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Fuzzy Information and Engineering

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ELSEVIER

ORIGINAL ARTICLE

Complement and Isomorphism on Bipolar Fuzzy Graphs

CrossMark

Ali Asghar Talebi ■ Hossein Rashmanlou

Received: 4 July 2013/ Revised: 5 August 2014/ Accepted: 22 October 2014/

Abstract In this paper, we discuss some properties of the self complement and self weak complement bipolar fuzzy graphs, and get a sufficient condition for a bipolar fuzzy graph to be the self weak complement bipolar fuzzy graph. Also we investigate relations between operations union, join, and complement on bipolar fuzzy graphs.

Keywords Isomorphism • Complement ■ Self complement ■ Bipolar fuzzy graph © 2014 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

1. Introduction

Presently, science and technology is featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models is based on an extension of the ordinary set theory, namely, fuzzy sets. Graph theory has numerous applications to problems in computer science, electrical engineering, system analysis, operations research, economics, networking routing, and transportation. The graph isomorphic transformations are reduced to redefinition of vertices and edges. This redefinition does not

Ali Asghar Talebi (E3)

Department of Mathematics, University of Mazandaran, Babolsar, Iran email: a.talebi@umz.ac.ir Hossein Rashmanlou

Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran

Peer review under responsibility of Fuzzy Information and Engineering Branch of the Operations Research

Society of China.

© 2014 Fuzzy Information and Engineering Branch of the Operations Research Society of China. Hosting by Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/jfiae.2015.01.007

change properties of the graph determined by an adjacent and an incidence of its vertices and edges. Fuzzy independent sets, domination fuzzy sets, and fuzzy chromatic sets are invariants concerning the isomorphism transformations of the fuzzy graphs and fuzzy hyper graph and allow theirs structural analysis [7]. In 1975, Zadeh [27] introduced the notion of fuzzy sets as a method for representing uncertainty. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines including logic, topology, algebra, analysis, information theory, artificial intelligence, operations research, neural networks and planning etc. [6, 10]. The fuzzy graphs theory as a generalization of Eulers graph theory was first introduced by Rosenfeld [14] in 1975. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs obtaining analogs of several graphs theoretical concepts. Later, Bhattacharya [5] gave some remarks on fuzzy graphs, and Mordeson and Peng [9] introduced some operation of fuzzy graphs. In 1994, Zhang [28,29] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. Bipolar fuzzy sets are 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 counter-property. Although bipolar fuzzy sets and intuitionistic fuzzy sets look similar to each other, they are essentially different sets. It is noted that positive information represents what is granted to be possible, while negative information represents what is considered to be impossible. This domain has recently motivated new research in several directions. For instance, when we assess the position of an object in space, we may have positive information expressed as a set of possible places and negative information expressed as a set of impossible places. This corresponds to the idea that the union of positive and negative information does not cover the whole space.

In 2011, Akram [2] defined bipolar fuzzy graphs. Akram and Davvaz [3] investigated strong intuitionistic fuzzy graphs. Then Akram and Karunambigai [4] defined length, distance, eccentricity, radius and diameter of a bipolar fuzzy graph and introduced the concept of self centered bipolar fuzzy graphs. Akram and Dudek [1] defined interval-valued fuzzy graphs. Samanta and Pal defined fuzzy tolerance graphs [19], fuzzy threshold graphs [20], irregular bipolar fuzzy graphs [21], fuzzy ^-competition graphs and p-competition fuzzy graphs [22] and some results on bipolar fuzzy sets and bipolar fuzzy intersection graphs [23]. Talebi and Rashmanlou [24] studied properties of isomorphism and complement on interval-valued fuzzy graphs. Likewise, they defined isomorphism on vague graphs [25]. Irregular interval-valued fuzzy graphs were studied by Pal and Rashmanlou [13]. Recently, Rashmanlou and Pal defined antipodal interval-valued fuzzy graphs [15], balanced interval-valued fuzzy graphs [16], some properties of highly irregular interval-valued fuzzy graphs [18] and isometry on interval-valued fuzzy graphs [17]. Nagoorgani and Malarvizhi [11, 12] investigated isomorphism properties on fuzzy graphs. Also they defined the self complementary fuzzy graphs. Bhutani [6] introduced the concept of weak isomorphism and isomorphism between fuzzy graphs.

Fuzzy Inf. Eng. (2014) 6: 505-522_507

In this paper, we defined the self complement and self weak complement bipolar fuzzy graphs with some properties of its discussed. We also study some properties of isomorphism and complement on bipolar fuzzy graphs.

2. Preliminaries

A fuzzy graph with V, a non-empty finite set as the underlying set is a pair G = (cr,//) where cr : V -»(0,1) is a fuzzy subset of V, n : V x V -»(0,1) is a symmetric fuzzy relation on the fuzzy subset cr such that fi(x,y) < min(cr(jf), cr(y)) for all x,y e V. A fuzzy relation n is symmetric if ¡i(x, y)= n(y, x) for all x, y e V. The underlying crisp graph of the fuzzy graph G = (cr,fi) is denoted as G* = (cj*,fi*) where cr* = {x e V \ cr(x) > 0} and //* = [(x, y) e V x V | fi(x,y) > 0}. If /i(x,y) > 0, then x and y are called neighbors. For simplicity, an edge (x, y) will be denoted by xy.

Let X be a non-empty set. A bipolar fuzzy set B in X is an object having the form B = <:),//*(*)) \ x e X], where ¡i' : X -> (0,1) and ¡j? : X -» (-1,0) are

mappings.

Let X be a non-empty set. Then we call a mapping A = : XxX -* (-1, l)x

(-1,1) a bipolar fuzzy relation on X such that ¡J.pA(x,y) e [0,1] and [¿¿(x, y) e (-1,0). Let A = be a bipolar fuzzy relation on a set X and B = (jig,fig) be bipolar

fuzzy set on X. Then, A = (Ha'Pa^ called a bipolar fuzzy relation on B = Qig,[ig) if ¡/A(x,y) < minGu£(*),/^(y)) and nNA (x,y) > tmxQi^(x),^(y)) for all x,y e X.

Let G = (cr,fi) be a fuzzy graph on underlying set V. The complement of G is defined as G = (cr,//) where fi(.x,y) = min(crw, cr^) - fi(x,y) for all x,y e V. When G = (cr,//) is a fuzzy graph, G = (cr,/T) is also a fuzzy graph. Let G = (cr,n) and G' = (cr7,//') be two fuzzy graphs with underlying sets V and V', respectively. A homomorphism from G to G' is a map h : V —> V' which satisfies cr(x) < cfihix)) for all x e V and //(*, y) < ¡i'{h(x), h(y)) for all x, y e V.

An isomorphism h from G to G' is a bijective homomorphism satisfing cr(x) = cr'(h(x)) for all x e V and fi(x, y) = fi(h(x), h(y)) for all x, y e V.

A weak isomorphism h from G to G' is a bijective homomorphism that satisfies cr{x) = cr'(h(x)) for all x 6 V.

A co-weak isomorphism h from G to G' is a bijective homomorphism that satisfies fi(x, y) = fx' (h(x), h(y)) for all x, y e V. Series system models are developed and work out under the following notations.

Definition 1 [2] A bipolar fuzzy graph with an underlying set V is defined to be a pair G = (A, B) where A = (ju^,//^) is a bipolar Juzzy set in V and B = Qig,fig) is a bipolar fuzzy set in E C V2 where V2 = {{x,y}\x,y e V, x + y} such that fig(x, y) < min(jiA(x),fig(x,y) > ma3i(fi%(x),fi%(y)) for all {x,y} e E. We call A the bipolar fuzzy vertex set of V, B the bipolar fuzzy edge set of E, respectively. We use the notation xy for an element of E. Thus, G = (A, B) is a bipolar fuzzy graph of G* = (V,E) ifnPB(xy) < mm(upA(x),^(y)), ^(xy) > maxGu»(x),u»(y)) for all xy eE.

Definition 2 The complement of a bipolar fuzzy graph G = (A, B) of a graph G* = (V, E) is a bipolar fuzzy graph G = (A, B) ofG* = (V, V2), where A = A = (p^,^)

and B = (jipB,nNB) is defined by fi?(xy) = min(jip(x),^(y)) - ¡J.pB{xy\ ¡4(xy) = max(/u%(x),$(y)) - n^(xy) for all x,y e V.

Throughout this paper, Gi = (At,B]) and G2 = (A2, B2) are taken to be the bipolar fuzzy graphs of G* = (V\,E\) and G'2 = (V2,E2), respectively.

Example 1 Consider a graph G* = (V, E) with V = {a, h, c, d), E = {ab, ad, bd, be). Let A = (¿1^,11%) be a bipolar fuzzy subset of V and let B = (jiB,fiB) be a bipolar fuzzy subset of E c V x V defined by

Table 1: Weight of vertices. Table 2: Weight of edges.

ab ad bd be

nP I ill „fiiil

f*A 2 1 2 5 r^B 8 9 9 4

UN -1 -1 -1 -1 UN -1 -1 -1 -1

"a PB 3 2 2 3

Fuzzy Inf. Eng. (2014) 6: 505-522_509

Definition 3 [2] Let G\ and G2 be the bipolar fuzzy graphs. A homomorphism f from Gi to G2 is a mapping f :Vi —» V2 which satisfies the following conditions:

(a) tfAi(Xl) </4(/(*i)),i<(*i) ><(/Ui))/<"-a«^i e V!.

(b) fipBl(xiyi) < 1) > <(/(*i)/6>i)) for all x01 6 E\.

Definition 4 [2] Let Gi and G2 be bipolar fuzzy graphs. An isomorphism f from Gi to Gi is a bijective mapping f :V1 —» V2 which satisfies the following conditions:

(a) /jpAi(x1)=ppAi(f(x1)),nl(xl)=nl(f(xl))forallx1 6 V,.

(b) ui(xm) = /%№i)/(yi)),<(w1) = <(/(x,)/(>>,)) for all xiyi e E\.

We denote Gi = G2 if there is an isomorphism from Gi to G2.

Definitions [2] Let Gi and G2 be the bipolar fuzzy graphs. Then, a weak isomorphism f from Gi to G2 is a bijective mapping f : Vi —» V2 which satisfies the following conditions:

(a) / is homomorphism,

(b) /jpAi(x1)=ppAi(J(x1)),nl(xl)=nl(f(xl))forallx1 6 V,.

Definition 6 [2] Let Gi and G2 be the bipolar fuzzy graphs. A co-weak isomorphism f from Gi to G2 is a bijective mapping f : Vi —» V2 which satisfies,

(a) f is homomorphism,

(b) t4S*iyi) = /4№I№)). <(*= <(/(*i)/(yi))/<"-al1 x<y> 6

Definition? [2] The unionGiUG2 = (AiUA2,B\UB2) of two bipolarfuzzy graphs Gi and G2 is defined as follows:

¡K U <)(*) = ifxeVuxiV2,

(A) u = ifxeV2,x$Vu

u HPM)(x) = = ma(*),//£(*)), ifxeV^Vr,

[K u <)(*) = ifxeVuxiV2,

(B) u <)(*) = ifxeV2,x£Vu

u<)« = = min^X W,<W), ifxeV^nVr,

(K if xy e E\,xy t E2,

(C) U<)(xy) = ttPB2(xy), if xye E2,xy g Eu

u HPBi)(xy) = maxin^ixyln^xy)), if xy e E1 n £2;

[K U e Euxy i E2,

CD) = ^(xy), if xy e E2,xy g £1,

WNBl U<)(ATy) = min^Oty),^ (xy)), if xy e Ei n E2.

Definition 8 [2] The join Gi + G2 = (Ai + M, + B2) of two bipolar fuzzy graphs Gi and G2 is defined as follows:

\{y.PA] + <)(x) = (MPAl U /£)(*), for all IGV.U V2, (A) +<)(*) = (<U<)W;

rm ¡(pPBl+fiPB2)(.xy) = (MPBl^fiPB2)(xy),

\ K + <X*y) = < u ¿/ y 6 n U V2;

(O i^fl!+/JS2XJ9') = min^A1(;c)'/iA2Cy))' if xyeE', + <)(*>■) = max^W.^OO);

where E' is the set of all edges joining the nodes ofV1 and V2.

3. Self Complement and Self Weak Complement Bipolar Fuzzy Graphs

In 1965, Zadeh [24] first introduced fuzzy sets as a mathematical way of representing impreciseness or vagueness in everyday life.

Definition 9 A bipolar fuzzy graph G = (A, B) of a graph G* = (V, E) is said to be self weak complement if G is weak isomorphism with its complement G, i.e., there exist a bijective homomorphism f from G to G such that for all x,y €V,

MPA(x) = = <(/(x))

< 7B(f(x)f(y))^NB(xy) > ¿f(/«/(y)). Definition 10 A bipolar fuzzy graph G is said to be self complement ifG = G.

Example 2 Consider a graph G* = (V, E) such that V = [a, b, c}, E = {ab, be}. Then a bipolar fuzzy graph G = (A, B), where

A _ /(_a_ b_ c_ a__b__c_ .

"0.2' 0.3' 0.3 -0.4' -0.5' -0.4 '' _ / ab be ab be \

~ \ oT' oils ' -02' -o!2 '

is self weak complement. In fact, identity bijective mapping is an weak isomorphism from G to G.

Fig. 2 Self weak complement bipolar fuzzy graph G

Example 3 Consider a graph G* = (V, E) with V = [a, b, c}, E = {ab, be). Then a bipolar fuzzy graph G = (A, B), where

a - ItJL — (—__-__

> 0.2' 0.3' 0.3 -0.4' -0.5' -0.5 ''

_ i ab be ab be \

~ '(02' 0I5'' -04' -0.25 '

is self complementary. In fact, bijective mapping / from G to G defined by a -* a, b -» c, c -» b is an isomorphism.

Fig. 3 Self complementary bipolar fuzzy graph G

Theorem 1 Let G = (A,B) be a self complement bipolar fuzzy graph of a graph G* = (V,E). Then

Proof Let G = (A, B) be a self complement bipolar fuzzy graph of a graph G* = (y,E). Then, there exist a weak isomorphism g from G to G such that for every x,y e V we have

= = ^(g(x)) for all* e V

= 7B(g(x)g(y)),i4(xy) = $(g(x)g(y))iai all x.y 6 V.

Now by definition of G, for every x,y e V, we have

i^feWgOO) = mmQi^gix^^igiy))) - p.pB(g(x)g(y)\ KCgMsOO) = max(^(g(x)),M^(g(y))) ~l4(s(x)g(y)),

lfipB(xy) = mm^pA(x),fipA(y))-fipB(g(x)g(y)), \4(xy) = maxGu»(x),nN(y))-4(g(x)g(y)),

Yj^) + XXfeMi'M) = ^min^U),/^)),

xiy xty x£y

2 Yi^BW = YjrmnQipA(x)^p(y)),

xiy x±y

xiy xiy

YjVn(xy) =

xty xiy

Example 4 In this example, we show that the reverse of the above theorem in the general is not true. We suppose that G = (A, B) is the bipolar fuzzy graph defined as follows.

Fuzzy Inf. Eng. (2014) 6: 505-522_513

Fig. 4 Bipolar fuzzy graphs G and G By a routine computations, it is easy to see that G satisfies the conditions

= i ^min^M^OO)

x£y x&y

Yj^xy) = ^ ^max^W^OO),

x£y xiy

but G is not self complement because there is not an isomorphism between G and G.

Theorem 2 Let G = (A, B) be a self weak complement bipolar fuzzy graph of a graph G* = (V,E). Then

xty xty

x£y xty

Proof Let G = (A, B) be a self weak complement bipolar fuzzy graph of a graph G* = (V, E). Then there exist a weak isomorphism h from G to G such that for all x, y e V, we have

= 7A(h(x)) = ,/AQi(x)\tiNA(x) = V»(h(x)) = ^(h(x))

fs(xy) < fipB(h(x)h(y)),4(xy) > nNB(h{x)h(y)).

Using the definition of complement in the above inequality, for all x,y e V, we have

SbW) <7B№)h(y)) = min(i£(Kx)).fiWy))) ~ AWxMy)),

l4(xy) > 4(Kx)h(y)) = maxO<№(*)),/^(y))) - fxNB(h(x)h(y)).

A(xy) +MB(h(x)h(y)) < min(ifA{h{x))yA(My)))

t4(xy) +4(Kx)h(y)) > max(jA»(h(x)),tf(h(y))).

Hence,

Y^(xy) + < ^ mm(jiA(h(x)), fiA(h(y)))

x±y x*y xiy

Yj^xy) + Y^xWy)) a: Yj (%))).

xty xty xty

Thus and

2 >]Tmax(^(x),<(x)Cy)).

Therefore, and

Remark 1 Example 4 shows the converse of above theorem is not true.

Theorem 3 Let G = (A, B) be a bipolar fuzzy graph of a graph G* = (V, E). If for allx,y € V, ¡/B{xy) < -mia(nPA(x),nPA(y)) and fi^(xy) > - max(ji%(x),/i%(y)), then G = (A, E) is a self weak complement bipolar fuzzy graph.

Proof Consider the identity map I \ V —> V, that for all x e V, fiA(x) = fiA(/(x)) and MA(x) = fiA(I(x)). By definition of jljj, we havenPB(xy) = min- fiB(xy) for all j,y€ V, and fiB(xy) = max(ju^(x),ju^(y)) - (¿B(xy). Hence, for every x,y e V,

we have

fip(xy) > xma(nPA(x),nFA(y)) - - rmn(p.pA(x),nPA(y)) = i mm(ppA(x\iipA(y)) > tig(xy),

= ^maiQ4(x),f4(y))<4(xy).

MbW) < 7BU(x)l(y)) for all x,y e V and^xy) > ^FB{I(x)I(y)).

4. Complement and Isomorphism on Bipolar Fuzzy Graphs

In this section, we discuss some of the properties of isomorphism and complement on bipolar fuzzy graphs.

Theorem 4 Let G\ = (Au B\) and Gi = (A2, B2) be two bipolar fuzzy graphs such that Vi n V2 = 0. Then Gi +G2 a G^ U (h.

Proof We shall prove that the identity map is the required isomorphism. Let / : Vi U V2 -» Vi U V2 be the identity map. We prove that for all x,y e V

(¿"a, + <)(x>') = (/^sf u = (Psf U jUgArjCxy).

Let V. Then

HPAl (x), ifxeVi

Hp2(x), if xeV2

p^ix), ifxeVi

jifxeV2

ifxeVi

if X 6 V2

J ifxe Vi

ifxe V2

(^Af = K +0

= (^Af U

(/¿Bf

= min (G< + + - K + MPB2)(xy)

(min((mpAi U(ßpAi Unp2)(y)) - Uif xy e £1 U £2 |min(0u^ UßpAi)(x),(jiAt U fip2)(y)) -min(/^(x),/^(y)),ifxy e E' min (fipAi (x),ßp2(y)) - fipBi (xy), if xy e Ei

= ■ min [fiPM (x),ßpAi(y)) - nFBl{xy), ifxeE2

min(^ (x),fip2(y)) - min(iipAi(x)yAi(y)), if xy 6 E'

IHbMy)> tfxyeE1 M^ixyl if xy e E2 = K u 0, ifxy 6 E'

(figN +MB»)(xy)

<- ((MA« + MA»)(X), (MA« + ßA«)(y)) ~(MB»+ f*B«)(xy) v («_ U ßNAi)(x), (ßl U n»2)(yj) - (nl U <)(xy), if xy 6 Et U E2 [max((^ U//£)«,(< U<)(y)) -max(^(x),^Cy)), ifxy e

if xy e £|

'(<«,<№)-<(*?), ifxye£2

max(^i(x),J<2(y)) - max^W.^M), ifxy e E' if Aye £i

if xyeE2 =(^U^)(xy). 0, ifxyeß'

= max^ [maxi

max^ maxi

Theorem 5 Let Gi = (Ai, B\) and G2 = (A2, B2) be two bipolar fuzzy graphs such that V,nV2 = 0. Then Gi U G2 = G^ + G^.

Proof We shall prove that the identity map is the required isomorphism. For all x, y e Vi U V2, then we have

(^Af U/^)(x) = U^)(x) - ^

= (Äif U Jj^)(x) = (sif + 0),

ifxe Vi ifxeV2

K«- ifxe Vi

K«> ifxe v2

ifxe Vi

ifxe v2

= UjUaar)(x) = (flAN +jUAp)(x),

(¿"sf U nBp)(xy) = min (guAl U fiAl)(x), (hpm U uAl)(y)) - (nPBl u M^)(.xy) min(fiPM(x),nPM(y)) -fipBi(xy), iixyeE, = min(^W./^OO) -iiBi{xy), if xy e E2 rmn(nPAi(x),tip2(y)) - 0, ifxe V,,y e V2

/¿s^xy), AijLfoO.

if xy e Ei if xy € E2

min^Ct),//j°2Cy)), ifxeVuyeV2

(¡4tU^)(xy), if xyeEl

(j^U vQ(xy), if xy e £2

min(//^(x),//^2Cy)), ifxye£'

'"(xy),

, u- Ku

Cy>)-A^C*y>. if*ye£i Kw^^wj-^cxy), if xy 6 £2

max (^(x),^2(y)) - 0, if x e Vlty e V2

if xy e £1

xy), if xy e £2

ifxeV,jel/2

Vii if xy e £1

if xy e £2

max^CxX^Cy)), if xy e £'

Remark 2 If there is a weak isomorphism between two bipolar fuzzy graphs Gi and G2, there need not to be a weak isomorphism between Gi and G2.

Consider the following example.

Example 5 Let V] = {a, b, c, d) and V2 = {«, v, x, w). Consider two bipolar fuzzy graphs G\ = (Ai, Bi) and G2 = (A2, B2) defined by

Table 3: Weight of vertices and edges of G\.

a b c d

< 0.2 0.3 0.3 0.2

^ -0.4 -0.5 -0.5 -0.6

ab ac be bd cd

0.1 0.1 0.2 0.1 0.1

< -0.3 -0.3 -0.3 -0.4 -0.4

Table 4: Weight of vertices and edges of G2.

u v x w

lfM 0.2 0.3 0.3 0.2 -0.4 -0.5 -0.5 -0.6

uv vx ux vw xw

A 0.15 0.25 0.15 0.1 0.1

< -0.4 -0.4 -0.4 -0.5 -0.5

W G 2 x

Fig. 5 Bipolar fuzzy graphs G\ and G2

It is easy to check that the mapping g : V\ -» V2, g(a) = u, g(b) = v, g(c) - x and g(d) = w, is a weak isomorphism from Gi to G2. Now by definition of complement, we have

ab) = 0.1 ^(ab) = -0.1,^(«c) = 0.1, ^(ac) = -0.1, ^(bc) = 0.1, be) = -0.2, = 0.1, ju^(M) = -0.1,<(cd) = 0.1,/j*(cd) = -0.1.

<(«v) = 0.05, ^(«v) = O.^vx) = 0.05,^(vx) = -0.1 ^(ux) = 0.05 <(«*) = 0, JJ^Jvw) = 0.1,^(vw) = 0,^(xw) = 0.1, <(xw) = 0.

Hence, there is not a weak isomorphism between Gi and G2, because

<(«v) = 0.05 < 0.1 = ^(xy) for all x,y e Vi.

Remark 3 If there is a co-weak isomorphism between bipolar fuzzy graphs Gi and G2, then there need not to be a co-weak isomorphism between Gi and G2.

The following example shows the above statement.

Example 6 Let V\ = {a, b, c) and V2 = {u, v, w}. Consider two bipolar fuzzy graphs Gi = (Ai, Bi) and G2 = (A2, B2) as follows.

Fig. 6 Bipolar fuzzy graphs Gi and

There is a co-weak isomorphism h : Vi -» V2 with h(a) - u, h(b) = v, h(c) = w. Also we have

rf^ab) = 0.1 = -0.1,^(ae) = 0.1, (ac) = -0.1,

= 0.1, ^(bc) = -0.2,^(uv) = 0.2 ,/4(«v) = -0.2, ^(«w) = 0.2 ,<(«w) = -0.2,/^(vw) = 0.2, ^(vw) = -0.3.

So, Gi is not a co-weak isomorphism with G2 because

/jJ(kw) = 0.2 * 0.1 = Ji^(xy) for all x,y e V,.

Theorem 6 Lei G = (A, B) ¿>e a bipolar fuzzy graph of a graph G* = (V, E). Then the automorphism group G and G are identical.

Proof We show that for any injective map h : V -» V, h e Aut(G) if and only if h e Aut(G), we have

7a(Kx)) = nA(h(x)) = fSA{x) = for all x eV, /£(*(*)) = fiNA(h(x)) = ¡ina{x) = ^A(x) for all * e V.

Also, for all x,y e V,

tf(h(x)h(y)) = 7B(xy)

« mill [fiA,(h(x)),fiAr(h(y))) - [ipB(h{x)h(y)) = min (¡iAp(x), nAp(y)) -MB(xy)

fiBr(h(x)h(y)) = fiBp(xy),

Fuzzy Inf. Eng. (2014) 6: 505-522_521

4(Kx)h(y)) =

<=>min{fiAN(h(x)),fiAN(h(y))j -nNB(h(x)h(y)) = min (/¿¿«(^./¿¿»Cy))

liBn(h(x)h(y)) = nB«{xy).

This complete the proof. 5. Conclusion

Graph theory is an extremely useful tool in solving the combinatorial problems in different areas including geometry, algebra, topology, optimization and computer science. In this paper, we discussed some properties of the self complement and self weak complement bipolar fuzzy graphs, and get a sufficient condition for a bipolar fuzzy graph to be the self-weak complement bipolar fuzzy graph. Also we investigated relations between operations union, join, and complement on bipolar fuzzy graphs.

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. References

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