeveryelement.Shannonand Atanassov[18]haveforthe firsttimedescribedtheconceptof intuitionisticfuzzysetrelationshipandintuitionisticfuzzygraphs. Theyhavepresentedseveralproprieties,theorems,andprovedin [18].Forfurtherstudyonintuitionisticfuzzygraphs,pleaserefer to[19–25].However,thefuzzygraphandintuitionisticfuzzy graphbothareemployedtorepresentmanyreal-lifeproblems,but uncertaintyduetotheinconsistentinformationandindeterminate informationofanyreal-lifedecision-makingproblemcannotbe handledpreciselybyafuzzygraphorintuitionisticfuzzygraph. Therefore,anexpertrequiresothernewconceptstohandlethese scenarios.

In[26],Smarandachehasdescribedtheneutrosophicset,by extendingtheideaofafuzzyset.Itcanmanagewith indeterminate,vague,uncertain,andinconsistentdataofany real-worldproblem.Theneutrosophicsetismainlyanextension oftheclassicalset,fuzzyset,andintuitionisticfuzzyset.A neutrosophicset[27–34]hasthreemembershipgrades:truth, indeterminate,andfalseofeachandeveryelement.Thosethree membershipgradesarealwaysindependentandliebetweenthe interval]0,1[.In[27],theauthorsintroducedamodifiedscore functionto findtherankofthesingle-valuedneutrosophicsetas wellastheinterval-valuedneutrosophicset.Theyhavealso describedadecision-makingmethodbasedontheproposed function.GargandNancy[28]introducedanon-linear programmingmodelto findthesolutionstoadecision-making probleminwhichparametersarerepresentedbyinterval neutrosophicnumbers.GargandNancy[29]describesomenew aggregatingoperatorsforneutrosophicinformation.Theyhave usedthoseoperatorsto findthesolutiontothemulti-criteria decision-makingproblems.GargandNancy[28]introducedsome hybridaggregationoperatorsusingarithmeticaggregation operatorsandgeometricaggregationoperators.Theyhavealso presentedadecision-makingapproachtosolvethemulti-criteria decision-makingproblem.

Thesingle-valuedneutrosophicsetsareappliedtographtheory andpresentanewtypeofgraphicalstructurewhichisdefinedasa single-valuedneutrosophicgraph.Theideaofneutrosophicgraph theory(NGT)isbasedonneutrosophicrelationships.TheNGT

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canbeappliedtomodeltherelationshipsbetweenseveral objects/individualsinreal-lifeproblems.Adecision-makermay usetheNGTinnumerousreal-lifeapplicationsindiverseareas suchasmodernengineeringandsciences,databasetheory, imageprocessing,datamining,artificialneuralnetworks,cluster analysis,expertsystems,andcontrolsystems.Itisanextension ofcrispgraphtheory,fuzzygraphtheory,andintuitionistic fuzzygraphtheory.TheideaofNGTisamoreefficient graphicalrepresentingmethodfordealingwithinconsistency, indeterminacy,anduncertaintiesinthereal-lifeinformation comparedtofuzzygraphandintuitionisticfuzzygraph. Neutrosophicgraph[9, 35]canefficientlyrepresentthereal-life problem.Severalresearchershaverecentlyresearchedneutrosophic networktheory,forinstance,Yang etal. [36],Borzooei etal.[37], Naz etal. [38],Ye[39],Wang[40],Arkam etal.[41],Arkam [42],PrasertpongandSiripitukdet[43],AkramandSiddique[44], andAkrama[45].

Thedominationnumberisusefulinformationtoanalyseany graph.Borzooei etal.[46]describedthenewideasofdominating setsforvaguegraphsanddescribedtheideaofstrongvague dominationnumbersforvaguegraphs.Theneutrosophicgraphis morepreciseand flexibletomodelthedecision-makingproblem whenitcompareswiththevaguegraphmodel.Thus,the importanceofconsideringneutrosophicnetworksareinevitable tomodelthemanydecision-makingproblems[39, 44, 47–52]in real-worldscenariosanditisrequiredtointroducetheideaof strongarc,weakarcstrongdominationnumbers,andstrong perfectdominationoftheneutrosophicnetwork.Inthisstudy, wehavedescribedseveralideasoftheneutrosophicnetwork. Then,wehaveintroducedtheideaofstrongarc,weakarcstrong dominationnumbers,andstrongperfectdominationofthe neutrosophicnetwork.Wehaveexploredtheuseofaneutrosophic networkformodellingthepassivemobile adhoc graph[mobile adhoc network(MANET)]andwirelesssensorgraph.Wehave describedthemethodto findthevaluesofstrongandstrong perfectdominationoftheneutrosophicnetwork.

2Preliminaries

Definition2: Aneutrosophicnetwork G=K, L () isdefinedasa strongneutrosophicnetworkif

(), IL k , l () = IK k () ^ IK l (), FL k , l ()= FK k () _ FK l ()∀k , l [ E (3)

Definition3: Let GK, L () beaneutrosophicnetwork.Anedge kl in G iscalledaneffectiveedgeif

T L kl ()=F K k () ^ F K l () T L kl ()=F K k () ^ F K l () F L kl ()=F K k () _ F K l () (4)

Definition4: Aneutrosophicnetwork G iscalledastrong neutrosophicnetworkif

T L kl ()=F K k () ^ F K l ()

T L kl ()=F K k () ^ F K l ()

F L kl ()=F K k () _ F K l ()∀kl [ E

Aneutrosophicnetworkisastrongneutrosophicnetworkifeachand everyedgehaseffectiveedges.

Definition5: Aneutrosophicnumber G iscalledacomplete neutrosophicnetworkifforevery k , l [ V ,

T L kl ()=F K k () ^ F K l () T L kl ()=F K k () ^ F K l () F L kl ()=F K k () _ F K l () (6)

Acompleteneutrosophicnetworkwith n verticesisrepresented by Kn

3Strongandweakneutrosophicedges

Definition6: Let GK, L () beaneutrosophicnetwork.

(i)Aneutrosophicpath p in G isacollectionofdifferentvertices y0 , y1 , y2 , , yl 1 , yl suchthat T L yi 1 yi , I L yi 1 yi , F L yi 1 yi . 0, i = 1, , l (7)

Here l describesthelengthofthepath p.Thetwonodes k and l areconnectedbyapathwithlength l suchas ˜ p:x = x0 , x1 , x2 , ..., xl 1 , xk = y then T k L kl (), I k L kl (),and F k L kl () are determinedby

T l L kl (), I l L kl (), F l L kl () = sup T L x, x1 ^ T L x1 , x2 ^ ^ T L xl 1 , y , inf I L x, x1 _ I L x1 , x2 _ _ I L xl 1 , y , inf F L x, x1 _ F L x1 , x2 _ ... _ F L xl 1 , y

(8)

Thestrengthofneutrosophicconnectednessbetweenanytwo neutrosophicnodes(k and l )withinaneutrosophicnetwork G is definedasfollows: T 1 L kl (), I 1 L kl (), F 1 L kl () = sup l [N T l L kl () ,inf l [N F l L kl () ,inf l [N F l L kl () (9)

Definition7: Anedge kl in G iscalledastrongedgeifandonly if T L kl ()≥T 1 L kl (), I L kl ()≤I 1 L kl (),and F L kl ()≤F 1 L kl (). Forany k, l [ V ,ifandonlyifthereexistsanystrongedge betweenthenodes k and l thenthenode k isdominatedbythe node l in G

Definition8: LetNDSbeaneutrosophicsubsetofthevertices V in G NDSissaidtobeaneutrosophicdominatingsetif,forall y [ V \NDS ,thereexistsomeneutrosophicnodes x [ NDS suchthatnode k dominatesnode y.Aneutrosophicdominating setNDSin G iscalledaminimalneutrosophicdominatingset ifthereexistsnopropersubsetofNDSinaneutrosophic dominatingset.

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Inthissection,theneutrosophicstrongandneutrosophic strongneighbourhooddominationnumbersaredefinedforthe neutrosophicnetwork.Wepresenttheideasofneutrosophicstrong sizeandneutrosophicstrongorderofaneutrosophicnetwork.

Definition9: Let GK, L () beaneutrosophicnetworkandnode k [ V .Thentheneutrosophicnode y [ V issaidtobea neutrosophicstrongneighbournodeofneutrosophicnode x,ifand onlyifneutrosophicedge kl isastrongedge.Thesetofall neutrosophicstrongneighbournodeof k isrepresentedby Ns k () Theneutrosophicclosedstrongneighbourhoodvertexof k is describedas Ns x [] = Ns k () < x {}

Definition10: Let GK, L () beaneutrosophicnetworkandthenode k [ V .Theneutrosophicstrongdegreeofanode l isdefinedas follows: ds l ()= k [Ns l () T L kl (), k [Ns l () I L kl (), k [Ns l () F L kl () (10)

Definition11: Let GK, L () beaneutrosophicnetworkandthenode k [ V .Theneutrosophicstrongneighbourhooddegreeofa neutrosophicnode l isdefinedasfollows: dsN l ()= k [Ns l () T K x (), k [Ns l () T K x (), k [Ns l () F K x () (11)

Here kl isastrongarc.Theneutrosophicstrongorderofthe neutrosophicnetwork G isdefinedasfollows:

Os G ()= kl [E

T K l ()+I K kl ()+ 1 −F K k () 3 (17)

Here, l isanendvertexofastrongedge.

Definition16: Let D beaneutrosophicdominatingsetina neutrosophicnetwork G.Thenodeweightandarcweightof D are definedasfollows:

Wx D ()= k [D,y[Ns k () ^ T K k () + _ I K k () + 3 _ F K k () 3

We D ()= k [D,y[Ns k () ^ T L kl () + _ I L kl () + 3 _ F L kl () 3

(18)

(19)

Theneutrosophicstrongdominationnumberofaneutrosophic network G isdefinedastheminimumneutrosophicarccostin G= (K, L)anditisdenotedby gs G ()

Theneutrosophicstrongneighbourhooddominationnumberof G isdefinedasminimumneutrosophicnodecostin G= (K, L)andit isdenotedby gsN G ()

T L kl ()+I L kl ()+ 3 −F L kl () 3 (12)

Definition12: Let GK, L () beaneutrosophicnetworkandthenode k [ V .Theneutrosophicstrongdegreecardinalityofnode l is definedasfollows: ds l () = k [Ns l ()

Definition13: Let GK, L () beaneutrosophicnetworkandthenode k [ V .Theneutrosophicstrongneighbourhooddegreecardinality ofnode l isdefinedasfollows: dsN l () = k [Ns l ()

T K k ()+I K k ()+ 3 −F K k () 3 (13)

Definition14: Themaximumandminimumneutrosophicstrong degreeof G= (K, L)aredeterminedasfollows: Ds G ()= _ ds l () ∀y [ V ds G ()= ^ ds l () ∀y [ V (14)

Theminimumandmaximumstrongneighbourhooddegreeof G= (K, L)aredeterminedasfollows: DNs G ()= _ dNs l () ∀y [ V dNs G ()= ^ dNs l () ∀y [ V (15)

Definition17: Let GK, L () beaneutrosophicnetwork.Thegraph G isacompletebipartiteneutrosophicnetworkifthevertexset V can bepartitionedintotwonon-emptysets p1 and p2 suchthat T L p1 p2 , I L p1 p2 , F L p1 p2 = 0,1 () for p1 , p2 [ p1 or p1 , p2 [ p2 Moreover T L kl (), I L kl (), F L kl () =T K k () ^ T K l (), I K k () ^ I K l (), F K k () _ F K l () (20) for k [ y1 and y [ V 2.

5Strongperfectdominationnumber

Inthissection,theideaofneutrosophicperfectdominatingsetand neutrosophicstrongperfectdominationnumberofaneutrosophic networkareintroduced.Thenweshowthatundersomespecific criteria,theneutrosophicstrongdominationnumberandthe neutrosophicstrongperfectdominationnumberinaneutrosophic network G= (K, L)arealwaysequalled.Finally,we finda maximumboundforstrongperfectdominationnumbersin neutrosophicnetworks.

Definition18: Let GK, L () beaneutrosophicnetwork.A neutrosophicsubset D ofvertexset V issaidtobeperfect neutrosophicdominatingsetor Dp in G= (K, L),ifallthe nodesof y [ V \D,thereisoneandonlynode k [ D suchthat node k dominatesnode y.The Dp iscalledtheminimalperfect neutrosophicdominatingsetifforeach y [ Dp , Dp \ v {} isnota perfectdominatingsetin G

T L kl () +I L kl () + 1 −F L kl () 3 (16)

Remark1: Let G= (K, L)beavaguegraph.If tA and fA areconstant functions,then ds G ()= dsN G () and Ds G ()= DsN G () Definition15: Theneutrosophicstrongsizeoftheneutrosophic network G isdefinedasfollows: Ss G ()= kl [E

Definition19: Thestrongperfectdominationnumberofavague graph G isdefinedastheminimumarcweightofperfect dominatingsetsof G= (K, L),whichisdenotedby gsp G ()

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Theideaofasemi-complementaryneutrosophicnetworkand semi-neutrosophicdominatingsetinaneutrosophicnetwork G are presentedinthissection.

Definition20: Let GK, L () beaneutrosophicnetwork,then semi-complementaryneutrosophicnetworkof G= (K, L), i.e. GSC =KSC , LSC ,isdescribedby

(i) T SC K l ()=T K l (), I SC K l ()=I K l (),and F SC K l ()=F K l () (ii) E SC = uv E , ∃w, uw, wv [ E {} whereforany kl [ E SC , T SC L kl () =T K k () ^ T K l (), I SC kl () =I K k () _ I K l (), and F SC kl ()=F K k () _ F K l ().

Definition21: Let GK, L () beaneutrosophicnetwork.

(i)Theneutrosophicsubset D isdefinedasasemi-neutrosophic dominatingsetinaneutrosophicnetwork G= (K, L)ifandonly iftheneutrosophicsubset D isaneutrosophicdominatingsetfor bothneutrosophicnetwork(G= (K, L))andsemi-complementary neutrosophicnetwork GSC (ii)Asemi-neutrosophicdominatingsetisdefinedasminimal semi-neutrosophicdominatingsetifandonlyif ∀y [ DS , DS \ v {} isnotaglobaldominatingsetin G

Definition22: Thestrongsemi-neutrosophicdominationnumberofa neutrosophicnetwork G isdescribedastheminimumedgecostof semi-neutrosophicdominatingsetin G.

whichisplacedinthemainbranchoftheserviceprovider’s(hub) andsomeopticalnetworkterminalsandopticalnetworkunits (ONUs),neartoend-userpoints.APONsystemcanreducethe costofthemainbranchequipmentand fibreneededcompared withapointtopointtechnology.Itisaspecialtypeof fibreoptic accesstelecommunicationgraph.PONisgenerallyusedasatree structure.Atreeisnothingbutaconnectedcircuitlessgraph.A serviceprovidercompanywantstoprovideaPONtoacity.We canrepresentthecitymapusinganeutrosophicnetwork.The roadsareusedasthecommunicationlinks(arc)andtheroad junctionstogetherwiththeONUandOLTsareconsideredasthe nodesoftheneutrosophicnetwork.Thedecisionmakerconsiders theweightsofthecommunicationlinksbasedonthelengthofthe correspondingroads.InthePONsystem,theOLTislocatedinthe mainbranchanditisusedasarootofthetree.TheONUis consideredastheleavesofthetreeandtheyareplacednearthe end-users.TheOLTandONUsbothareconsideredaspassive elementsofthe fibresandtheopticalsplittersarespread.Inthe PON,useraccesspointsarelinkedtotheONUsviaclassical engineeringsuchascoaxlinesorcopper.AnexampleofaPONis showninFig. 1.Mostoftheprojectcosthastobeusedfor diggingthelineductstoinstallanewPONinacity.Thedecision makerhasto findtheshortestpaths,whichconnectalltheONUs withthesingleOLTinthePON.Ithelpsto findminimisethe projectcost.Fromtheneutrosophicnetworkrepresentation,this neutrosophicoptimisationproblemcanbeconvertedintothe neutrosophicminimumSteinertreeproblem.Thedecision-maker wantstodetermineaneutrosophictreeoftheneutrosophic network,whichconsistsofasubsetofneutrosophicvertices whosesumoftheweightsoftheselectedarcsbecomesminimal.

8Someapplicationsofstrongdomination numbers

AneutrosophicsetisgeneralisedofAtanassov’sintuitionisticfuzzy setwhichconsistsofthreemembershipgrades:truthmembership, indeterminatemembership,andfalsemembership.The neutrosophicnetworkisanextensionofavaguegraphand intuitionisticfuzzygraphwhichprovidesmore flexibility, precision,andcompatibilitytodesignthereal-lifeproblemwhen Fig.1

ThePONisa fibreoptictelecommunicationgraphthatusesthe singlepointtomultipointtechnologyandopticalsplittertosend databetweenasinglesourcepointtomultipleend-users.APON systemmainlyconsistsofasingleopticallineterminal(OLT),

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comparedwiththeintuitionisticfuzzygraphs.Neutrosophicgraph modelsarerecentlyusingtomodelmanyreal-lifeproblemsin severaldifferentareasofengineeringandscience.Inthisstudy, weintroducetheideaofadominatingsetinneutrosophicnetwork theory.Thedominationintheneutrosophicnetworkcanbeused tosolvemuchreal-lifeproblem.Here,insociety,aswellasin administration,theinfluenceoftheindividualdependsonthe strengththathederivesfromhissupporters,andtheseeffectsmay benoteffective.Besides,theindividualhastodependmoreonhis supportersthanonhimself.

Now,weexpressanapplicationofdominatingset.Anoffice consistsofsevenemployeesandelectionsarebeingheldto determinethenewhead.Weshowthatafewemployeescanselect aperson G= (K, L)(whodoesnothaveconsiderableinfluenceon allemployees)astheheadbyusingadominationsetofavague graph.First,werepresenttheofficewithavaguegraph G asinFig. 2

Inthisvaguegraph,thenodesandarcsrepresentemployeesand friendshipsbetweenthem,respectively.Truemembershipfunction foreachnodeisconsideredasthesignificanceofthenodeinthe office,includinglevelofeducation,workexperienceetc.,andfalse membershipfunctionforeachnodeisevaluatedaslackof compatibilitybetweeneducationalmajorandoccupation,lackof ability,andothercases.Inthisexample,weseethat ac, cd, bd, de, cf,and cg arestrongarcs,andthereisastrongrelationship betweenthem.Hence, D1 = c, d {}, D2 = c, b, d {},and D3 = c, d , e {} aredominatingsetsinthisvaguegraphandweightsare

We D1 = 0 2 + 0 5 + 0 5 + 0 6 = 1 8

We D2 = 0 2 + 0 5 + 0 5 + 0 6 + 0 5 + 0 8 = 3 1 We D3 = 0 2 + 0 5 + 0 5 + 0 6 + 0 5 + 0 6 = 2 9

Also,so D1 isaminimaldominationsetinthisexample.Sincethe nodesindominatingsethavethemostinfluenceontheother

memberswhoarenotin D1 ,therefore,bythisinfluence,theycan selectoneoftheirownas G= (K, L)oranyoneelseasheadof office.(However,weseethatthecrispgraph,whichismadeup ofemployeesandrelationshipsbetweenthem, D = c, e {} isa minimaldominationsetthathaslessinfluenceontheother membersascomparedto D1 )

Inthefollowing,wehavesomemoreapplicationsofstrong dominationnumbersineverydaylife.

8.1Mobile

adhoc graph(MANET)

Amobile adhoc graph(MANET)isadecentralisemobilegraph, whichiscontinuouslyself-organising,self-configuring,and infrastructurelessgraphusedtocommunicateoverwireless channel.InthisMANET,eachmobiledevicecanfreelymovein anydirectionandthelinksbetweenothermobiledeviceswill changefrequently.EachdeviceinaMANETmustsendthetraffic totheirownuse.ThemainchallengeinmodellingaMANETisto equipeachmobiledevicetoprovidetheexactinformationneeded tohandletheroutetrafficproperly.WecanrepresenttheMANET asaneutrosophicnetworkwheretheverticesrepresentthemobile deviceandedgesdescribepatternofmessagingamongthemobile devices.Theideaofneutrosophicnetworkcanhandlethe uncertaintyofMANET.Wecanusethedominatingsetto find theroutingtableinMANET.Thesmalldominatingsetis consideredasabackboneforMANET.Ifadeviceisnotpresent inthedominatingsetthenthedevicetransmitsthemessagesusing theneighbourdevicesintheset.Thestrongedgesareonly consideredinthedominatingsetofaneutrosophicnetworkandif anydeviceisnotpresentindominatingsetofgraph(neutrosophic network)thenithasalwaysastrongneighbourinthedominating set.Wecantransmitmessagefastertoitsneighbourintheset. Thestrongdominationnumberdescribesthesmallestnumberof mobiledeviceindominatingsetoftheneutrosophicnetwork.The transmissionspeedandroutingofmessagecanbeimprovedusing dominatingset,anditcanhelptominimisetheconstructioncost oftheMANET.

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8.2Computercommunicationgraphs

Theideaofneutrosophicdominatingsetplaysanimportantrolein computerandcommunicationgraphstoroutetheinformation betweenthenodes.Neutrosophicgraphcanbeusedtomodelthe computergraphwheretheeachnoderepresentsthecomputerand twonodesareconnectedbyadirectedarcifthereexistsanydirect communicationlinebetweenthetwocomputers.Informationis collectedfromthecomputerandeachcomputercantransmitthe informationtoothercomputers,whichhavedirectcommunication linewithit.Inthisproblem,weneedto findasmallsetof computersinthegraph,whichdirectlycommunicatewithallother computers.Wecansolvethisproblemusingtheideaofminimum neutrosophicdominatingsetofthecorrespondingneutrosophic network.Theinformationmaybecollectedbymakingeach processortorouteitsinformationtooneofthesmallsetsof collectingprocessor,whichisconstructedbyadominatingsetof thecorrespondinggraph.

8.3Wirelesssensorgraph

Awirelesssensorgraph[wirelesssensornetwork(WSN)]isan infrastructurelessandself-configurewirelessgraph.Itiscollection oftransducers,whichcanmonitorandrecordtheconditionsof differentlocations.Itgenerallymonitorstheparameterssuchas pressure,temperature,humidity,pollutantlevels,windspeed, powerlinevoltage,winddirection,chemicalconcentrations,sound intensity,andilluminationintensity.TheWSNismainly constructedofsensornodes,whicharegenerallyspreadinthe graph.Thesensornodesgathertheinformationfromothersensor nodesandgraphs.Thetopologycontrolisthemostbasicproblem inWSN.WecanmodeltheWSNasaneutrosophicnetworksince theplaceofeachnodeisindefiniteandcanconveyeverypieceof information,evendestructive,toothernodesand,thereforeinduce interventioninthegraph.Asaresult,wecandefinetrue (regardinglevelofimportance,necessity,effect,paceof conveyanceetc.)andfalse(regardingthedegreeofintervention, vaguenessetc.)membershipfunctionsandalsovaluateseacharc consideringimportanceandnecessityofconveyanceof informationetc.Accordingly,sinceeachdominatingsetinavague graphisgainedusingstrongarcs,wecan,insodoing,makethe smallestandthemosteffectiveminimalbackbonesetbygaining minimaldominatingset(byusingminimumarcweight).Virtual backboneisnecessaryforfaulttoleranceandrouting flexibility.

9Conclusion

Inthispaper,wehaveexploredtheuseofneutrosophicnetworkfor modellingthePON,mobile adhoc graph(MANET),andwireless sensorgraph.Wehavepresentedtheideaofstrongarc,weakarc strongdominationnumbers,andstrongperfectdominationof neutrosophicnetwork.Wehavedescribedthemethodto findthe valuesofstrongandstrongperfectdominationofneutrosophic network.Finally,weusetheideaofstrongarcstrongdomination numberinMANETandwirelesssensorgraph.

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