IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 3 Ver. I1 (May - June 2017), PP 46-54 www.iosrjournals.org
An Introduction to Fuzzy Soft Digraph Sarala N 1 Tharani S 2 1
2
(Department of Mathematics, A.D.M.College for Women, Nagapattinam, Tamilnadu, India) (Department of Mathematics, E.G.S.Pillay Engineering College, Nagapattinam, Tamilnadu, India)
Abstract: Fuzzy sets and soft sets are two different tools for representing uncertainty and vagueness. We introduce the notions of fuzzy soft digraphs, fuzzy soft walk, fuzzy soft trail in digraph and some operations in fuzzy soft in fuzzy soft digraph. Keywords: fuzzy soft graph, fuzzy soft digraph, walk in fuzzy soft digraph, trail in fuzzy soft digraph, union and intersection in fuzzy soft digraph.
I.
Introduction
The Concept of soft set theory was initiated by Molodtsov [1] for dealing uncertainties. A Rosenfeld [2] developed the theory of fuzzy graphs in 1975 by considering fuzzy relation on fuzzy set, which was developed by Zadeh [3] in the year 1965. Some operations on fuzzy graphs are studied by Mordeson a C.S. Peng [4] .Later Ali et al. discussed about fuzzy sets and fuzzy soft sets induced by soft sets. M.Akram and S Nawaz [5] introduced fuzzy soft graphs in the year 2015. Sumit mohinta and T K samanta [6] also introduced fuzzy soft graphs independently. The notion of fuzzy soft graph and few properties related to it are presented in their paper. In this paper, fuzzy soft digraph, walk in fuzzy soft digraph, trail in fuzzy soft digraph and some operations are introduced.
II.
Preliminaries
We now review some elementary concepts of digraph and fuzzy soft graph Definition: 2.1 Let U be an initial universe set and E be the set of parameters. Let A ∠đ??¸ , A pair (F,A) is called fuzzy soft set over U where F is a mapping given by F : A → đ??ź đ?‘ˆ , where đ??ź đ?‘ˆ denotes the collection of all fuzzy subsets of U. Definition: 2.2 Let V be a nonempty finite set and đ?œŽ âˆś đ?‘‰ → 0, 1 . Again , let đ?œ‡ âˆś đ?‘‰ đ?‘‹ đ?‘‰ → [0,1] such that đ?œ‡ đ?‘Ľ, đ?‘Ś ≤ đ?œŽ đ?‘Ľ ∧ đ?œŽ đ?‘Ś for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‰ đ?‘‹ đ?‘‰. Then the pair đ??ş = (đ?œŽ , đ?œ‡) is called a fuzzy graph over the set V. Here đ?œŽ đ?‘Žđ?‘›đ?‘‘ đ?œ‡ are respectively called fuzzy vertex and fuzzy edge of the fuzzy graph G = (đ?œŽ , đ?œ‡ ) Definition: 2.3 A fuzzy digraph GD = (đ?œŽđ??ˇ , đ?œ‡đ??ˇ ) is a pair of function đ?œŽđ??ˇ âˆś đ?‘‰ → 0, 1 and đ?œ‡đ??ˇ âˆś đ?‘‰ đ?‘‹ đ?‘‰ → [0,1] Where đ?œ‡đ??ˇ đ?‘Ľ, đ?‘Ś ≤ đ?œŽđ??ˇ đ?‘Ľ ∧ đ?œŽđ??ˇ đ?‘Ś for all đ?‘Ľ, đ?‘Ś ∈ đ?‘‰ đ?‘‹ đ?‘‰ and đ?œ‡đ??ˇ is a set of fuzzy directed edges are called fuzzy arcs. Definition: 2.4 The degree of any vertex đ?œŽ(đ?‘Ľđ?‘– ) of a fuzzy graph is sum of degree of membership of all those edges which are incident on a vertex đ?œŽ(đ?‘Ľđ?‘– ) and is denoted by deg(đ?œŽ(đ?‘Ľđ?‘– )). Definition: 2.5 In a fuzzy digraph the number of arcs directed away from the vertex đ?œŽ đ?‘Ľ is called the outdegree of vertex, it is denoted by od( đ?œŽ đ?‘Ľ ). The number of arcs directed to the vertex đ?œŽ đ?‘Ľ is called indegree of vertex, it is denoted by id(đ?œŽ đ?‘Ľ ). The degree of vertex đ??ˆ đ?’™ in a fuzzy digraph is deined to be deg(đ?œŽ đ?‘Ľ ) = id(đ?œŽ đ?‘Ľ ) + đ?‘œđ?‘‘( đ?œŽ đ?‘Ľ ). Definition: 2.6 Let đ??ş = (đ?œŽ , đ?œ‡) be a fuzzy graph. The Order of đ?‘Ž = (đ?œŽ , đ?œ‡) is defined as đ?‘‚ đ??ş = đ?œŽ đ?‘˘ u ∈đ?‘‰ and the size of đ?‘Ž = (đ?œŽ , đ?œ‡) is defined as S đ??ş =
đ?œ‡ đ?‘˘, đ?‘Ł . u,v ∈ đ?‘‰
DOI: 10.9790/5728-1303024654
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An Introduction to Fuzzy Soft Digraph Definition: 2.7 A directed fuzzy walk in a fuzzy graph is an alternating sequence of vertices an edges, đ?‘Ľ0 , e1 , đ?‘Ľ1 , e2 , ‌ ‌ ‌ ‌ . . , eđ?‘› , đ?‘Ľđ?‘› in which each edge eđ?‘– is đ?‘Ľđ?‘–−1 , đ?‘Ľđ?‘– . Definition : 2.8 A fuzzy path is a fuzzy walk in which all vertices are distinct. Definition : 2.9 Let đ??ş1 = (đ?œŽ1 , đ?œ‡1 ) and đ??ş2 = (đ?œŽ2 , đ?œ‡2 ) be two fuzzy graphs over the set V. Then the union of đ??ş1 đ?‘Žđ?‘›d đ??ş2 is another fuzzy graph đ??ş3 = (đ?œŽ3 , đ?œ‡3 ) over the set V, where đ?œŽ3 = đ?œŽ1 âˆŞ đ?œŽ2 and đ?œ‡3 = đ?œ‡1 âˆŞ đ?œ‡2 , đ?‘–. e. đ?œŽ3 đ?‘Ľ = đ?‘šđ?‘Žđ?‘Ľ đ?œŽ1 đ?‘Ľ , đ?œŽ2 đ?‘Ľ ∀ đ?‘Ľ ∈ đ?‘‰ and đ?œ‡3 đ?‘Ľ, đ?‘Ś = đ?‘šđ?‘Žđ?‘Ľ đ?œ‡1 đ?‘Ľ, đ?‘Ś , đ?œ‡2 đ?‘Ľ, đ?‘Ś ∀ đ?‘Ľ, đ?‘Ś ∈ đ?‘‰. Definition : 2.10 Let đ??ş1 = (đ?œŽ1 , đ?œ‡1 ) and đ??ş2 = (đ?œŽ2 , đ?œ‡2 ) be two fuzzy graphs over the set V. Then the intersection of đ??ş1 đ?‘Žđ?‘›d đ??ş2 is another fuzzy graph đ??ş3 = (đ?œŽ3 , đ?œ‡3 ) over the set V, where đ?œŽ3 = đ?œŽ1 ∊ đ?œŽ2 and đ?œ‡3 = đ?œ‡1 ∊ đ?œ‡2 , đ?‘–. e. đ?œŽ3 đ?‘Ľ = đ?‘šđ?‘–đ?‘› đ?œŽ1 đ?‘Ľ , đ?œŽ2 đ?‘Ľ ∀ đ?‘Ľ ∈ đ?‘‰ and đ?œ‡3 đ?‘Ľ, đ?‘Ś = đ?‘šđ?‘–đ?‘› đ?œ‡1 đ?‘Ľ, đ?‘Ś , đ?œ‡2 đ?‘Ľ, đ?‘Ś ∀ đ?‘Ľ, đ?‘Ś ∈ đ?‘‰.
III.
Fuzzy Soft Digraph
We now introduce some basic concepts of fuzzy soft digraph Definition: 3.1 Let V = { đ?‘Ľ1, đ?‘Ľ2, đ?‘Ľ3, đ?‘Ľ4, đ?‘Ľ5, ‌ ‌ đ?‘Ľđ?‘› } (non empty set), E(parameters set) and A ≤ E Also let (i) đ?œŒđ??ˇ : đ??´ → đ??š đ?‘‰ ( collection of all fuzzy subsets in V) (đ?‘’) đ?‘’ â&#x;ź đ?œŒđ??ˇ = đ?œŒđ??ˇđ?‘’ (đ?‘ đ?‘Žđ?‘Ś) And đ?œŒđ??ˇđ?‘’ âˆś đ?‘‰ → 0,1 (A, đ?œŒđ??ˇ ) : Fuzzy soft vertex
đ?‘Ľđ?‘– â&#x;ź đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘– )
(ii) đ?œ‡đ??ˇ : đ??´ → đ??š đ?‘‰ đ?‘‹ đ?‘‰ ( Collection of all Fuzzy subsets in V X V ) (đ?‘’) đ?‘’ â&#x;ź Âľđ??ˇ = Âľđ?‘’đ??ˇ (đ?‘ đ?‘Žđ?‘Ś) đ?‘’ And Âľđ??ˇ âˆś đ?‘‰ đ?‘‹ đ?‘‰ → 0,1 (đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ) â&#x;ź Âľđ?‘’đ??ˇ (đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ) đ?‘’ And Âľđ??ˇ is a set of fuzzy directed edges are called fuzzy arcs. ( A, đ?œ‡đ??ˇ ) : Fuzzy soft edge Then ((A, đ?œŒđ??ˇ ) , ( A, đ?œ‡đ??ˇ )) is called fuzzy soft digraph iff Âľđ?‘’đ??ˇ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ≤ đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘– ) É… đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘— ) ∀ e ∈ đ??´ đ?‘Žđ?‘›đ?‘‘ ∀đ?‘–, đ?‘— = 1,2,3, ‌ ‌ . đ?‘› đ?‘Žđ?‘›đ?‘‘ đ?‘Ąâ„Žđ?‘–đ?‘ đ?’‡đ?’–đ?’›đ?’›đ?’š đ?’”đ?’?đ?’‡đ?’• đ?’…đ?’Šđ?’ˆđ?’“đ?’‚đ?’‘đ?’‰ is denoted by đ??ˇđ??´,đ?‘‰ Example: 3 .1 Consider a fuzzy soft digraph đ??ˇđ??´,đ?‘‰ where V = { đ?‘Ľ1, đ?‘Ľ2, đ?‘Ľ3, đ?‘Ľ4 } đ?‘Žđ?‘›đ?‘‘ đ??¸ = { đ?‘’1, đ?‘’2, đ?‘’3 } đ??ˇđ??´,đ?‘‰ Described by Table 3.1 and Âľđ?‘’đ??ˇ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— = 0 đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ đ?‘‹ đ?‘‰ / { đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ2 , đ?‘Ľ3 , đ?‘Ľ3 , đ?‘Ľ4 , đ?‘Ľ4 , đ?‘Ľ1 } And for all e ∈ đ??¸ Table 3.1 đ?œŒđ??ˇ đ?‘’1 đ?‘’2 đ?‘’3
đ?‘Ľ1 0.2 0.1 0.4
đ?‘Ľ2 0.8 0.3 0.5
đ?‘Ľ3 0.6 0.7 0.9
đ?‘Ľ4 0.4 0.5 0
đ?œ‡đ??ˇ đ?‘’1 đ?‘’2 đ?‘’3
đ?‘Ľ1 , đ?‘Ľ2 0.1 0.1 0.2
đ?‘Ľ2 , đ?‘Ľ3 0.5 0.2 0.4
đ?‘Ľ3 , đ?‘Ľ4 0.4 0.4 0
đ?‘Ľ4 , đ?‘Ľ1 0.2 0.1 0
Fig. 3.1 Fuzzy Soft Digraph DA,V DOI: 10.9790/5728-1303024654
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An Introduction to Fuzzy Soft Digraph Definition: 3.2 Let DA,V = ((A, đ?œŒđ??ˇ ) , ( A, đ?œ‡đ??ˇ )) deg DA,V
be a Fuzzy Soft digraph . The degree of a vertex ‘u’ is defined as đ?‘œđ?‘‘ đ?‘˘ + đ?‘–đ?‘‘(đ?‘˘)
= e∈đ??´ u∈ đ?‘‰
Definition: 3.3 Let DA,V = ((A, đ?œŒđ??ˇ ) , ( A, đ?œ‡đ??ˇ )) be a Fuzzy Soft digraph . Then the Order of DA,V is defined as O( DA,V ) = đ?œŒđ??ˇđ?‘’ đ?‘Ľđ?‘– e ∈ đ??´ đ?‘Ľđ?‘– ∈ đ?‘‰ Definition: 3.4 The Size of DA,V is defined as S(DA,V ) = Âľđ?‘’đ??ˇ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— e ∈ đ??´ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ Example from the above Fig 3.1 DA,V e1 e2 e3 Deg
đ?‘œđ?‘‘ đ?‘Ľ1 + đ?‘–đ?‘‘(đ?‘Ľ1 ) 0.1 + 0.2 = 0.3 0.1 + 0.1` = 0.2 0.2 + 0 = 0.2 Deg(đ?‘Ľ1 ) = 0.7
đ?‘œđ?‘‘ đ?‘Ľ2 + đ?‘–đ?‘‘ đ?‘Ľ2 0.5 + 0.1 = 0.6 0.2 + 0.1 = 0.3 0.4 + 0.2 = 0.6 Deg(đ?‘Ľ2 ) = 1.5
đ?‘œđ?‘‘ đ?‘Ľ3 + đ?‘–đ?‘‘ đ?‘Ľ3 0.4 + 0.5 = 0.9 0.4 + 0.2 = 0.6 0 + 0.4 = 0.4 Deg(đ?‘Ľ3 ) = 1.9
đ?‘œđ?‘‘ đ?‘Ľ4 + đ?‘–đ?‘‘(đ?‘Ľ4 ) 0.1 + 0.2 = 0.3 0.1 + 0.4 = 0.5 0+0=0 Deg(đ?‘Ľ4 ) = 0.8
The degree of the vertex đ?‘Ľđ?‘– is deg DA,V (đ?‘Ľđ?‘– ) = đ?‘œđ?‘‘ đ?‘˘ + đ?‘–đ?‘‘(đ?‘˘) e∈đ??´ u∈ đ?‘‰ Therefore, deg DA,V (đ?‘Ľ1 ) = 0.7 deg DA,V (đ?‘Ľ2 ) = 1.5 deg DA,V (đ?‘Ľ3 ) = 1.9 deg DA,V (đ?‘Ľ4 ) = 0.8 The Order of fuzzy soft digraph DA,V is O( DA,V ) = đ?œŒđ??ˇđ?‘’ đ?‘Ľđ?‘– e ∈ đ??´ đ?‘Ľđ?‘– ∈ đ?‘‰ = ( 0.2+ 0.8 + 0.6 + 0.4) + ( 0.1+ 0.3 + 0.7 + 0.5) + ( 0.4+0.5 + 0.9) = 2.0 + 1.6 + 1.8 Therefore, O( DA,V ) = 5.4 The Size of fuzzy soft digraph DA,V is S(DA,V ) = Âľđ?‘’đ??ˇ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— e ∈ đ??´ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ (i.e) Size = Sum of all the fuzzy arcs
Therefore,
S(DA,V )
= ( 0.1+ 0.5 + 0.4 + 0.2) + ( 0.1+ 0.2 + 0.4+ 0.1)+ (0.2+ 0.4) = 1.2 + 0.8 + 0.6 = 2.6
Definition: 3.5 A fuzzy soft arc joining a fuzzy soft vertex to itself is called a fuzzy soft loop in digraph ( i.e) id = od Definition: 3.6 Let DA,V = ((A, đ?œŒđ??ˇ ) , ( A, đ?œ‡đ??ˇ )) be a fuzzy soft digraph. If for all e ∈ đ??´ there is more than one fuzzy soft arc joining two fuzzy soft vertices, then the fuzzy soft digraph is called a fuzzy soft pseudo digraph an these arcs are called fuzzy soft multiple arcs. Definition: 3.7 Let DA,V = ((A, đ?œŒđ??ˇ ) , ( A, đ?œ‡đ??ˇ )) be a fuzzy soft simple digraph if it has neither fuzzy soft loops nor fuzzy soft multiple arcs for all e ∈ đ??´ Example: 3.2 Consider a fuzzy soft digraph đ??ˇđ??´,đ?‘‰ where V = { đ?‘Ž1, đ?‘Ž2, đ?‘Ž3, đ?‘Ž4 } đ?‘Žđ?‘›đ?‘‘ đ??¸ = đ?‘’1, đ?‘’2, đ?‘’3 đ??ˇđ??´,đ?‘‰ described by Table 3.2 and Âľđ?‘’đ??ˇ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— = 0 đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ đ?‘‹ đ?‘‰ / { đ?‘Ž1 , đ?‘Ž2 , đ?‘Ž2 , đ?‘Ž3 , đ?‘Ž3 , đ?‘Ž4 , đ?‘Ž4 , đ?‘Ž1 , đ?‘Ž2 , đ?‘Ž1 , đ?‘Ž1 , đ?‘Ž4 , DOI: 10.9790/5728-1303024654
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An Introduction to Fuzzy Soft Digraph đ?‘Ž1 , đ?‘Ž1 } and for all e ∈ đ??¸ Table 3.2 đ?œŒđ??ˇ đ?‘’1 đ?‘’2 đ?‘’3 đ?œ‡đ??ˇ đ?‘’1 đ?‘’2 đ?‘’3
đ?‘Ž1 , đ?‘Ž2 0.1 0.1 0.3
đ?‘Ž1, 0.5 0.4 0.5
đ?‘Ž2 0.2 0.7 0.9
đ?‘Ž2 , đ?‘Ž3 0.2 0.2 0.2
đ?‘Ž3 0.4 0.6 0.3
đ?‘Ž3 , đ?‘Ž4 0.1 0.1 0.2
đ?‘Ž4 0.3 0.5 0.4
đ?‘Ž4 , đ?‘Ž1 0.2 0.1 0.3
đ?‘Ž2 , đ?‘Ž1 0.2 0 0.4
đ?‘Ž1 , đ?‘Ž4 0.2 0.1 0
đ?‘Ž1 , đ?‘Ž1 0 0 0.2
Fig 3.2 Pseudo Fuzzy Soft Digraph DA,V The figure above contains a fuzzy soft loops corresponding to parameters e1 and e3.It also contains fuzzy soft multiple arcs corresponding to the parameter e 1, e2 and e3. Therefore, DA,V is Pseudo Fuzzy soft digraph, but it is not simple fuzzy soft digraph. Definition: 3.8 A fuzzy soft walk in DA,V = ((A, đ?œŒđ??ˇ ) , ( A, đ?œ‡đ??ˇ )) is an alternating sequence of W = đ?œŒđ??ˇđ?‘’ đ?‘Ľ1 Âľđ?‘’đ??ˇ đ?‘Ž1 đ?œŒđ??ˇđ?‘’ đ?‘Ľ2 Âľđ?‘’đ??ˇ đ?‘Ž2 đ?œŒđ??ˇđ?‘’ đ?‘Ľ3 Âľđ?‘’đ??ˇ đ?‘Ž4 ‌ ‌ ‌ ‌ ‌ ‌ đ?œŒđ??ˇđ?‘’ đ?‘Ľđ?‘˜âˆ’1 Âľđ?‘’đ??ˇ đ?‘Žđ?‘˜ −1 đ?œŒđ??ˇđ?‘’ đ?‘Ľđ?‘˜ of fuzzy soft vertices đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘– ) and fuzzy soft directed edges Âľđ?‘’đ??ˇ đ?‘Žđ?‘– from D such tha the tail of Âľđ?‘’đ??ˇ đ?‘Žđ?‘– is đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘– ) and the head of Âľđ?‘’đ??ˇ đ?‘Žđ?‘– is đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘–+1 ) for every I = 1,2,3,‌‌‌‌ k-1. A fuzzy soft walk is closed if đ?œŒđ??ˇđ?‘’ (đ?‘Ľ1 ) = đ?œŒđ??ˇđ?‘’ (đ?‘Ľ đ?‘˜ ) and open otherwise. We say that W is fuzzy soft walk from đ?œŒđ??ˇđ?‘’ (đ?‘Ľ1 ) to đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘˜ ) or an Âľđ?‘’đ??ˇ đ?‘Ľ1 , đ?‘Ľđ?‘˜ - fuzzy soft walk. Definition: 3.9 The length of the fuzzy soft walk in D is L D (W) = Âľđ?‘’đ??ˇ đ?‘Žđ?‘– where a i = ( đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘– ), đ?œŒđ??ˇđ?‘’ (đ?‘Ľđ?‘— )) e ∈ đ??´ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ Example: 3.3 Consider the V = { đ?‘Ľ1 , đ?‘Ľ2, đ?‘Ľ3, đ?‘Ľ4 , đ?‘Ľ5 } đ?‘Žđ?‘›đ?‘‘ đ??¸ = { đ?‘’1, đ?‘’2, đ?‘’3 , đ?‘’4 }. Let đ??´ = { đ?‘’1, đ?‘’2, đ?‘’3 } đ??ˇđ??´,đ?‘‰ defined by Table 3.3 and Âľđ?‘’đ??ˇ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— = 0 đ?‘“đ?‘œđ?‘&#x; đ?‘Žđ?‘™đ?‘™ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ đ?‘‹ đ?‘‰ / { đ?‘Ľ1 , đ?‘Ľ2 , đ?‘Ľ2 , đ?‘Ľ5 , đ?‘Ľ5 , đ?‘Ľ4 , đ?‘Ľ5 , đ?‘Ľ1 , đ?‘Ľ4 , đ?‘Ľ3 , đ?‘Ľ3 , đ?‘Ľ2 , đ?‘Ľ3 , đ?‘Ľ5 } and for all e ∈ đ??´ Table 3.3 đ?œŒđ??ˇ đ?‘’1 đ?‘’2 đ?‘’3 đ?œ‡đ??ˇ đ?‘’1 đ?‘’2 đ?‘’3
đ?‘Ľ1 , đ?‘Ľ2 (đ?‘Ž1 ) 0.1 0.2 0.5
DOI: 10.9790/5728-1303024654
đ?‘Ľ2 , đ?‘Ľ5 (đ?‘Ž6 ) 0.2 0.1 0.2
đ?‘Ľ1 0.1 0.8 0.6
đ?‘Ľ2 0.2 0.3 0.8 đ?‘Ľ5 , đ?‘Ľ4 (đ?‘Ž4 ) 0.2 0.7 0
đ?‘Ľ3 0.8 0.3 0.7 đ?‘Ľ5 , đ?‘Ľ1 (đ?‘Ž5 ) 0.2 0.6 0.3
đ?‘Ľ4 0.2 0.9 0
đ?‘Ľ5 0.3 0.9 0.5
đ?‘Ľ4 , đ?‘Ľ3 (đ?‘Ž3 ) 0.1 0.2 0
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đ?‘Ľ3 , đ?‘Ľ2 (đ?‘Ž2 ) 0.4 0.1 0.4
đ?‘Ľ3 , đ?‘Ľ5 (đ?‘Ž7 ) 0.2 0.2 0.3
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An Introduction to Fuzzy Soft Digraph
Fig 3. 3 D = { H (e1) , H(e2) , H(e3) } is fuzzy soft digraph.
Fig, 3.4 W( D) = { W(e1) , W(e2) , W(e3) } is the fuzzy soft walk in digraph. The length of the fuzzy soft walk Âľđ?‘’đ??ˇ đ?‘Žđ?‘–
L D (W) = e ∈ đ??´ đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰ = 0.8 + 1.3 + 1.4 L D (W) = 3.5 Definition: 3.10
If the fuzzy soft directed edges are distinct in a fuzzy soft walk is called a trail of fuzzy soft digraph.
Definition: 3.11 If the fuzzy soft vertices are distinct in a fuzzy soft walk is called a path of fuzzy soft digraph. From the above Example .3.3
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An Introduction to Fuzzy Soft Digraph
Fig, 3.5 W( D) = { W(e1) , W(e2) , W(e3) } is the fuzzy soft path in digraph.
IV.
Operations On Fuzzy Soft Digraph
We now introduce some basic operations on fuzzy soft digraph Definition: 4.1 Let 𝑉1 , 𝑉2 ∁ 𝑉 and A , B ∁ 𝐸 then the union of two fuzzy soft digraphs . 1 𝐷𝐴,𝑉 = ((A, 𝜌𝐷𝑒 1 ), (A, 𝜇𝐷𝑒 1 ) ) and 1 2 𝐷𝐵,𝑉 = ((B, 𝜌𝐷𝑒 2 ), (B, 𝜇𝐷𝑒 2 ) ) 2 3 𝐷𝐶,𝑉 = ((C, 𝜌𝐷𝑒 3 ), (C, 𝜇𝐷𝑒 3 ) ) 3
Where C = A U B ,
is defined to be say.
𝑉3 = 𝑉1 𝑈 𝑉2
𝜌𝐷𝑒 3 ( 𝑥𝑖 ) =
𝜌𝐷𝑒 1 ( 𝑥𝑖 )
∀ 𝑥𝑖 ∈ 𝑉1 − 𝑉2 & 𝑒 ∈ 𝐴 − 𝐵
0
∀ 𝑥𝑖 ∈ 𝑉2 − 𝑉1 & 𝑒 ∈ 𝐴 − 𝐵
𝜌𝐷𝑒 1 ( 𝑥𝑖 )
∀ 𝑥𝑖 ∈ 𝑉1 ∩ 𝑉2 & 𝑒 ∈ 𝐴 − 𝐵
𝜌𝐷𝑒 2 ( 𝑥𝑖 )
∀ 𝑥𝑖 ∈ 𝑉2 − 𝑉1 & 𝑒 ∈ 𝐵 − 𝐴
0
∀ 𝑥𝑖 ∈ 𝑉1 − 𝑉2 & 𝑒 ∈ 𝐵 − 𝐴
𝜌𝐷𝑒 2 ( 𝑥𝑖 )
∀ 𝑥𝑖 ∈ 𝑉1 ∩ 𝑉2 & 𝑒 ∈ 𝐵 − 𝐴
𝑚𝑎𝑥 { 𝜌𝐷𝑒 1 ( 𝑥𝑖 ) ∪ 𝜌𝐷𝑒 2 ( 𝑥𝑖 )} ∀ 𝑥𝑖 ∈ 𝑉1 ∩ 𝑉2 & 𝑒 ∈ 𝐴 ∩ 𝐵 𝜌𝐷𝑒 1 ( 𝑥𝑖 )
∀ 𝑥𝑖 ∈ 𝑉1 − 𝑉2 & 𝑒 ∈ 𝐴 ∩ 𝐵
𝜌𝐷𝑒 2 ( 𝑥𝑖 )
∀ 𝑥𝑖 ∈ 𝑉2 − 𝑉1 & 𝑒 ∈ 𝐴 ∩ 𝐵
And
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An Introduction to Fuzzy Soft Digraph 𝜇𝐷𝑒 1 ( 𝑥𝑖 , 𝑥𝑗 )
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 − (𝑉2 𝑋𝑉2 ) & 𝑒 ∈ 𝐴 − 𝐵 if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉2 𝑋𝑉2 − 𝑉1 𝑋 𝑉1 & 𝑒 ∈ 𝐴 − 𝐵
0
𝜇𝐷𝑒 3 ( 𝑥𝑖 , 𝑥𝑗 ) =
𝜇𝐷𝑒 1 ( 𝑥𝑖 , 𝑥𝑗 )
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 ∩ (𝑉2 𝑋𝑉2 ) & 𝑒 ∈ 𝐴 − 𝐵
𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 )
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉2 𝑋𝑉2 − 𝑉1 𝑋 𝑉1 & 𝑒 ∈ 𝐵 − 𝐴
0
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 ∩ (𝑉2 𝑋𝑉2 ) & 𝑒 ∈ 𝐵 − 𝐴
𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 )
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 ∩ (𝑉2 𝑋𝑉2 ) & 𝑒 ∈ 𝐵 − 𝐴
Max { 𝜇𝐷𝑒 1 ( 𝑥𝑖 , 𝑥𝑗 ) ∪ 𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 )} if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 ∩ (𝑉2 𝑋𝑉2 ) & 𝑒 ∈ 𝐴 ∩ 𝐵 𝜇𝐷𝑒 1 ( 𝑥𝑖 , 𝑥𝑗 )
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 − (𝑉2 𝑋𝑉2 ) & 𝑒 ∈ 𝐴 ∩ 𝐵
𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 )
if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉2 𝑋𝑉2 − 𝑉1 𝑋 𝑉1 & 𝑒 ∈ 𝐴 ∩ 𝐵
Where 𝜇𝐷𝑒 1 ( 𝑥𝑖 , 𝑥𝑗 ) , 𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 ) and 𝜇𝐷𝑒 3 ( 𝑥𝑖 , 𝑥𝑗 ) = called fuzzy arcs.
𝜇𝐷𝑒 1 ∪ 𝜇𝐷𝑒 2 are the set of fuzzy directed edges are
Example: 4.1 Consider V = { 𝑥1 , 𝑥2, 𝑥3, 𝑥4 } 𝑎𝑛𝑑 𝐸 = { 𝑒1, 𝑒2, 𝑒3 }. Let V1 = { 𝑥1 , 𝑥2, 𝑥3 } , A = { 𝑒1, 𝑒2 }, 1 V2 = { 𝑥2, 𝑥3, 𝑥4 } & 𝐵 = { 𝑒2, 𝑒3 }.𝐷𝐴,𝑉 is defined 𝑏𝑦 𝑇𝑎𝑏𝑙𝑒 4.1 and 1 𝑒 𝜇𝐷 1 (𝑥𝑖 , 𝑥𝑗 ) = 0 for all (𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉1 𝑋 𝑉1 \ 𝑥1 , 𝑥2 , 𝑥2 , 𝑥3 , 𝑥3 , 𝑥1 } and for all e ∈ 𝐴. 2 𝐷𝐵,𝑉 is defined by Table 4.2 and 2 𝑒 𝜇𝐷 2 (𝑥𝑖 , 𝑥𝑗 ) = 0 for all (𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉2 𝑋𝑉2 \ { 𝑥2 , 𝑥3 , 𝑥3 , 𝑥4 , 𝑥3 , 𝑥2 , 𝑥2 , 𝑥4 , 𝑥4 , 𝑥2 } and for all e 1 2 3 ∈ 𝐵. Then the Union of 𝐷𝐴,𝑉 and 𝐷𝐵,𝑉 is 𝐷𝐶,𝑉 given by the Table 4.3 and 𝜇𝐷𝑒 3 ( 1 2 3 𝑥𝑖 , 𝑥𝑗 ) = 0 for all 𝑥𝑖 , 𝑥𝑗 ∈ 𝑉3 𝑋 𝑉3 \ { 𝑥2 , 𝑥3 , 𝑥3 , 𝑥4 , 𝑥3 , 𝑥2 , 𝑥3 , 𝑥1 , 𝑥1 , 𝑥2 , 𝑥2 , 𝑥4 , 𝑥4 , 𝑥2 } and for all e ∈ 𝐶. Table 4.1
Table 4.2 𝜌𝐷 2 𝑒2 𝑒3
𝑥2 0.6 0.7
𝑥3 0.5 0.5
𝑥4 0.4 0.4
𝜇𝐷 2 𝑒2 𝑒3
Fig 4.1 DOI: 10.9790/5728-1303024654
𝑥2 , 𝑥3 0 0.3
𝑥3 , 𝑥2 0.4 0
𝑥3 , 𝑥4 0.2 0
𝑥4 , 𝑥2 0.2 0
𝑥2 , 𝑥4 0 0.2
𝑫𝟏𝑨,𝑽𝟏 www.iosrjournals.org
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An Introduction to Fuzzy Soft Digraph
𝑫𝟐𝑩,𝑽𝟐
Fig 4.2
Table 4.3 𝜌𝐷 𝑒1 𝑒2 𝑒3 𝜇𝐷 𝑒1 𝑒2 𝑒3
𝑥1 , 𝑥2 0.4 0.1 0
𝑥2 , 𝑥3 0.3 0.2 0.3
𝑥1 0.2 0.8 0.7
𝑥2 0.8 0.6 0.5
𝑥3 , 𝑥2 0 0.4 0
Fig 4.3
𝑥3 0.6 0.9 0.4 𝑥3 , 𝑥1 0.2 0 0
𝑥4 0 0.4 0 𝑥3 , 𝑥4 0 0.2 0
𝑥4 , 𝑥2 0 0.2 0
𝑥2 , 𝑥4 0 0 0.2
3 𝐷𝐶,𝑉 3
Definition: 4. 2 Let 𝑉1 , 𝑉2 ∁ 𝑉 and A , B ∁ 𝐸 then the Intersection of two fuzzy soft digraphs . 1 𝐷𝐴,𝑉 = ((A, 𝜌𝐷𝑒 1 ), (A, 𝜇𝐷𝑒 1 ) ) and 1 2 𝐷𝐵,𝑉 = ((B, 𝜌𝐷𝑒 2 ), (B, 𝜇𝐷𝑒 2 ) ) 2 3 𝐷𝐶,𝑉 = ((C, 𝜌𝐷𝑒 3 ), (C, 𝜇𝐷𝑒 3 ) ) 3
is defined to be say.
Where C = A ∩ B , 𝑉3 = 𝑉1 ∩ 𝑉2 and 𝜌𝐷𝑒 3 ( 𝑥𝑖 ) = 𝜌𝐷𝑒 1 ( 𝑥𝑖 ) ∩ 𝜌𝐷𝑒 2 ( 𝑥𝑖 ) for all 𝑥𝑖 ∈ 𝑉3 and 𝑒 ∈ 𝐶, and 𝜇𝐷𝑒 3 ( 𝑥𝑖 , 𝑥𝑗 ) = 𝜇𝐷𝑒 1 ( 𝑥𝑖 , 𝑥𝑗 ) 𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 ) if ( 𝑥𝑖 , 𝑥𝑗 ) ∈ 𝑉3 & 𝑒 ∈ 𝐶.
Example : 4.2 Consider V = { 𝑥1 , 𝑥2, 𝑥3, 𝑥4 } 𝑎𝑛𝑑 𝐸 = { 𝑒1, 𝑒2, 𝑒3 }. Let V1 = { 𝑥1 , 𝑥2, 𝑥3 } ,𝐴 = { 𝑒1, 𝑒2 }, V2 = { 1 𝑥2, 𝑥3, 𝑥4 } & 𝐵 = { 𝑒2, 𝑒3 }. 𝐷𝐴,𝑉 is defined 𝑏𝑦 𝑇𝑎𝑏𝑙𝑒 4.4 and 𝜇𝐷𝑒 1 (𝑥𝑖 , 𝑥𝑗 ) = 0 for all (𝑥𝑖 , 𝑥𝑗 ) ∈ 1 2 𝑉1 𝑋 𝑉1 \ 𝑥1 , 𝑥2 , 𝑥2 , 𝑥3 , 𝑥3 , 𝑥1 } and for all e ∈ 𝐴. 𝐷𝐵,𝑉 is defined by Table 4.5 and 𝜇𝐷𝑒 2 ( 𝑥𝑖 , 𝑥𝑗 ) = 2 DOI: 10.9790/5728-1303024654
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An Introduction to Fuzzy Soft Digraph 0 for all (đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ) ∈ đ?‘‰2 đ?‘‹đ?‘‰2 \ { đ?‘Ľ2 , đ?‘Ľ3 , đ?‘Ľ2 , đ?‘Ľ4 , đ?‘Ľ4 , đ?‘Ľ2 } and for all e ∈ đ??ľ. So đ?‘‰3 = đ?‘Ľ2 , đ?‘Ľ3 {đ?‘’2 }. 1 2 3 Then the intersection of đ??ˇđ??´,đ?‘‰ and đ??ˇđ??ľ,đ?‘‰ is đ??ˇđ??ś,đ?‘‰ given by the Table 4.6 and 1 2 3 đ?‘’ đ?œ‡đ??ˇ 3 ( đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ) = 0 for all đ?‘Ľđ?‘– , đ?‘Ľđ?‘— ∈ đ?‘‰3 đ?‘‹ đ?‘‰3 \ { đ?‘Ľ2 , đ?‘Ľ2 , đ?‘Ľ3 , đ?‘Ľ2 , đ?‘Ľ3 , đ?‘Ľ3 } and for all e ∈ đ??ś.
đ?‘Žđ?‘›đ?‘‘ đ??ś =
Table 4.4 đ?œŒđ??ˇ 1 đ?‘’1 đ?‘’2
đ?‘Ľ1 0.7 0.5
đ?‘Ľ2 0.5 0.7
đ?œ‡đ??ˇ 1 đ?‘’1 đ?‘’2
đ?‘Ľ3 0.3 0.6
đ?‘Ľ1 , đ?‘Ľ2 0.3 0.4
đ?‘Ľ2 , đ?‘Ľ3 0 0.3
đ?‘Ľ3 , đ?‘Ľ1 0.2 0
Table 4.5 đ?œŒđ??ˇ 2 đ?‘’2 đ?‘’3
đ?‘Ľ2 0.8 0.5
đ?‘Ľ3 0.6 0.4
đ?‘Ľ4 0.4 0.3
đ?œ‡đ??ˇ 2 đ?‘’2 đ?‘’3
đ?‘Ľ2 , đ?‘Ľ3 0.5 0.3
đ?‘Ľ4 , đ?‘Ľ2 0..2 0
đ?‘Ľ2 , đ?‘Ľ4 0 0.1
đ?‘Ťđ?&#x;?đ?‘¨,đ?‘˝đ?&#x;?
Fig 4.4
đ?‘Ťđ?&#x;?đ?‘Š,đ?‘˝đ?&#x;?
Fig 4.5 Table 4.6 đ?œŒđ??ˇ 3 đ?‘’2
đ?‘Ľ2 0.7
đ?‘Ľ3 0.6
đ?œ‡đ??ˇ 3 đ?‘’2
đ?‘Ľ2 0.8
đ?‘Ľ2 , đ?‘Ľ3 0.3
(0.6)đ?‘Ľ3 Fig 4.6
V.
đ?‘Ťđ?&#x;‘đ?‘Ş,đ?‘˝đ?&#x;‘
Conclusion
Finally, we have analyzed some concepts of fuzzy soft digraph, union and intersection of fuzzy soft digraph. In future it can be applied to real applications in mobile networks, one way transportation problem and decision making problems.
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DOI: 10.9790/5728-1303024654
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