bipolar fuzzy graphs

Information Sciences 181 (2011) 5548–5564

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Information Sciences

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Bipolar fuzzy graphs

Muhammad AkramPunjab University College of Information Technology, University of the Punjab, Old Campus, Lahore 54000, Pakistan

a r t i c l e i n f o

Article history:Received 28 May 2010Received in revised form 10 June 2011Accepted 21 July 2011Available online 29 July 2011

Keywords:Bipolar fuzzy graphsIsomorphismsStrong bipolar fuzzy graphsSelf complementary

0020-0255/$ - see front matter � 2011 Elsevier Incdoi:10.1016/j.ins.2011.07.037

E-mail addresses: [email protected], m.a

a b s t r a c t

In this paper, we introduce the notion of bipolar fuzzy graphs, describe various methods oftheir construction, discuss the concept of isomorphisms of these graphs, and investigatesome of their important properties. We then introduce the notion of strong bipolar fuzzygraphs and study some of their properties. We also discuss some propositions of self com-plementary and self weak complementary strong bipolar fuzzy graphs.

� 2011 Elsevier Inc. All rights reserved.

1. Introduction

In 1965, Zadeh [51] introduced the notion of a fuzzy subset of a set as a method for representing uncertainty. Since then,the theory of fuzzy sets has become a vigorous area of research in different disciplines including medical and life sciences,management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multi-agent systems, pattern recognition, robotics, computer networks, expert systems, decision making and automata theory.

In 1994, Zhang [56,57] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. Bipolar fuzzy sets are anextension of fuzzy sets whose membership degree range is [�1,1]. In a bipolar fuzzy set, the membership degree 0 of an ele-ment means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indi-cates that the element somewhat satisfies the property, and the membership degree [�1,0) of an element indicates that theelement somewhat satisfies the implicit counter-property. Although bipolar fuzzy sets and intuitionistic fuzzy sets look sim-ilar to each other, they are essentially different sets [27]. In many domains, it is important to be able to deal with bipolar infor-mation. It is noted that positive information represents what is granted to be possible, while negative information representswhat is considered to be impossible. This domain has recently motivated new research in several directions. In particular, fuz-zy and possibilistic formalisms for bipolar information have been proposed [18], because when we deal with spatial informa-tion in image processing or in spatial reasoning applications, this bipolarity also occurs. For instance, when we assess theposition of an object in a space, we may have positive information expressed as a set of possible places and negative informa-tion expressed as a set of impossible places. As another example, let us consider the spatial relations. Human beings consider‘‘left’’ and ‘‘right’’ as opposite directions. But this does not mean that one of them is the negation of the other. The semantics of‘‘opposite’’ captures a notion of symmetry rather than a strict complementation. In particular, there may be positions whichare considered neither to the right nor to the left of some reference object, thus leaving some room for indetermination. Thiscorresponds to the idea that the union of positive and negative information does not cover the whole space.

In 1975, Rosenfeld [45] introduced the concept of fuzzy graphs. The fuzzy relations between fuzzy sets were also consid-ered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts.Later on, Bhattacharya [8] gave some remarks on fuzzy graphs, and some operations on fuzzy graphs were introduced by

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M. Akram / Information Sciences 181 (2011) 5548–5564 5549

Mordeson and Peng [32]. The complement of a fuzzy graph was defined by Mordeson and Nair [34] and further studied bySunitha and Vijayakumar [46]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [10] and studiedsome of their properties. The concept of strong arcs in fuzzy graphs was discussed in [12]. Recently, Akram [1] has introducedthe notion of cofuzzy graphs and investigated several of their properties. Shannon and Atanassov [47] introduced the conceptof intuitionistic fuzzy relations and intuitionistic fuzzy graphs, and investigated some of their properties in [48]. Parvathi et al.defined operations on intuitionistic fuzzy graphs in [40]. In this paper, we introduce the notion of bipolar fuzzy graphs, de-scribe various methods of their construction, discuss the concept of isomorphism of these graphs, and investigate some oftheir important properties. We then introduce the notion of strong bipolar fuzzy graphs and study some of their properties.We also discuss some propositions of self complementary and self weak complementary strong bipolar fuzzy graphs.

We have used standard definitions and terminologies in this paper. For other notations, terminologies and applicationsnot mentioned in the paper, the readers are referred to [2–7,9–17,19,21–25,27–31,33,35–39,41–44,48–50,53–58].

2. Preliminaries

In this section, we first review some definitions of undirected graphs that are necessary for this paper.

Definition 2.1 [20]. Recall that a graph is an ordered pair G⁄ = (V,E), where V is the set of vertices of G⁄ and E is the set ofedges of G⁄. Two vertices x and y in an undirected graph G⁄ are said to be adjacent in G⁄ if {x,y} is an edge of G⁄. A simple graphis an undirected graph that has no loops and no more than one edge between any two different vertices.

Definition 2.2 [20]. A subgraph of a graph G⁄ = (V,E) is a graph H = (W,F), where W # V and F # E.

Definition 2.3 [20]. The complementary graph G� of a simple graph has the same vertices as G⁄. Two vertices are adjacent inG� if and only if they are not adjacent in G⁄.

Definition 2.4 [20]. Consider the Cartesian product G� ¼ G�1 � G�2 ¼ ðV ; EÞ of graphs G�1 and G�2. Then V = V1 � V2 and

E ¼ ðx; x2Þðx; y2Þjx1 2 V1; x2y2 2 E2f g [ ðx1; zÞðy1; zÞjz 2 V2; x1y1 2 E1f g:

Definition 2.5 [20]. Let G�1 ¼ ðV1; E1Þ and G�2 ¼ ðV2; E2Þ be two simple graphs. Then, the composition of graph G�1 with G�2 isdenoted by G�1½G

�2� ¼ ðV1 � V2; E

0Þ, where E0 = E [ {(x1,x2) (y1,y2)jx1y1 2 E1,x2 – y2} and E is defined in G�1 � G�2. Note thatG�1½G

�2�– G�2½G

�1�.

Definition 2.6 [20]. The union of two simple graphs G�1 ¼ ðV1; E1Þ and G�2 ¼ ðV2; E2Þ is the simple graph with the vertex setV1 [ V2 and edge set E1 [ E2. The union of G�1 and G�2 is denoted by G� ¼ G�1 [ G�2 ¼ ðV1 [ V2; E1 [ E2Þ.

Definition 2.7 [20]. The join of two simple graphs G�1 ¼ ðV1; E1Þ and G�2 ¼ ðV2; E2Þ is the simple graph with the vertex setV1 [ V2 and edge set E1 [ E2 [ E0, where E0 is the set of all edges joining the nodes of V1 and V2 and assume that V1 \ V2 – ;.The join of G�1 and G�2 is denoted by G� ¼ G�1 þ G�2 ¼ ðV1 [ V2; E1 [ E2 [ E0Þ.

Definition 2.8 [20]. An isomorphism of the graphs G�1 and G�2 is a bijection between the vertex sets of G�1 and G�2 such that anytwo vertices v1 and v2 of G�1 are adjacent in G�1 if and only if f(v1) and f(v2) are adjacent in G�2. If an isomorphism exists betweentwo graphs, then the graphs are called isomorphic and we write G�1 ’ G�2. An automorphism of a graph is a graph isomorphismwith itself, i.e., a mapping from the vertices of the given graph G⁄ back to vertices of G⁄ such that the resulting graph G⁄ isisomorphic with G⁄.

Definition 2.9 ([51,52]). A fuzzy subset l on a set X is a map l : X ? [0,1]. A map m : X � X ? [0,1] is called a fuzzy relation onX if m(x,y) 6min (l(x),l(y)) for all x, y 2 X. A fuzzy relation m is symmetric if m(x,y) = m(y,x) for all x, y 2 X.

Definition 2.10 ([26,56]). Let X be a nonempty set. A bipolar fuzzy set B in X is an object having the form

B ¼ ðx;lPðxÞ;lNðxÞÞjx 2 X� �

;

where lP : X ? [0,1] and lN : X ? [�1,0] are mappings.

We use the positive membership degree lP(x) to denote the satisfaction degree of an element x to the property corre-sponding to a bipolar fuzzy set B, and the negative membership degree lN(x) to denote the satisfaction degree of an elementx to some implicit counter-property corresponding to a bipolar fuzzy set B. If lP(x) – 0 and lN(x) = 0, it is the situation that x

5550 M. Akram / Information Sciences 181 (2011) 5548–5564

is regarded as having only positive satisfaction for B. If lP(x) = 0 and lN(x) – 0, it is the situation that x does not satisfy theproperty of B but somewhat satisfies the counter property of B. It is possible for an element x to be such that lP(x) – 0 andlN(x) – 0 when the membership function of the property overlaps that of its counter property over some portion of X.

For the sake of simplicity, we shall use the symbol B = (lP,lN) for the bipolar fuzzy set B = {(x,lP(x),lN(x))jx 2 X}.

Definition 2.11 [26]. For every two bipolar fuzzy sets A ¼ lPA;lN

A

� �and B ¼ lP

B;lNB

� �in X, we define

� AT

Bð ÞðxÞ ¼ min lPAðxÞ;lP

BðxÞ� �

;max lNA ðxÞ;lN

B ðxÞ� ��

,

� AS

Bð ÞðxÞ ¼ max lPAðxÞ;lP

BðxÞ� �

;min lNA ðxÞ;lN

B ðxÞ� ��

.

Definition 2.12 [56]. Let X be a nonempty set. Then we call a mapping A ¼ ðlPA;lN

A Þ : X � X ! ½�1;1� � ½�1;1� a bipolar fuzzyrelation on X such that lP

Aðx; yÞ 2 ½0;1� and lNA ðx; yÞ 2 ½�1;0�.

Definition 2.13 [56]. Let A ¼ lPA;lN

A

� �and B ¼ lP

B;lNB

� �be bipolar fuzzy sets on a set X. If A ¼ lP

A;lNA

� �is a bipolar fuzzy rela-

tion on a set X, then A ¼ lPA;lN

A

� �is called a bipolar fuzzy relation on B ¼ lP

B;lNB

� �if lP

Aðx; yÞ 6min lPBðxÞ;lP

BðyÞ� �

andlN

A ðx; yÞP max lNB ðxÞ;lN

B ðyÞ� �

for all x, y 2 X. A bipolar fuzzy relation A on X is called symmetric if lPAðx; yÞ ¼ lP

Aðy; xÞ andlN

A ðx; yÞ ¼ lNA ðy; xÞ for all x, y 2 X.

Throughout this paper, G⁄ will be a crisp graph, and G a bipolar fuzzy graph.

3. Bipolar fuzzy graphs

Definition 3.1. A bipolar fuzzy graph with an underlying set V is defined to be a pair G = (A,B) where A ¼ ðlPA;lN

A Þ is a bipolarfuzzy set in V and B ¼ ðlP

B;lNB Þ is a bipolar fuzzy set in E # V � V such that

lPBðfx; ygÞ 6min lP

AðxÞ;lPAðyÞ

� �and lN

B ðfx; ygÞP max lNA ðxÞ;lN

A ðyÞ� �

for all {x,y} 2 E. We call A the bipolar fuzzy vertex set of V, B the bipolar fuzzy edge set of E, respectively. Note that B is a sym-metric bipolar fuzzy relation on A. We use the notation xy for an element of E. Thus, G = (A,B) is a bipolar graph of G⁄ = (V,E) if

lPBðxyÞ 6 min lP

AðxÞ;lPAðyÞ

� �and lN

B ðxyÞP max lNA ðxÞ;lN

A ðyÞ� �

for all xy 2 E:

Example 3.2. Consider a graph G⁄ = (V,E) such that V = {a,b,c}, E = {ab,bc,ca}. Let A ¼ lPA;lN

A

� �be a bipolar fuzzy subset of V

and let B ¼ lPB;lN

B

� �be a bipolar fuzzy subset of E # V � V defined by

By routine computations, it is easy to see that G = (A,B) is a bipolar fuzzy graph of G⁄.� � � �
We now discuss the operations on bipolar fuzzy graphs.Definition 3.3 Let A1 ¼ lP
A1;lN

A1and A2 ¼ lP

A2;lN

A2be bipolar

fuzzy subsets of V1 and V2 and let B1 ¼ lPB1;lN

B1

� �and B2 ¼ lP

B2;lN

B2

� �be bipolar fuzzy subsets of E1 and E2, respectively. Then,

we denote the Cartesian product of two bipolar fuzzy graphs G1 and G2 of the graphs G�1 and G�2 by G1 � G2 = (A1 � A1,B1 � B2),and define as follows:


ðiÞ lPA1� lP

A2

� �ðx1; x2Þ ¼min lP

A1ðx1Þ;lP

A2ðx2Þ

� �lN

A1� lN

A2

� �ðx1; x2Þ ¼max lN

A1ðx1Þ;lN

A2ðx2Þ

� �for all ðx1; x2Þ 2 V ;

ðiiÞ lPB1� lP

B2

� �ððx; x2Þðx; y2ÞÞ ¼min lP

A1ðxÞ;lP

B2ðx2y2Þ

� �;

lNB1� lN

B2

� �ððx; x2Þðx; y2ÞÞ ¼max lN

A1ðxÞ;lN

B2ðx2y2Þ

� �for all x 2 V1; for all x2y2 2 E2;

ðiiiÞ lPB1� lP

B2

� �ððx1; zÞðy1; zÞÞ ¼min lP

B1ðx1y1Þ;lP

A2ðzÞ

� �lN

B1� lN

B2

� �ððx1; zÞðy1; zÞÞ ¼max lN

B1ðx1y1Þ;lN

A2ðzÞ

� �for all z 2 V2; for all x1y1 2 E1:

Proposition 3.4. If G1 and G2 are the bipolar fuzzy graphs, then G1 � G2 is a bipolar fuzzy graph.

Proof. Let x 2 V1, x2y2 2 E2. Then we have

lPB1� lP

B2

� �ðx; x2Þðx; y2Þð Þ ¼ min lP

A1ðxÞ;lP

B2ðx2y2Þ

� �6 min lP

A1ðxÞ;min lP

A2ðx2Þ;lP

A2ðy2Þ

� �� ¼ min min lP

A1ðxÞ;lP

A2ðx2Þ

� �;min lP

A1ðxÞ;lP

A2ðy2Þ

� �� ¼ min lP

A1� lP

A2

� �ðx; x2Þ; lP

A1� lP

A2

� �ðx; y2Þ

� �;

lNB1� lN

B2

� �ðx; x2Þðx; y2Þð Þ ¼ max lN

A1ðxÞ;lN

B2ðx2y2Þ

� �P max lN

A1ðxÞ;max lN

A2ðx2Þ;lN

A2ðy2Þ

� �� ¼ max max lN

A1ðxÞ;lN

A2ðx2Þ

� �;max lN

A1ðxÞ;lN

A2ðy2Þ

� �� ¼ max lN

A1� lN

A2

� �ðx; x2Þ; lN

A1� lN

A2

� �ðx; y2Þ

� �:

Let z 2 V2, x1y1 2 E1. Then, we have

lPB1� lP

B2


B1ðx1y1Þ;lP

A2ðzÞ

� �6min min lP

A1ðx1Þ;lP

A1ðy1Þ

� �;lP

A2ðzÞ

� �¼min min lP

A1ðxÞ;lP

A2ðzÞ

� �;min lP

A1ðy1Þ;lP

A2ðzÞ

� �� ¼min lP

A1� lP

A2

� �ðx1; zÞ; lP

A1� lP

A2

� �ðy1; zÞ

� �;

lNB1� lN

B2


B1ðx1y1Þ;lN

A2ðzÞ

� �P max max lN

A1ðx1Þ;lN

A1ðy1Þ

� �;lN

A2ðzÞ

� �¼max max lN

A1ðxÞ;lN

A2ðzÞ

� �;max lN

A1ðy1Þ;lN

A2ðzÞ

� �� ¼max lN

A1� lN

A2

� �ðx1; zÞ; lN

A1� lN

A2

� �ðy1; zÞ

� �:

This completes the proof. h

Definition 3.5. Let A1 ¼ lPA1;lN

A1

� �and A2 ¼ lP

A2;lN

A2

� �be bipolar fuzzy subsets of V1 and V2 and let B1 ¼ lP

B1;lN

B1

� �and

B2 ¼ lPB2;lN

B2

� �be bipolar fuzzy subsets of E1 and E2, respectively. Then, we denote the composition of two bipolar fuzzy

graphs G1 and G2 of the graphs G�1 and G�2 by G1[G2] = (A1 � A2,B1 � B2) and define as follows:

ðiÞ lPA1� lP

A2


A1ðx1Þ;lP

A2ðx2Þ

� �lN

A1� lN

A2

� �ðx1; x2Þ ¼ max lN

A1ðx1Þ;lN

A2ðx2Þ

� �for all ðx1; x2Þ 2 V ;

ðiiÞ lPB1� lP

B2


A1ðxÞ;lP

B2ðx2y2Þ

� �;

lNB1� lN

B2

� �ððx; x2Þðx; y2ÞÞ ¼max lN

A1ðxÞ;lN

B2ðx2y2Þ

� �for all x 2 V1; for all x2y2 2 E2;


ðiiiÞ lPB1� lP

B2


B1ðx1y1Þ;lP

A2ðzÞ

� �lN

B1� lN

B2


B1ðx1y1Þ;lN

A2ðzÞ

� �for all z 2 V2; for all x1y1 2 E1;

ðivÞ lPB1� lP

B2

� �ððx1; x2Þðy1; y2ÞÞ ¼ min lP

A2ðx2Þ;lP

A2ðy2Þ;lP

B1ðx1y1Þ

� �;

lNB1� lN

B2

� �ððx1; x2Þðy1; y2ÞÞ ¼max lN

A2ðx2Þ;lN

A2ðy2Þ;lN

B1ðx1y1Þ

� �; for all ðx1; x2Þðy1; y2Þ 2 E0 � E:

Proposition 3.6. If G1 and G2 are the bipolar fuzzy graphs, then G1[G2] is a bipolar fuzzy graph.

Proof. Let x 2 V1 and x2y2 2 E2. Then, we have

lPB1� lP

B2

� �ððx; x2Þðx; y2ÞÞ ¼ min lP

A1ðxÞ;lP

B2ðx2y2Þ

� �6 min lP

A1ðxÞ;min lP

A2ðx2Þ;lP

A2ðy2Þ

� �� ¼ min min lP

A1ðxÞ;lP

A2ðx2Þ

� �;min lP

A2ðxÞ;lP

A2ðy2Þ

� �� ¼ min lP

A1� lP

A2


A1� lP

A2

� �ðx; y2Þ

� �;

lNB1� lN

B2

� �ðx; x2Þðx; y2Þð Þ ¼ max lN

A1ðxÞ;lN

B2ðx2y2Þ

� �P max lN

A1ðxÞ;max lN

A2ðx2Þ;lN

A2ðy2Þ

� �� ¼ max min lN

A1ðxÞ;lN

A2ðx2Þ

� �;max lN

A1ðxÞ;lN

A2ðy2Þ

� �� ¼ max lN

A1� lN

A2


A1� lN

A2

� �ðx; y2Þ

� �:

Let z 2 V2 and x1y1 2 E1. Then we deduce the following:

lPB1� lP

B2


B1ðx1y1Þ;lP

A2ðzÞ

� �6min min lP

A1ðx1Þ;lP

A1ðy1Þ

� �;lP

A2ðzÞ

� �¼min min lP

AðxÞ;lPAðzÞ

� �;min lP

Aðy1Þ;lPA2ðzÞ

� �� ¼min lP

A1� lP

A2

� �ðx1; zÞ; lP

A1� lP

A2

� �ðy1; zÞ

� �;

lNB1� lN

B2


B1ðx1y1Þ;lN

A2ðzÞ

� �P max max lN

A1ðx1Þ;lN

A1ðy1Þ

� �;lN

A2ðzÞ

� �¼max max lAðxÞ;lN

A ðzÞ� �

;max lNA ðy1Þ;lN

A2ðzÞ

� �� ¼max lN

A1� lN

A2

� �ðx1; zÞ; lN

A1� lN

A2

� �ðy1; zÞ

� �:

Let (x1,x2)(y1,y2) 2 E0 � E, so x1y1 2 E1, x2 – y2. Then we have

lPB1� lP

B2

� �ððx1; x2Þðy1; y2ÞÞ ¼min lP

A2ðx2Þ;lP

A2ðy2Þ;lP

B1ðx1y1Þ

� �6 min lP

A2ðx2Þ;lP

A2ðy2Þ;min lP

A1ðx1Þ;lP

A1ðy1Þ

� �� ¼min min lP

A1ðx1Þ;lP

A2ðx2Þ

� �;min lP

A1ðy1Þ;lP

A2ðy2Þ

� �� ¼min lP

A1� lP

A2

� �ðx1; x2Þ; lP

A1� lP

A2

� �ðy1; y2Þ

� �;

lNB1� lN

B2

� �ððx1; x2Þðy1; y2ÞÞ ¼max lN

A2ðx2Þ;lN

A2ðy2Þ;lN

B1ðx1y1Þ

� �P max lN

A2ðx2Þ;lN

A2ðy2Þ;max lN

A1ðx1Þ;lN

A1ðy1Þ

� �� ¼max max lN

A1ðx1Þ;lN

A2ðx2Þ

� �;max lN

A1ðy1Þ;lN

A2ðy2Þ

� �� ¼max lN

A1� lN

A2

� �ðx1; x2Þ; lN

A1� lN

A2

� �ðy1; y2Þ

� �:



A1

� �and A2 ¼ lP

A2;lN

A2


B1;lN

B1

� �and

B2 ¼ lPB2;lN

B2

� �be bipolar fuzzy subsets of E1 and E2, respectively. Then, we denote the union of two bipolar fuzzy graphs

G1 and G2 of the graphs G�1 and G�2 by G1 [ G2 = (A1 [ A2,B1 [ B2) and define as follows:


ðAÞ lPA1[ lP

A2

� �ðxÞ ¼ lP

A1ðxÞ if x 2 V1 \ V2;

lPA1[ lP

A2

� �ðxÞ ¼ lP


lPA1[ lP

A2

� �ðxÞ ¼max lP

A1ðxÞ;lP

A2ðxÞ

� �if x 2 V1 \ V2:

ðBÞ lNA1\ lN

A2

� �ðxÞ ¼ lN


lNA1\ lN

A2

� �ðxÞ ¼ lN


lNA1\ lN

A2

� �ðxÞ ¼min lN

A1ðxÞ;lN

A2ðxÞ

� �if x 2 V1 \ V2:

ðCÞ lPB1[ lP

B2

� �ðxyÞ ¼ lP

B1ðxyÞ if xy 2 E1 \ E2;

lPB1[ lP

B2

� �ðxyÞ ¼ lP


lPB1[ lP

B2

� �ðxyÞ ¼max lP

B1ðxyÞ;lP

B2ðxyÞ

� �if xy 2 E1 \ E2:

ðDÞ lNB1\ lN

B2

� �ðxyÞ ¼ lN


lNB1\ lN

B2

� �ðxyÞ ¼ lN


lNB1\ lN

B2

� �ðxyÞ ¼min lN

B1ðxyÞ;lN

B2ðxyÞ

� �if xy 2 E1 \ E2:

Example 3.8. Consider the bipolar fuzzy graphs.


Proposition 3.9. If G1 and G2 are bipolar fuzzy graphs, then G1 [ G2 is a bipolar fuzzy graph.

Proof. Let xy 2 E1 \ E2. Then

lPB1[ lP

B2


B1ðxyÞ;lP

B2ðxyÞ

� �6max min lP

A1ðxÞ;lP

A1ðyÞ

� �;min lP

A2ðxÞ;lP

A2ðyÞ

� �� ¼min max lP

A1ðxÞ;lP

A2ðxÞ

� �;max lP

A1ðyÞ;lP

A2ðyÞ

� �� ¼min lP

A1[ lP

A2

� �ðxÞ; lP

A�1 [ lPA2

� �ðyÞ

� �;

lNB1[ lN

B2


B1ðxyÞ;lN

B2ðxyÞ

� �P min max lN

A1ðxÞ;lN

A1ðyÞ

� �;max lN

A2ðxÞ;lN

A2ðyÞ

� �� ¼max min lN

A1ðxÞ;lN

A2ðxÞ

� �;min lN

A1ðyÞ;lN

A2ðyÞ

� �� ¼max lN

A1[ lN

A2

� �ðxÞ; lN

A�1 [ lNA2

� �ðyÞ

� �:

Similarly, we can show that if xy 2 E1 \ E2, then

lPB1[ lP

B2

� �ðxyÞ 6min lP

A1[ lP

A2

� �ðxÞ; lP

A1[ lP

A2

� �ðyÞ

� �;

lNB1[ lN

B2

� �ðxyÞP max lN

A1[ lN

A2

� �ðxÞ; lN

A1[ lN

A2

� �ðyÞ

� �:

If xy 2 E2 \ E1, then

lPB1[ lP

B2

� �ðxyÞ 6min lP

A1[ lP

A2

� �ðxÞ; lP

A1[ lP

A2

� �ðyÞ

� �;

lNB1[ lN

B2


A1[ lN

A2

� �ðxÞ; lN

A1[ lN

A2

� �ðyÞ

� �:


Proposition 3.10. Let {Gi : i 2K} be a family of bipolar fuzzy graphs with the underlying set V. ThenT

Gi is a bipolar fuzzy graph.

Proof. For any x, y 2 V, we have

\lPBðxyÞ ¼ inf

i2KlP

BðxyÞ 6 infi2K

min lPAiðxÞ;lP

AiðyÞ

n o¼min inf

i2KlP

AiðxÞ; inf

i2KlP

AiðyÞ

� ¼min \lP

AiðxÞ;\lP

AiðyÞ

n o;

\lNB ðxyÞ ¼ sup

i2KlN

B ðxyÞP supi2K

max lNAiðxÞ;lN

AiðyÞ

n o¼max sup

i2KlN

AiðxÞ; sup

i2KlN

AiðyÞ

� ¼max \lN

AiðxÞ;\lN

AiðyÞ

n o:

HenceT

Gi is a bipolar fuzzy graph. h


A1

� �and A2 ¼ lP

A2;lN

A2


B1;lN

B1

� �and

B2 ¼ lPB2;lN

B2

� �be bipolar fuzzy subsets of E1 and E2, respectively. Then, we denote the join of two bipolar fuzzy graphs

G1 and G2 of the graphs G�1 and G�2 by G1 + G2 = (A1 + A2,B1 + B2) and define as follows:

ðiÞ lPA1þ lP

A2

� �ðxÞ ¼ lP

A1[ lP

A2

� �ðxÞ;

lNA1þ lN

A2

� �ðxÞ ¼ lN

A1\ lN

A2

� �ðxÞ if x 2 V1 [ V2;

ðiiÞ lPB1þ lP

B2

� �ðxyÞ ¼ lP

B1[ lP

B2

� �ðxyÞ ¼ lP

B1ðxyÞ;

lNB1þ lN

B2

� �ðxyÞ ¼ lN

B1\ lN

B2

� �ðxyÞ ¼ lN


ðiiiÞ lPB1þ lP

B2


A1ðxÞ;lP

A2ðyÞ

� �;

lNB1þ lN

B2


A1ðxÞ;lN

A2ðyÞ

� �;

if xy 2 E0, where E0 is the set of all edges joining the nodes of V1 and V2.


Proposition 3.12. If G1 and G2 are the bipolar fuzzy graphs, then G1 + G2 is a bipolar fuzzy graph.

Proof. Let xy 2 E0. Then

lPB1þ lP

B2


A1ðxÞ;lP

A2ðyÞ

� �6 max lP

A1[ lP

A2

� �ðxÞ; lP

A1[ lP

A2

� �ðyÞ

� �¼max lP

A1þ lP

A2

� �ðxÞ; lP

A1þ lP

A2

� �ðyÞ

� �;

lNB1þ lN

B2


A1ðxÞ;lN

A2ðyÞ

� �P min lN

A1\ lN

A2

� �ðxÞ; lN

A1\ lN

A2

� �ðyÞ

� �¼min lN

A1þ lN

A2

� �ðxÞ; lN

A1þ lN

A2

� �ðyÞ

� �:

Let xy 2 E1 [ E2. Then the result follows from Proposition 3.9. This completes the proof. h

We formulate the following characterizations.

Proposition 3.13. Let G�1 ¼ ðV1; E1Þ and G�2 ¼ ðV2; E2Þ be crisp graphs and let V1 \ V2 = ;. Let A1, A2, B1 and B2 be bipolar fuzzysubsets of V1, V2, E1 and E2, respectively. Then G1 [ G2 = (A1 [ A2,B1 [ B2) is a bipolar fuzzy graph of G⁄ if and only if G1 = (A1,B1) andG2 = (A2,B2) are bipolar fuzzy graphs of G�1 and G�2, respectively.

Proof. Suppose that G1 [ G2 is a bipolar fuzzy graph. Let xy 2 E1. Then xy R E2 and x, y 2 V1 � V2. Thus

lPB1ðxyÞ ¼ lP

B1[ lP

B2

� �ðxyÞ 6 min lP

A1[ lP

A2

� �ðxÞ; lP

A1[ lP

A2

� �ðyÞ

� �¼ min lP

A1ðxÞ;lP

A1ðyÞ

� �;

lNB1ðxyÞ ¼ lN

B1\ lN

B2


A1\ lN

A2

� �ðxÞ; lN

A1\ lN

A2

� �ðyÞ

� �¼max lN

A1ðxÞ;lN

A1ðyÞ

� �:

This shows that G1 = (A1,B1) is a bipolar fuzzy graph. Similarly, we can show that G2 = (A2,B2) is a bipolar fuzzy graph.The converse the above proposition is given by Proposition 3.9. h

As a consequence of Propositions 3.12 and 3.13, we obtain.

Proposition 3.14. Let G�1 ¼ ðV1; E1Þ and G�2 ¼ ðV2; E2Þ be crisp graphs and let V1 \ V2 = ;. Let A1, A2, B1 and B2 be bipolar fuzzysubsets of V1, V2, E1 and E2, respectively. Then G1 + G2 = (A1 + A2,B1 + B2) is a bipolar fuzzy graph of G⁄ if and only if G1 = (A1,B1) andG2 = (A2, B2) are bipolar fuzzy graphs of G�1 and G�2, respectively.

4. Automorphic bipolar fuzzy graphs

Definition 4.1. Let G1 and G2 be the bipolar fuzzy graphs. A homomorphism f : G1 ? G2 is a mapping f : V1 ? V2 which satisfiesthe following conditions:

(a) lPA1ðx1Þ 6 lP

A2ðf ðx1ÞÞ; lN

A1ðx1ÞP lN

A2ðf ðx1ÞÞ,

(b) lPB1ðx1y1Þ 6 lP

B2ðf ðx1Þf ðy1ÞÞ; lN

B1ðx1y1ÞP lN

B2ðf ðx1Þf ðy1ÞÞ

for all x1 2 V1, x1y1 2 E1.

Definition 4.2. Let G1 and G2 be bipolar fuzzy graphs. An isomorphism f : G1 ? G2 is a bijective mapping f : V1 ? V2 whichsatisfies the following conditions:

(c) lPA1ðx1Þ ¼ lP

A2ðf ðx1ÞÞ; lN

A1ðx1Þ ¼ lN

A2ðf ðx1ÞÞ,

(d) lPB1ðx1y1Þ ¼ lP


B1ðx1y1Þ ¼ lN


for all x1 2 V1, x1y1 2 E1.

Definition 4.3. Let G1 and G2 be bipolar fuzzy graphs. Then, a weak isomorphism f : G1 ? G2 is a bijective mapping f : V1 ? V2

which satisfies the following conditions:

(e) f is homomorphism,(f) lP

A1ðx1Þ ¼ lP

A2ðf ðx1ÞÞ; lN

A1ðx1Þ ¼ lN

A2ðf ðx1ÞÞ,

for all x1 2 V1. Thus, a weak isomorphism preserves the weights of the nodes but not necessarily the weights of the arcs.


Example 4.4. Consider bipolar fuzzy graphs G1 and G2 of G�1 and G�2, respectively.

A map f : V1 ? V2 defined by f(a1) = b2 and f(b1) = a2. Then we see that:

� lPA1ða1Þ ¼ lP

A2ðb2Þ; lN

A1ða1Þ ¼ lN

A2ðb2Þ; lP

A1ðb1Þ ¼ lP

A2ða2Þ; lN

A1ðb1Þ ¼ lN

A2ða2Þ,

� lNB1ða1b1Þ ¼ lN

B2ða2b2Þ, but lP

B1ða1b1Þ– lP

B2ðf ða1Þf ðb1ÞÞ ¼ lP

B2ða2b2Þ. Hence the map is a weak isomorphism but not

isomorphism.

Definition 4.5. Let G1 and G2 be the bipolar fuzzy graphs. A co-weak isomorphism f : G1 ? G2 is a bijective mapping f : V1 ? V2

which satisfies

(g) f is homomorphism,(h) lP

B1ðx1y1Þ ¼ lP


B1ðx1y1Þ ¼ lN


for all x1y1 2 V1. Thus a co-weak isomorphism preserves the weights of the arcs but not necessarily the weights of thenodes.

Example 4.6. Consider bipolar fuzzy graphs G1 and G2 of G�1 and G�2, respectively.

A map f : V1 ? V2 defined by f(x1) = x2 and f(y1) = y2. By routine computations, we can show that the map is a co-weakisomorphism but not isomorphism since lP

A1ðx1Þ – lP

A2ðy2Þ; lN

A1ðx1Þ – lN

A2ðy2Þ.

Remark.

1. If G1 = G2 = G, then the homomorphism f over itself is called an endomorphism. An isomorphism f over G is called anautomorphism.

2. Let A ¼ ðlPA;lN

A Þ be a bipolar fuzzy graph with an underlying set V. Let Aut (G) be the set of all bipolar automorphisms of G.Let e : G ? G be a map defined by e(x) = x for all x 2 V. Clearly, e 2 Aut(G).

3. If G1 = G2, then the weak and co-weak isomorphisms actually become isomorphic.4. If f : V1 ? V2 is a bijective map, then f�1 : V2 ? V1 is also bijective map.


Definition 4.7. A bipolar fuzzy set A ¼ lPA;lN

A

� �in a semigroup S is called a bipolar subsemigroup of S if it satisfies:

lPAðxyÞP min lP

AðxÞ;lPAðyÞ

� �;lN

A ðxyÞ 6 max lNA ðxÞ;lN

A ðyÞ� �

for all x; y 2 S:

A bipolar fuzzy set A ¼ lPA;lN

A

� �in a group G is called a bipolar fuzzy subgroup of a group G if it is a bipolar fuzzy subsemi-

group of G and satisfies:

lPAðx�1Þ ¼ lP

AðxÞ; lNA ðx�1Þ ¼ lN

A ðxÞ for all x 2 G:
We now show how to associate a bipolar fuzzy group with a bipolar fuzzy graph in a natural way.
Proposition 4.8. Let G = (A,B) be a bipolar fuzzy graph and let Aut(G) be the set of all automorphisms of G. Then (Aut(G),�) forms agroup.

Proof. Let /, w 2 Aut(G) and let x, y 2 V. Then

lPB ð/ � wÞðxÞð/ � wÞðyÞð Þ ¼ lP

B ð/ðwðxÞð Þ /ðwÞð ÞðyÞÞP lPB wðxÞwðyÞð ÞP lP

BðxyÞ;

lNB ð/ � wÞðxÞð/ � wÞðyÞð Þ ¼ lN

B ð/ðwðxÞÞð/ðwÞÞðyÞÞð Þ 6 lNB ðwðxÞwðyÞÞ 6 lN

B ðxyÞ;

lPBðð/ � wÞðxÞÞ ¼ lP

B ð/ðwðxÞÞÞð ÞP lPBðwðxÞÞP lP

BðxÞ;

lPNðð/ � wÞðxÞÞ ¼ lN

B ð/ðwðxÞÞÞð Þ 6 lNB ðwðxÞÞ 6 lN

B ðxÞ:

Thus / � w 2 Aut(G). Clearly, Aut(G) satisfies associativity under the operation �; / � e ¼ / ¼ e � /; lPAð/

�1Þ ¼lP

Að/Þ; lNA ð/

�1Þ ¼ lNA ð/Þ for all / 2 Aut(G). Hence (Aut(G),�) forms a group. h

We sate propositions without their proofs.

Proposition 4.9. Let G = (A,B) be a bipolar fuzzy graph and let Aut(G) be the set of all automorphisms of G. Let g ¼ ðlPg ;lN

g Þ be abipolar fuzzy set in Aut(G) defined by

lPgð/Þ ¼ sup lP

Bð/ðxÞ;/ðyÞÞ : ðx; yÞ 2 V � V� �

;

lNg ð/Þ ¼ inf lN

B ð/ðxÞ;/ðyÞÞ : ðx; yÞ 2 V � V� �

for all / 2 Aut(G). Then g ¼ lPg ;lN

g

� �is a bipolar fuzzy group on Aut(G).

Proposition 4.10. Every bipolar fuzzy group has an embedding into the bipolar fuzzy group of the group of automorphisms ofsome bipolar fuzzy graph.

We now prove that the isomorphism (resp. weak isomorphism) between bipolar fuzzy graphs is an equivalence relation(resp. partial order relation).

Proposition 4.11. Let G1, G2 and G3. Then the isomorphism between these bipolar fuzzy graphs is an equivalence relation.

Proof. The reflexivity is obvious. To prove the symmetry, we let f : V1 ? V2 be an isomorphism of G1 onto G2. Then f is abijective map defined by f(x1) = x2, for all x1 2 V1 � � � (1) satisfying

lPA1ðx1Þ ¼ lP

A2ðf ðx1ÞÞ;lN

A1ðx1Þ ¼ lN

A2ðf ðx1ÞÞ for all x1 2 V1;

lPB1ðx1y1Þ ¼ lP

B2ðf ðx1Þf ðy1ÞÞ;lN

B1ðx1y1Þ ¼ lN

B2ðf ðx1Þf ðy1ÞÞ for all x1y1 2 E1:

Since f is bijective, from (1) it follows that f�1(x2) = x1 for all x2 2 V2. Thus

lPA1ðf�1ðx2ÞÞ ¼ lP

A2ðx2Þ;lN

A1ðf�1ðx2ÞÞ ¼ lN

A2ðx2Þ for all x2 2 V2;

lPB1ðf�1ðx2y2ÞÞ ¼ lP

B2ðx2y2Þ;lN

B1ðf�1ðx2y2ÞÞ ¼ lN

B2ðx2y2Þ for all x2y2 2 E2:

Hence a bijective map f�1 : V2 ? V1 is an isomorphism from G2 onto G1.To prove the transitivity, we let f : V1 ? V2 and g : V2 ? V3 be the isomorphisms of G1 onto G2 and G2 onto G3, respectively.

Then g � f : V1 ? V3 is a bijective map from V1 to V3, where (g� f)(x1) = g(f(x1)) for all x1 2 V1. Since a map f : V1 ? V2 defined byf(x1) = x2 for x1 2 V1 is an isomorphism, so we have


lPA1ðx1Þ ¼ lP

A2ðf ðx1ÞÞ ¼ lP

A2ðx2Þ;lN

A1ðx1Þ ¼ lN

A2ðf ðx1ÞÞ ¼ lN

A2ðx2Þ for all x1 2 V1 � � � ðAÞ;

lPB1ðx1y1Þ ¼ lP

B2ðf ðx1Þf ðy1ÞÞ ¼ lP

B2ðx2y2Þ;

lNB1ðx1y1Þ ¼ lN

B2ðf ðx1Þf ðy1ÞÞ ¼ lN

B2ðx2y2Þ for all x1y1 2 E1 � � � ðBÞ:

Since a map g : V2 ? V3 defined by g(x2) = x3 for x2 2 V2 is an isomorphism, so

lPA2ðx2Þ ¼ lP

A3ðgðx2ÞÞ ¼ lP

A3ðx3Þ;lN

A2ðx2Þ ¼ lN

A3ðgðx2ÞÞ ¼ lN

A3ðx3Þ for all x2 2 V2 � � � ðCÞ;

lPB2ðx2y2Þ ¼ lP

B3ðgðx2Þgðy2ÞÞ ¼ lP

B3ðx3y3Þ;

lNB2ðx2y2Þ ¼ lN

B3ðgðx2Þgðy2ÞÞ ¼ lN

B3ðx3y3Þ for all x2y2 2 E2 � � � ðDÞ:

From (A), (C) and f(x1) = x2, x1 2 V1, we have

lPA1ðx1Þ ¼ lP


A2ðx2Þ ¼ lP

A3ðgðx2ÞÞ ¼ lP

A3ðgðf ðx1ÞÞÞ;

lNA1ðx1Þ ¼ lN


A2ðx2Þ ¼ lN

A3ðgðx2ÞÞ ¼ lN


for all x1 2 V1,From (B) and (D), we have

lPB1ðx1y1Þ ¼ lP

B2f ðx1Þf ðy1Þð Þ ¼ lP

B2ðx2y2Þ ¼ lP

B3gðx2Þgðy2Þð Þ ¼ lP

B3gðf ðx1ÞÞgðf ðy1ÞÞð Þ;

lNB1ðx1y1Þ ¼ lN

B2f ðx1Þf ðy1Þð Þ ¼ lN

B2ðx2y2Þ ¼ lN

B3gðx2Þgðy2Þð Þ ¼ lN

B3gðf ðx1ÞÞg f ðy1Þð Þð Þ

for all x1y1 2 E1.Therefore, g � f is an isomorphism between G1 and G3. This completes the proof. h

Proposition 4.12. Let G1, G2 and G3 be bipolar fuzzy graphs. Then the weak isomorphism between these bipolar fuzzy graphs is apartial order relation

Proof. The reflexivity is obvious. To prove the anti symmetry, we let f : V1 ? V2 be a weak isomorphism of G1 onto G2. Then fis a bijective map defined by f(x1) = x2 for all x1 2 V1

satisfying

lPA1ðx1Þ ¼ lP

A2ðf ðx1ÞÞ;lN

A1ðx1Þ ¼ lN


lPB1ðx1y1Þ 6 lP

B2ðf ðx1Þf ðy1ÞÞ;lN

B1ðx1y1ÞP lN

B2ðf ðx1Þf ðy1ÞÞ for all x1y1 2 E1 � � � ðEÞ:

Let g : V2 ? V1 be a weak isomorphism of G2 onto G1. Then g is a bijective map defined by g(x2) = x1 for all x2 2 V2

satisfying

lPA2ðx2Þ ¼ lP

A1ðgðx2ÞÞ;lN

A2ðx2Þ ¼ lN

A1ðgðx2ÞÞ for all x2 2 V2;

lPB2ðx2y2Þ 6 lP

B1ðgðx2Þgðy2ÞÞ;lN

B2ðx2y2ÞP lN

B1ðgðx2Þgðy2ÞÞ for all x2y2 2 E2 � � � ðFÞ:

The inequalities (E) and (F) hold on the finite sets V1 and V2 only when G1 and G2 have the same number of edges and thecorresponding edges have same weight. Hence G1 and G2 are identical.

To prove the transitivity, we let f : V1 ? V2 and g : V2 ? V3 be weak isomorphisms of G1 onto G2 and G2 onto G3,respectively. Then g � f : V1 ? V3 is a bijective map from V1 to V3, where (g � f)(x1) = g(f(x1)) for all x1 2 V1. Since a map f :V1 ? V2 defined by f(x1) = x2 for x1 2 V1 is a weak isomorphism, therefore,

lPA1ðx1Þ ¼ lP


A2ðx2Þ;lN

A1ðx1Þ ¼ lN


A2ðx2Þ for all x1 2 V1 � � � ðGÞ;

lPB1ðx1y1Þ 6 lP


B2ðx2y2Þ;

lNB1ðx1y1ÞP lN


B2ðx2y2Þ for all x1y1 2 E1 � � � ðHÞ:


Since a map g : V2 ? V3 defined by g(x2) = x3 for x2 2 V2 is a weak isomorphism, so

lPA2ðx2Þ ¼ lP

A3ðgðx2ÞÞ ¼ lP

A3ðx3Þ;lN

A2ðx2Þ ¼ lN

A3ðgðx2ÞÞ ¼ lN

A3ðx3Þ for all x2 2 V2 � � � ðIÞ;

lPB2ðx2y2Þ 6 lP


B3ðx3y3Þ;

lNB2ðx2y2ÞP lN


B3ðx3y3Þ for all x2y2 2 E2 � � � ðJÞ:

From (G), (I) and f(x1) = x2, x1 2 V1, we have

lPA1ðx1Þ ¼ lP


A2ðx2Þ ¼ lP

A3ðgðx2ÞÞ ¼ lP


lNA1ðx1Þ ¼ lN


A2ðx2Þ ¼ lN

A3ðgðx2ÞÞ ¼ lN


for all x1 2 V1,From (H) and (J), we have

lPB1ðx1y1Þ 6 lP


B2ðx2y2Þ ¼ lP


B3ðgðf ðx1ÞÞgðf ðy1ÞÞÞ;

lNB1ðx1y1ÞP lN


B2ðx2y2Þ ¼ lN


B3ðgðf ðx1ÞÞgðf ðy1ÞÞÞ

for all x1y1 2 E1.Therefore, g � f is a weak isomorphism between G1 and G3. This completes the proof. h

5. Strong bipolar fuzzy graphs

Definition 5.1. A bipolar fuzzy graph G = (A,B) is called strong if

lPBðxyÞ ¼minðlP

AðxÞ;lPAðyÞÞ and lN

B ðxyÞ ¼maxðlNA ðxÞ;lN

A ðyÞÞ for all xy 2 E:

Example 5.2. Consider a graph G⁄ such that V = {x,y,z}, E = {xy,yz,zx}. Let A be a bipolar fuzzy subset of V and let B be a bipo-lar fuzzy subset of E defined by

By routine computations, it is easy to see that G is a strong bipolar fuzzy graph of G⁄.

Proposition 5.3. If G1 and G2 are the strong bipolar fuzzy graphs, then G1 � G2, G1[G2] and G1 + G2 are strong bipolar fuzzy graphs.

Proof. The proof follows from Propositions 3.4, 3.6 and 3.12. h

Remark. The union of two strong bipolar fuzzy graphs is not necessary a strong bipolar fuzzy graph.


Example 1.

Proposition 5.4. If G1 � G2 is strong bipolar fuzzy graph, then at least G1 or G2 must be strong.

Proof. Suppose that G1 and G2 are not strong bipolar fuzzy graphs. Then there exist x1y1 2 E1 and x2y2 2 E2 such that

lPB1ðx1y1Þ < min lP

A1ðxÞ;lP

A1ðyÞ

� �;lP

B2ðx1y1Þ < min lP

A2ðxÞ;lP

A2ðyÞ

� �� ð1Þ

lNB1ðx1y1Þ > max lN

A1ðxÞ;lN

A1ðyÞ

� �;lN

B2ðx1y1Þ > max lN

A2ðxÞ;lN

A2ðyÞ

� �� ð2Þ

Assume that

lPB2ðx2y2Þ 6 lP

B1ðx1y1Þ < min lP

A1ðx1Þ;lP

A1ðy1Þ

� �6 lP

A1ðx1Þ � � � ð3Þ

Let

E ¼ ðx; x2Þðx; y2Þjx1 2 V1; x2y2 2 E2f g [ ðx1; zÞðy1; zÞjz 2 V2; x1y1 2 E1f g:

Consider (x,x2)(x,y2) 2 E, we have

lPB1� lP

B2


A1ðxÞ;lP

B2ðx2y2Þ

� �< min lP

A1ðxÞ;lP

A2ðx2Þ;lP

A2ðy2Þ

� �
and
lPA1� lP

A2


A1ðx1Þ;lP

A2ðx2Þ

� �; lP

A1� lP

A2

� �ðx1; y2Þ ¼min lP

A1ðx1Þ;lP

A2ðy2Þ

� �:

Therefore,

min lPA1� lP

A2


A1� lP

A2

� �ðx; y2Þ

� �¼min lP

A1ðxÞ;lP

A2ðx2Þ;lP

A2ðy2Þ

� �:

Hence

lPB1� lP

B2

� �ððx; x2Þðx; y2ÞÞ < min lP

A1� lP

A2


A1� lP

A2

� �ðx; y2Þ

� �:

Similarly, we can easily show that

lNB1� lN

B2

� �ððx; x2Þðx; y2ÞÞ > max lN

A1� lN

A2


A1� lN

A2

� �ðx; y2Þ

� �:

That is, G1 � G2 is not strong bipolar fuzzy graph, a contradiction. Hence if G1 � G2 is strong bipolar fuzzy graph, then atleast G1 or G2 must be strong bipolar fuzzy graph. h

Proposition 5.5. If G1[G2] is strong bipolar fuzzy graph, then at least G1 or G2 must be strong.

Proof. Obvious. h


Definition 5.6. The complement of a strong bipolar fuzzy graph G = (A,B) of G⁄ = (V,E) is a strong bipolar fuzzy graphG ¼ ðA;BÞ on G�, where A ¼ lP

A;lNA

� �and B ¼ lP

B;lNB

� �are defined by

(i)

V ¼ V ;

(ii)

lPAðxÞ ¼ lP

AðxÞ;lNA ðxÞ ¼ lN

A ðxÞ for all x 2 V ;

(iii)

lPBðxyÞ ¼

0 if lPBðxyÞ > 0;

min lPAðxÞ;lP

AðyÞ� �

if if lPBðxyÞ ¼ 0;

(

lNB ðxyÞ ¼

0 if lNB ðxyÞ > 0;

max lNA ðxÞ;lN

A ðyÞ� �

if if lNB ðxyÞ ¼ 0:

(

Definition 5.7. A strong bipolar fuzzy graph G is called self complementary if G G.

Example 5.8. Consider a graph G⁄ = (V,E) such that V = {a,b,c}, E = {ab,bc}. Consider a strong bipolar fuzzy graph G

Clearly, G ¼ G. Hence G is self complementary.

Proposition 5.9. Let G be a self complementary strong bipolar fuzzy graph. Then
Xx–y
lPBðxyÞ ¼

Xx–y

min lPAðxÞ;lP

AðyÞ� �

;

Xx–y

lNB ðxyÞ ¼

Xx–y

max lNA ðxÞ;lN

A ðyÞ� �

:

Proof. Let G be a self complementary strong bipolar fuzzy graph. Then there exists an automorphism f : V ? V such thatlP

Aðf ðxÞÞ ¼ lPAðxÞ and lN

A ðf ðxÞÞ ¼ lNA ðxÞ for all x 2 V and lP

Bðf ðxÞf ðyÞÞ ¼ lPBðxyÞ and lN

B ðf ðxÞf ðyÞÞ ¼ lNB ðxyÞ for all x, y 2 V. By def-

inition of G, we have

lPBðf ðxÞf ðyÞÞ ¼min lP

Aðf ðxÞÞ;lPAðf ðyÞÞ

� �lP

BðxyÞ ¼ min lPAðxÞ;lP

AðyÞ� �

;Xx–y

lPBðxyÞ ¼

Xx–y

min lPAðxÞ;lP

AðyÞ� �

;

lNBðf ðxÞf ðyÞÞ ¼max lN

Aðf ðxÞÞ;lNAðf ðyÞÞ

� �;lN

B ðxyÞ ¼max lNA ðxÞ;lN

A ðyÞ� �

;Xx–y

lNB ðxyÞ ¼

Xx–y

max lNA ðxÞ;lN

A ðyÞ� �

:



Proposition 5.10. Let G be a strong bipolar fuzzy graph. If lPBðxyÞ ¼min lP

AðxÞ;lPAðyÞ

� �and lN

B ðxyÞ ¼ max lNA ðxÞ;lN

A ðyÞ� �

for all x,y 2 V, then G is self complementary.

Proof. Let G be a strong bipolar fuzzy graph such that lPBðxyÞ ¼minðlP

AðxÞ;lPAðyÞÞ and lN

B ðxyÞ ¼maxðlNA ðxÞ;lN

A ðyÞÞ for all x,y 2 V. Then G G under the identity map I : V ? V. Hence G is self complementary. h

Proposition 5.11. Let G1 and G2 be strong bipolar fuzzy graphs. Then G1 ffi G2 if and only if G1 ffi G2.

Proof. Assume that G1 and G2 are isomorphic, there exists a bijective map f : V1 ? V2 satisfying

lPA1ðxÞ ¼ lP

A2ðf ðxÞÞ; lN

A1ðxÞ ¼ lN

A2ðf ðxÞÞ for all x 2 V1;

lPB1ðxyÞ ¼ lP

B2ðf ðxÞf ðyÞÞ; lN

B1ðxyÞ ¼ lN

B2ðf ðxÞf ðyÞÞ for all xy 2 E1:

By definition of complement, we have

lPB1 ðxyÞ ¼min lP

A1ðxÞ;lP

A1ðyÞ

�¼min lP

A2ðf ðxÞÞ; lP

A2ðf ðyÞÞ

� �¼ lP

B2 ðf ðxÞf ðyÞÞ;

lNB1 ðxyÞ ¼ max lN

A1ðxÞ;lN

A1ðyÞ

�¼max lN

A2ðf ðxÞÞ;lN

A2ðf ðyÞÞ

� �¼ lN

B2 ðf ðxÞf ðyÞÞ for all xy 2 E1:

Hence G1 ffi G2.The proof of converse part is straightforward. This completes the proof. h

Proposition 5.12. Let G1 and G2 be strong bipolar fuzzy graphs. If there is a weak isomorphism between G1 and G2, then there is aweak isomorphism between G1 and G2.

Proof. Let f be a weak isomorphism between G1 and G2, then f : V1 ? V2 is a bijective map that satisfies f(x1) = x2 for allx1 2 V1,

lPA1ðx1Þ ¼ lP

A2ðf ðx1ÞÞ;lN

A1ðx1Þ ¼ lN


lPA1ðx1y1Þ 6 lP

A2ðf ðx1Þf ðy1ÞÞ;lN

A1ðx1y1ÞP lN

A2ðf ðx1Þf ðy1ÞÞ for all x1y1 2 E1:

Since f : V1 ? V2 is a bijective map, f�1 : V2 ? V1 is also bijective map such that f�1(x2) = x1 for all x2 2 V2. Thus

lPA1ðf�1ðx2ÞÞ ¼ lP

A2ðx2Þ; lN

A1ðf�1ðx2ÞÞ ¼ lN

A2ðx2Þ for all x2 2 V2:

By definition of complement, we have

lPB1 ðx1y1Þ ¼min lP

A1ðx1Þ;lP

A1ðy1Þ

� �P min lP

A2ðf ðx2ÞÞ;lP

A2ðf ðy2ÞÞ

� �¼ min lP

A2ðx2Þ;lP

A2ðy2Þ

� �¼ lP

B2 ðx2y2Þ;

lNB1 ðx1y1Þ ¼ max lN

A1ðx1Þ;lN

A1ðy1Þ

� �6 max lN

A2ðf ðx2ÞÞ;lN

A2ðf ðy2ÞÞ

� �¼max lN

A2ðx2Þ;lN

A2ðy2Þ

� �¼ lN

B2 ðx2y2Þ:

Thus, f�1 : V2 ? V1 is a bijective map which is a weak isomorphism between G1 and G2. This ends the proof. h

Proposition 5.13. Let G1 and G2 be strong bipolar fuzzy graphs. If there is a co-weak isomorphism between G1 and G2, then there isa homomorphism between G1 and G2.

Proof. Obvious. h

6. Conclusions

Graph theory is an extremely useful tool in solving the combinatorial problems in different areas including geometry,algebra, number theory, topology, operations research, optimization and computer science. The bipolar fuzzy sets constitutea generalization of Zadeh’s fuzzy set theory. The bipolar fuzzy models give more precision, flexibility and compatibility to thesystem as compared to the classical and fuzzy models. We have introduced the concept of bipolar fuzzy graphs in this paper.The concept of bipolar fuzzy graphs can be applied in various areas of engineering, computer science: database theory, expertsystems, neural networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, and


medical diagnosis. We plan to extend our research of fuzzification to (1) bipolar fuzzy hypergraphs, (2) intuitionistic fuzzyhypergraphs, (3) vague hypergraphs, (4) interval-valued hypergraphs, and (5) rough hypergraphs.

Acknowledgements.

The author is highly thankful to the Editor-in-Chief and the referees for their valuable comments and suggestions forimproving the paper. The author would also like to pay his gratitude to Professor Syed Mansoor Sarwar who gave invaluablesuggestions.

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