# Cayley Bipolar Fuzzy Graphs

Authors: Noura O. Alshehri; Muhammad Akram
Journal: The Scientific World Journal
 (2013)

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## Abstract

We introduce the concept of Cayley bipolar fuzzy graphs and investigate some of their properties. We present some interesting properties of bipolar fuzzy graphs in terms of algebraic structures. We also discuss connectedness in Cayley bipolar fuzzy graphs.


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## Body

## 1. Introduction

Graph theory is an extremely useful tool in solving the combinatorial problems in different areas. Point-to-point interconnection networks for parallel and distributed systems are usually modeled bydirected graphs (or digraphs). A digraph is a graph whose edges have direction and are called arcs (edges). Arrows on the arcs are used to encode the directional information: an arc from vertex (node) x to vertex y indicates that one may move from x to y but not from y to x. The Cayley graph was first considered for finite groups by Cayley in 1878. Max Dehn in his unpublished lectures on group theory from 1909 to 1910 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point [1].The notion of fuzzy sets was introduced by Zadeh [2] as a method of representing uncertainty and vagueness. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines. In 1994, Zhang [3] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets [4]. A bipolar fuzzy set is an extension of fuzzy sets whose membership degree range is [-1,1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1,0) of an element indicates that the element somewhat satisfies the implicit counterproperty.Kaufmann's initial definition of a fuzzy graph [5] was based on Zadeh's fuzzy relations [2]. Mordeson and Nair [6] introduced the fuzzy analogue of several basic graph-theoretic concepts. Kauffman [5] defined the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. Akram and Dudek [7–9] introduced many new concepts, including bipolar fuzzy graphs, complete bipolar fuzzy graphs, regular bipolar fuzzy graphs, and irregular bipolar fuzzy graphs. Wu [4] discussed fuzzy digraphs. Shahzamanian et al. [10] introduced the notion of roughness in Cayley graphs and investigated several properties. Namboothiri et al. [11] discussed Cayley fuzzy graphs. In this paper, we introduce the concept of Cayley bipolar fuzzy graphs and investigate some of their properties. We present some interesting properties of bipolar fuzzy graphs in terms of algebraic structures. We also discuss connectedness in Cayley bipolar fuzzy graphs.


## 2. Preliminaries

A digraph is a pairD*=(V,E), where V is a finite set and E⊆V×V. Let D1*=(V1,E1) and D2*=(V2,E2) be two digraphs. The Cartesian product of D1* and D2* gives a digraph D1*×D2*=(V,E) with V=V1×V2 and
(1)E={(x,x2)⟶(x,y2)∣x∈V1,x2⟶y2∈E2}∪{(x1,z)⟶(y1,z)∣x1⟶y1∈E1,z∈V2}.
In this paper, we will write xy∈E to mean x→y∈E, and if e=xy∈E, we say x and y are adjacent such that x is a starting node and y is an ending node.The study of vertex transitive graphs has a long and rich history in discrete mathematics. Prominent examples of vertex transitive graphs are Cayley graphs which are important in both theory and applications; for example, Cayley graphs are good models for interconnection networks.Definition 1.
LetG be a finite group and let S be a minimal generating set of G. A Cayley graph (G,S) has elements of G as its vertices; the edge set is given by {(g,gs):g∈G,s∈S}. Two vertices g1 and g2 are adjacent if g2=g1·s, where s∈S. Note that a generating set S is minimal if S generates G but no proper subset of S does.Theorem 2.
All Cayley graphs are vertex transitive.Definition 3.
Let(V,*) be a group and let A be any subset of V. Then the Cayley graph induced by (V,*,A) is the graph G=(V,R), where R={(x,y):x-1y∈A}.Definition 4 (see [2]).
A fuzzy subsetμ on a set X is a map μ:X→[0,1]. A fuzzy binary relation on X is a fuzzy subset μ on X×X. By a fuzzy relation we mean a fuzzy binary relation given by μ:X×X→[0,1].Definition 5 (see [11]).
Let(V,*) be a group and let μ be a fuzzy subset of V. Then the fuzzy relation R on V defined by
(2)R(x,y)={μ(x-1*y)∀x,y∈V}
induces a fuzzy graph G=(V,R), called the Cayley fuzzy graph induced by the (V,*,μ).Definition 6 (see [3]).
LetX be a nonempty set. A bipolar fuzzy setB in X is an object having the form
(3)B={(x,μBP(x),μBN(x))∣x∈X},
where μBP:X→[0,1] and μBN:X→[-1,0] are mappings.We use the positive membership degreeμBP(x) to denote the satisfaction degree of an element x to the property corresponding to a bipolar fuzzy set B and the negative membership degree μBN(x) to denote the satisfaction degree of an element x to some implicit counterproperty corresponding to a bipolar fuzzy set B. If μBP(x)≠0 and μBN(x)=0, it is the situation that x is regarded as having only positive satisfaction for B. If μBP(x)=0 and μBN(x)≠0, it is the situation that x does not satisfy the property of B but somewhat satisfies the counterproperty of B. It is possible for an element x to be such that μBP(x)≠0 and μBN(x)≠0 when the membership function of the property overlaps that of its counterpro   perty over some portion of X.For the sake of simplicity, we shall use the symbolB=(μBP,μBN) for the bipolar fuzzy set
(4)B={(x,μBP(x),μBN(x))∣x∈X}.
A nice application of bipolar fuzzy concept is a political acceptation (map to [0, 1]) and nonacceptation (map to [-1,0]).Definition 7 (see [3]).
A bipolar fuzzy relationR=(μRP(x,y),μRN(x,y)) in a universe X×Y (R(X→Y), for short) is a bipolar fuzzy set of the form
(5)R={〈(x,y),μRP(x,y),μRN(x,y)〉∣(x,y)∈X×Y},
where μRP:X×Y→[0,1] and μRN:X×Y→[-1,0].Definition 8.
LetR be a bipolar fuzzy relation on universe X. Then R is called a bipolar fuzzy equivalence relation on X if it satisfies the following conditions:  (a)
Ris bipolar fuzzy reflexive; that is, R(x,x) = (1, −1) for each x∈X;(b)
R is bipolar fuzzy symmetric; that is, R(x,y)=R(y,x) for any x, y∈X;(c)
R is bipolar fuzzy transitive; that is, R(x,z)≥⋁y(R(x,y)∧R(y,z)).Definition 9.
LetR be a bipolar fuzzy relation on universe X. Then R is called a bipolar fuzzy partial order relation on X if it satisfies the following conditions:(a)
R is bipolar fuzzy reflexive; that is, R(x,x)=(1,-1) for each x∈X;(b)
R is bipolar fuzzy antisymmetric; that is, R(x,y)≠R(y,x) for any x, y∈X;(c)
R is bipolar fuzzy transitive; that is, R(x,z)≥⋁y(R(x,y)∧R(y,z)).Definition 10.
LetR be a bipolar fuzzy relation on universe X. Then R is called a bipolar fuzzy linear order relation on X if it satisfies the following conditions:(a)
R is bipolar fuzzy partial relation;(b)
(μRP∨μR-1P)(x,y)>0, (μRN∧μR-1N)(x,y)<0 for all x, y∈X.


## 3. Cayley Bipolar Fuzzy Graphs

Definition 11.
Abipolar fuzzy digraph of a digraph D* is a pair D=(A,B) where A=(μAP,μAN) is a bipolar fuzzy set in V and B=(μBP,μBN) is a bipolar fuzzy relation on E such that
(6)μBP(xy)≤min(μAP(x),μAP(y)),μBN(xy)≥max(μAN(x),μAN(y))
for all xy∈E. We note that B need not to be symmetric.Definition 12.
LetD be a bipolar fuzzy digraph. The indegree of a vertex x in D is defined by ind(x)=(indμP(x),indμN(x)), where indμP(x)=∑y≠x‍μAP(xy) and indμN(x)=∑y≠x‍μAN(xy). Similarly, the outdegree of a vertex x in D is defined by outd(x)=(outdμP(x),outdμN(x)), where outdμP(x)=∑y≠x‍μAP(xy) and outdμN(x)=∑y≠x‍μAN(xy). A bipolar fuzzy digraph in which each vertex has the same outdegree r is called an outregular digraph with index of outregularity r. In-regular digraphs are defined similarly.Definition 13.
Let(V,*) be a group and let A=(μAP,μAN) be the bipolar fuzzy subset of V. Then the bipolar fuzzy relation R defined on V by
(7)R(x,y)=(μRP(x-1y),μRN((x-1y))),∀x,y∈V,
induces a bipolar fuzzy graph G=(V,R) called the Cayley bipolar fuzzy graph induced by the (V,*,μRP,μRN).We now introduce Cayley bipolar fuzzy graphs and prove that all Cayley bipolar fuzzy graphs are regular.Definition 14.
Let(V,*) be a group and let A=(μAP,μAN) be a bipolar fuzzy subset of V. Then the bipolar fuzzy relation R on V defined by
(8)R(x,y)={μRP(x-1y),μRN(x-1y)∀x,y∈V}
induces a bipolar fuzzy graph G=(V,R), called the Cayley bipolar fuzzy graph induced by the (V,*,A).Example 15.
Consider the groupZ3 and take V={0,1,2}. Define μAP:V→[0,1] and μAN:V→[-1,0] by μAP(0)=μAP(1)=μAP(2)=0.5, μAN(0)=μAN(1)=μAN(2)=-0.4. Then the Cayley bipolar fuzzy graph G=(V,R) induced by (Z3,+,A) is given by Table 1 and Figure 1.Table 1
R
(
a
,
b
) for Cayley bipolar fuzzy graph.
a
0
0
0
1
1
1
2
2
2
b
0
1
2
0
1
2
0
1
2
(
-
a
)
+
b
0
1
2
2
0
1
1
2
0
R
(
a
,
b
)
(0.5, −0.4)
(0.3, −0.2)
(0.3, −0.2)
(0.3, −0.2)
(0.5, −0.4)
(0.3, −0.2)
(0.3, −0.2)
(0.3, −0.2)
(0.5, −0.4)Figure 1
Cayley bipolar fuzzy graph.We see that Cayley bipolar fuzzy graphs are actually bipolar fuzzy digraphs. Furthermore, the relationR in the above definition describes the strength of each directed edge. Let G denote a bipolar fuzzy graph G=(V,R) induced by the triple (V,*,A).Theorem 16.
The Cayley bipolar fuzzy graphG is vertex transitive.Proof.
Leta,b∈V. Define ψ:V→V by ψ(x)=ba-1x for all x∈V. Clearly, ψ is a bijective map. For each x, y∈V,(9)R(ψ(x),ψ(y))=(RμP(ψ(x),ψ(y)),RμN(ψ(x),ψ(y))).NowRμP(ψ(x),ψ(y))=RμP(ba-1x,ba-1y)=μAP((ba-1x)-1(ba-1x))=μAP(x-1y)=RμP(x,y),RμN(ψ(x),ψ(y))=RμN(ba-1x,ba-1y)=μAN((ba-1x)-1(ba-1x))=μAN(x-1y)=RμN(x,y).
Therefore, R(ψ(x),ψ(y))=R(x,y). Hence ψ is an automorphism on G. Also ψ(a)=b. Hence G is vertex transitive.Theorem 17.
Every vertex transitive bipolar fuzzy graph is regular.Proof.
LetG=(V,R) be any vertex transitive bipolar fuzzy graph. Let u,v∈V. Then there is an automorphism f on G such that f(u)=v. Note that
(10)ind(u)=∑x∈VR(x,u)=∑x∈V(RμP(x,u),RμN(x,u))=∑x∈V(RμP(f(x),f(u)),RμN(f(x),f(u)))=∑x∈V(RμP(f(x),v),RμN(f(x),v))=∑x∈V(RμP(y,v),RμN(y,v))=ind(v),outd(u)=∑x∈VR(x,u)=∑x∈V(RμP(u,x),RμN(u,x))=∑x∈V(RμP(f(u),f(x)),RμN(f(u),f(x)))=∑x∈V(RμP(v,f(x)),RμN(v,f(x)))=∑x∈V(RμP(v,y),RμN(v,y))=outd(v).
Hence G is regular.Theorem 18.
Cayley bipolar fuzzy graphs are regular.Proof.
Proof follows from Theorems16 and 17.Theorem 19.
LetG=(V,R) denote bipolar fuzzy graph. Then bipolar fuzzy relation R is reflexive if and only if μAP(1)=1 and μAN(1)=-1.Proof.
R is reflexive if and only if R(x,x)=(1,-1) for all x∈V. Now
(11)R(x,x)=(μAP(x-1x),μAN(x-1x))=(μAP(1),μAN(1)),∀x∈V.
Hence R is reflexive if and only if μAP(1)=1 and μAN(1)=-1.Theorem 20.
LetG=(V,R) denote bipolar fuzzy graph. Then bipolar fuzzy relation R is symmetric if and only if (μAP(x),μAN(x))=(μAP(x-1),μAN(x-1)) for all x∈V.Proof.
Suppose thatR is symmetric. Then for any x∈V,
(12)(μAP(x),μAN(x))=(μAP(x-1x2),μAN(x-1x2))=R(x,x2)=R(x2,x)(sinceRissymmetric)=(μAP((x2)-1x),μAN(x2)-1x)=μAP(x-2x),μAN(x-2x)=μAP(x-1),μAN(x-1).
Conversely, suppose that (μAP(x),μAN(x))=(μAP(x-1),μAN(x-1)) for all x∈V. Then for all x,y∈V,
(13)R(x,y)=(μAP(x-1y),μAN(x-1y))=(μAP(y-1x),μAN(y-1x))=R(y,x).
Hence R is symmetric.Theorem 21.
A bipolar fuzzy relationR is antisymmetric if and only if {x:(μAP(x),μAN(x))=(μAP(x-1),μAN(x-1))=(1,-1).Definition 22.
Let(S,*) be a semigroup. Let A=(μAP,μAN) be a bipolar fuzzy subset of S. Then A is said to be a bipolar fuzzy subsemigroup of S if for all x,y∈S, μBP(xy)≥μAP(x)∧μAP(y) and μBN(xy)≤μAN(x)∨μAN(y).Theorem 23.
A bipolar fuzzy relationR is transitive if and only if A=(μAP,μAN) is a bipolar fuzzy subsemigroup of (V,*).Proof.
Suppose thatR is transitive and let x,y∈V. Then R2≤R. Now for any x∈V, we have R(1,x)=(μAP(x),μAN(x)). This implies that {R(1,z)∧R(z,xy):z∈V}=R2(1,xy)≤R(1,xy). That is ∨{μAP(z)∧μAP(z-1xy):z∈V}≤μAP(xy) and ∧{μAN(z)∨μAN(z-1xy):z∈V}≥μAN(xy). Hence μAP(xy)≥μAP(x)∧μAP(y) and μAN(xy)≤μAP(x)∨μAN(y). Hence A=(μAP,μAN) is a bipolar fuzzy subsemigroup of (V,*).
Conversely, suppose thatA=(μAP,μAN) is a bipolar fuzzy subsemigroup of (V,*). That is, for all x,y∈VμBP(xy)≥μAP(x)∧μAP(y) and μBN(xy)≤μAN(x)∨μAN(y). Then for any x,y∈V,
(14)R2(x,y)=(RμP2(x,y),RμN2(x,y)),RμP2(x,y)=∨{RμP(x,z)∧RμP(z,y):z∈V}=∨{μAP(x-1z)∧μAP(z-1y):z∈V}≤μAP(x-1y)=RμP(x,y),RμN2(x,y)=∧{RμN(x,z)∨RμN(z,y):z∈V}=∧{μAN(x-1z)∨μAN(z-1y):z∈V}≥μAN(x-1y)=RμN(x,y).
Hence RμP2(x,y)≤RμP(x,y) and RμN2(x,y)≥RμN(x,y). Hence R is transitive.We conclude that.Theorem 24.
A bipolar fuzzy relationR is a partial order if and only if A=(μAP,μAN) is a bipolar fuzzy subsemigroup of (V,*) satisfying(i)
μAP(1)=1 and μAN(1)=-1,(ii)
{x:(μAP(x),μAN(x))=(μAP(x-1),μAN(x-1))={1,-1}}.Theorem 25.
A bipolar fuzzy relationR is a linear order if and only if (μAP,μAN) is a bipolar fuzzy subsemigroup of (V,*) satisfying(i)
μAP(1)=1 and μAN(1)=-1,(ii)
{x:(μAP(x),μAN(x))=(μAP(x-1),μAN(x-1))={1,-1}},(iii)
{x:μAP(x)∨μAP(x-1)>0,μAN(x)∧μAN(x-1)<0}=V.Proof.
SupposeR is a linear order. Then by Theorem 24, conditions (i), (ii), and (iii) are satisfied. For any x∈V, (R∨R-1)(1,x)>0. This implies that R(1,x)∨R(x,1)>0. Hence {x:μAP(x)∨μAP(x-1)>0,μAN(x)∧μAN(x-1)<0}.
Conversely, suppose that conditions (i), (ii), and (iii) hold. Then by Theorem24, R is partial order. Now for any x,y∈V, we have (x-1y), (y-1x)∈V. Then by condition (iv), {x:μAP(x)∨μAP(x-1)>0,μAN(x)∧μAN(x-1)<0}. Therefore R is linear order.Theorem 26.
A bipolar fuzzy relationR is a equivalence relation if and only if (μAP,μAN) is a bipolar fuzzy subsemigroup of (V,*) satisfying(i)
μAP(1)=1 and μAN(1)=-1,(ii)
(μAP(x),μAN(x))=(μAP(x-1),μAN(x-1)) for all x∈V.Theorem 27.
G is a Hasse diagram if and only if for any collection x1,x2,x3,…,xn of vertices in V with n≥2 and μAP(xi)>0,μAN(xi)<0, for i=1,2,3,…,n, we have, μAP(x1x2⋯xn)=0 and μAN(x1x2⋯xn)=0.Proof.
SupposeG is a Hasse diagram and let x1,x2,…,xn be vertices in V with n≥2 and μAP(xi)>0,μAN(xi)<0, for i=1,2,3,…,n. Then it is obvious that R(x1x2⋯xi-1,x1x2⋯xi)=(μAP(xi),μAN(xi)), for i=1,2,…,n, where x0=1. Therefore (1,x1,x1x2,…,x1x2⋯xn) is a path from 1 to x1x2⋯xn. Since G is a Hasse diagram, we have R(1,x1x2⋯xn)=0. This implies that μAP(x1x2⋯xn)=0 and μAN(x1x2⋯xn)=0. Conversely, suppose that for any collection x1,x2,…xn of vertices in V with n≥2 and μAP(xi)>0,μAN(xi)<0, for i=1,2,3,…,n, we have, μAP(x1x2⋯xn)=0 and μAN(x1x2⋯xn)=0. Let (x0,x1,x2,…xn) be a path in G from x0 to xn with n≥2. Then R(xi-1,xi)>0, for i=1,2,…,n. Therefore μAP(xi-1-1xi)>0,μAN(xi-1-1xi)<0, for i=1,2,…,n. Now consider the elements x0-1x1,x1-1x2,…,xn-1-1xn in V. Then by assumption μAP(x0-1x1x1-1x2⋯xn-1-1xn)=0 and μAN(x0-1x1x1-1x2⋯xn-1-1xn)=0. That is, μAP(x0-1xn)=0 and μAN(x0-1xn)=0. Hence, R(x0,xn)=0. Thus G is a Hasse diagram.LetG=(V,R) be any bipolar fuzzy graph; then G is connected (weakly connected, semiconnected, locally connected, or quasi-connected) if and only if the induce fuzzy graph (V,R0+) is connected (weakly connected, semiconnected, locally connected, or quasi-connected).Definition 28.
Let(S,*) be a semigroup and let A=(μAP,μAN) be a bipolar fuzzy subset of S. Then the subsemigroup generated by A is the meeting of all bipolar fuzzy subsemigroups of S which contains A. It is denoted by 〈A〉.Lemma 29.
Let(S,*) be a semigroup and A=(μAP,μAN) be a bipolar fuzzy subset of S. Then bipolar fuzzy subset 〈A〉 is precisely given by 〈μAP〉(x)=∨{μAP(x1)∧μAP(x2)∧⋯∧μAP(xn):x=x1x2⋯xn withμAP(xi)>0 for i=1,2,…,n}, 〈μAN〉(x)=∧{μAN(x1)∨μAN(x2)∨⋯∨μAN(xn):x=x1x2⋯xn with μAN(xi)<0fori=1,2,…,n} for any x∈S.Proof.
LetA′=(μ´AP,μ´AN) be a bipolar fuzzy subset of S defined by μ´AP(x)=∨{μAP(x1)∧μAP(x2)∧⋯∧μAP(xn):x=x1x2⋯xn with μAP(xi)>0 for i=1,2,…,n}, μ´AN(x)=∧{μAN(x1)∨μAN(x2)∨⋯∨μAN(xn):x=x1x2⋯xn with μAN(xi)<0 for i=1,2,…,n}, for any x∈S. Let x,y∈S. If μAP(x)=0 or μAP(y)=0, then μAP(x)∧μAP(y)=0 and μAN(x)=0 or μAN(y)=0, and then μAN(x)∨μAN(y)=0. Therefore, μ´BP(xy)≥μAP(x)∧μAP(y) and μ´BN(xy)≤μAN(x)∨μAN(y). Again, if μAP(x)≠0,μAN(x)≠0, then by definition of μ´AP(x) and μ´AN(x), we have μ´BP(xy)≥μAP(x)∧μAP(y) and μ´BN(xy)≤μAN(x)∨μAN(y). Hence (μ´AP,μ´AN) is a bipolar fuzzy subsemigroup of S containing (μAP,μAN). Now let L be any bipolar fuzzy subsemigroup of S containing (μAP,μAN). Then for any x∈S with x=x1x2⋯xn with μAP(xi)>0,μAN(xi)<0, for i=1,2,…,n, we have μLP(xi)≥μLP(x1)∧μLP(x2)∧⋯∧μLP(xn)≥μAP(x1)∧μAP(x2)∧⋯∧μAP(xn) and μLN(xi)≥μLN(x1)∧μLN(x2)∧⋯∧μLN(xn)≥μAN(x1)∧μAN(x2)∧⋯∧μAN(xn). Thus μLP(x)≥∨{μAP(x1)∧μAP(x2)∧⋯∧μAP(xn):x=x1x2⋯xn with μAP(xi)>0 for i=1,2,…,n}, μLN(x)≤∧{μAN(x1)∨μAN(x2)∨⋯∨μAN(xn):x=x1x2⋯xn with μAN(xi)<0 for i=1,2,…,n}, for any x∈S. Hence μLP(x)≥μ´AP(x),μLN(x)≤μ´AN(x), for all x∈S. Thus μ´AP(x)≤μLP(x), μ´AN(x)≥μAP(x). Thus A′=(μ´AP,μ´AN) is the meeting of all bipolar fuzzy subsemigroups containing (μAP,μAN).Theorem 30.
Let(S,*) be a semigroup and A=(μAP,μAN) be a bipolar fuzzy subset of S. Then for any α∈[0,1], (〈μαP〉,〈μαN〉)=(〈μP〉α,〈μN〉α) and (〈(μ+)αP〉,〈(μ+)αN〉)=(〈μP〉α+,〈μN〉α+), where (〈μαP〉,〈μαN〉) denotes the subsemigroup generated by (μαP,μαN) and 〈(μP,μN)〉 denotes bipolar fuzzy subsemigroup generated by (μP,μN).Proof.
(15)
x
∈
(
〈
μ
P
〉
α
,
〈
μ
N
〉
α
)
⟺
there
exists
x
1
,
x
2
,
…
,
x
n
in
(
μ
α
P
,
μ
α
N
)
such
that
x
=
x
1
x
2
⋯
x
n
⟺
there
exists
x
1
,
x
2
,
…
,
x
n
in
S
such
that
μ
P
(
x
i
)
≥
α
,
μ
N
(
x
i
)
≤
α
,
∀
i
=
1,2
,
…
,
n
,
x
=
x
1
x
2
⋯
x
n
⟺
〈
μ
P
〉
(
x
)
≥
α
,
〈
μ
N
〉
(
x
)
≤
α
⟺
x
∈
〈
μ
P
〉
α
,
x
∈
〈
μ
N
〉
α
.
Therefore (〈μαP〉,〈μαN〉)=(〈μP〉α,〈μN〉α). Similarly, we have (〈(μ+)αP〉,〈(μ+)αN〉)=(〈μP〉α+,〈μN〉α+).Remark 31.
Let(S,*) be a semigroup and A=(μAP,μAN) be a bipolar fuzzy subset of S. Then by Theorem 30, we have 〈supp(A)=A+〉=supp〈A〉.LetG denote the Cayley bipolar fuzzy graphs G=(V,R) induced by (V,*,μP,μN). Then we have the following results.Theorem 32.
LetA be any subset of V′ and G′=(V′,R′) be the Cayley graph induced by (V′,*,A). Then G′ is connected if and only if 〈A〉⊇V-v1.Theorem 33.
G is connected if and only ifsupp〈A〉⊇V-v1.Theorem 34.
LetA be any subset of a set V′ and let G′=(V′,R′) be the Cayley graph induced by the triplet (V′,*,A). Then G′ is weakly connected if and only if 〈A∪A-1〉⊇V-v1, where A-1={x-1:x∈A}.Definition 35.
Let(S,*) be a group and let A be a bipolar fuzzy subset of S. Then we define A-1 as bipolar fuzzy subset of S given by A-1(x)=A(x-1) for all x∈S.Theorem 36.
G is weakly connected if and only if supp(〈A∪A-1〉)⊇V-v1.Proof.
(16)
G
is
weakly
connected
⟺
(
V
,
R
0
+
)
is
weakly
connected
⟺
〈
A
0
+
∪
(
A
0
+
)
-
1
〉
⊇
V
-
v
1
⟺
〈
supp
(
A
)
∪
supp
(
A
)
-
1
〉
⊇
V
-
v
1
⟺
supp
〈
A
∪
(
A
)
-
1
〉
⊇
V
-
v
1
⟺
supp
〈
A
∪
A
-
1
〉
⊇
V
-
v
1
.Theorem 37.
LetA be any subset of a set V′ and let G′=(V′,R′) be the Cayley graph induced by the triplet (V′,*,A). Then G′ is semiconnected if and only if 〈A〉∪〈A-1〉⊇V-v1, where A-1={x-1:x∈A}.Theorem 38.
G is semi-connected if and only if supp(〈A〉∪〈A-1〉)⊇V-v1.Proof.
(17)
G
is
semiconnected
⟺
(
V
,
R
0
+
)
is
semi
connected
⟺
〈
A
0
+
〉
∪
〈
(
A
0
+
)
-
1
〉
⊇
V
-
v
1
⟺
〈
supp
(
A
)
〉
∪
〈
supp
(
A
)
-
1
〉
⊇
V
-
v
1
⟺
supp
〈
A
〉
∪
〈
(
A
)
-
1
〉
⊇
V
-
v
1
⟺
supp
(

〈

A

〉

∪

〈

A
-
1

〉

)
⊇
V
-
v
1
.Theorem 39.
LetG′=(V′,R′) be the Cayley graph induced by the triplet (V′,*,A). Then G′ is locally connected if and only if 〈A〉=〈A-1〉, where A-1=(x-1:x∈A).Theorem 40.
LetG is locally connected if and only if supp(〈A〉)=supp(〈A-1〉).Proof.
(18)
G
is
locally
connected
⟺
(
V
,
R
0
+
)
is
locally
connected
⟺
〈
A
0
+
〉
=
〈
(
A
0
+
)
-
1
〉
⟺
〈
supp
(
A
)
〉
=
〈
supp
(
A
)
-
1
〉
⟺
supp
〈
A
〉
=
supp
〈
A
-
1
〉
.Theorem 41.
LetG′=(V′,R′) be the Cayley graph induced by the triplet (V′,*,A), where V′ is finite. Then G′ is quasi-connected if and only if it is connected.Theorem 42.
A finite Cayley bipolar fuzzy graphG is quasi-connected if and only if it is connected.Proof.
(19)
G
is
quasi-connected
⟺
(
V
,
R
0
+
)
is
quasi-connected
⟺
(
V
,
R
0
+
)
is
connected
⟺
G
is
connected
.Definition 43.
TheμP strength of a path P=v1,v2,…,vn is defined as min(μ2P(vi,vj)) for all i and j and is denoted by SμP. The μN strength of a path P=v1,v2,…,vn is defined as max(μ2N(vi,vj)) for all i and j and is denoted by SμN.Definition 44.
LetG=(V,μP,μN) be a bipolar fuzzy graph. Then G is said to be(1)
α-connected if for every pair of vertices x,y∈G, there is a path P from x to y such that strength (P)≥α,(2)
weaklyα-connected if a bipolar fuzzy graph (V,R∨R-1) is α-connected,(3)
semi-α-connected if for every x,y∈V, there is a path of strength greater than or equal to α from x to y or from y to x in G,(4)
locallyα-connected if for every pair of vertices x and y, there is a path P of strength greater than or equal to α from x to y whenever there is a path P′ of strength greater than or equal to α from y to x,(5)
quasi-α-connected if for every pair x,y∈V, there is some z∈V such that there is directed path from z to x of strength greater than or equal to α and there is a directed path from z to y of strength greater than or equal to α.Remark 45.
LetG=(V,R) be any bipolar fuzzy graph; then G is α-connected (weakly α-connected, semi α-connected, locally α-connected or quasi α-connected) if and only if the induce fuzzy graph (V,R0+) is connected (weakly connected, semiconnected, locally connected, or quasi-connected).LetG denote the Cayley bipolar fuzzy graphs G=(V,R) induced by (V,*,μP,μN). Also for any α∈[-1,1], we have the following results.Theorem 46.
G is α-connected if and only if 〈A〉α⊇V-v1.Proof.
(20)
G
is
connected
⟺
(
V
,
R
α
)
is
connected
⟺
〈
A
α
〉
⊇
V
-
v
1
⟺
〈
A
〉
α
⊇
V
-
v
1
.Theorem 47.
G is weakly α-connected if and only if 〈A∪A-1〉α⊇V-v1.Proof.
(21)
G
is
weakly
connected
⟺
(
V
,
R
α
)
is
weakly
connected
⟺
〈
A
α
∪
(
A
α
)
-
1
〉
⊇
V
-
v
1
⟺
〈
(
A
∪
A
-
1
)
α
〉
⊇
V
-
v
1
⟺
〈
A
∪
(
A
)
-
1
〉
α
⊇
V
-
v
1
.Theorem 48.
G is semi-α-connected if and only if (〈A〉α∪〈A-1〉α)⊇V-v1.Theorem 49.
LetG be locally α-connected if and only if 〈A〉α=〈Aα-1〉.Theorem 50.
A finite Cayley bipolar fuzzy graphG is quasi-α-connected if and only if it is α-connected.


## 4. Conclusions

Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim of reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. A bipolar fuzzy set is a generalization of the notion of a fuzzy set. We have introduced the notion of Cayley bipolar fuzzy graphs in this paper. The natural extension of this research work is application of bipolar fuzzy digraphs in the area of soft computing including neural networks, decision making, and geographical information systems.



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*Source: 156786-2013-12-12.xml*