Regular bipolar fuzzy graphs - Springer Link
Neural Comput & Applic (2012) 21 (Suppl 1):S197–S205 DOI 10.1007/s00521-011-0772-6
ORIGINAL ARTICLE
Regular bipolar fuzzy graphs Muhammad Akram • Wieslaw A. Dudek
Received: 14 July 2011 / Accepted: 30 November 2011 / Published online: 27 December 2011 Springer-Verlag London Limited 2011
Abstract We introduce the concepts of regular and totally regular bipolar fuzzy graphs. We prove necessary and sufficient condition under which regular bipolar fuzzy graph and totally bipolar fuzzy graph are equivalent. We introduce the notion of bipolar fuzzy line graphs and present some of their properties. We state a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We examine when an isomorphism between two bipolar fuzzy graphs follows from an isomorphism of their corresponding bipolar fuzzy line graphs. Keywords Bipolar fuzzy sets Bipolar fuzzy graphs Bipolar fuzzy line graphs Regular bipolar fuzzy graphs Totally regular bipolar fuzzy graphs
1 Introduction In 1736, Euler first introduced the notion of graph theory. In the history of mathematics, the solution given by Euler of the well-known Ko¨nigsberg bridge problem is considered to be the first theorem of graph theory. This has now become a subject generally regarded as a branch of combinatorics. The theory of graph is an extremely useful tool for solving combinatorial problems in different areas such
M. Akram (&) Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore 54000, Pakistan e-mail: [email protected]; [email protected] W. A. Dudek Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wrocław, Poland e-mail: [email protected]
as geometry, algebra, number theory, topology, operations research, optimization, and computer science. In 1965, Zadeh [39] introduced the notion of a fuzzy subset of a set. 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, multiagent systems, pattern recognition, robotics, computer networks, expert systems, decision making, and automata theory. In 1994, Zhang [43, 44] 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 [25]. In many domains, it is important to be able to deal with bipolar information. 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. In particular, fuzzy and possibilistic formalisms for bipolar information have been proposed [18], because when we deal with spatial information in image processing or in spatial reasoning applications, this bipolarity also occurs. For instance, when we assess the position of an object in a space, we may have positive information expressed as a set of possible places and negative information expressed as a set of impossible
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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 that are considered neither to the right nor to the left of some reference object, thus leaving some room for indetermination. This corresponds to the idea that the union of positive and negative information does not cover the whole space. In 1975, Rosenfeld [34] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [21] in 1973. The fuzzy relations between fuzzy sets were also considered by Rosenfeld, and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. Later on, Bhattacharya [11] gave some remarks on fuzzy graphs. The notion of fuzzy line graph was introduced by Mordeson in [28]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [15] and studied some of their properties. The concept of strong arcs in fuzzy graphs was discussed in [13]. Shannon and Atanassov [35] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs, and investigated some of their properties in [36]. Parvathi et al. defined operations on intuitionistic fuzzy graphs in [32]. Akram and Dudek [5] discussed intervalvalued fuzzy graphs. Recently, Akram has introduced the notions of bipolar fuzzy graphs in [3] and interval-valued fuzzy line graphs in [2]. In this paper, we introduce the concepts of regular and totally regular bipolar fuzzy graphs. We prove necessary and sufficient condition under which regular bipolar fuzzy graph and totally bipolar fuzzy graph are equivalent. We introduce the notion of bipolar fuzzy line graphs and present some of their properties. We give a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We examine when an isomorphism between two bipolar fuzzy graphs follows from an isomorphism of their corresponding bipolar fuzzy line graphs.ient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [1, 4, 7, 8, 20, 22, 23, 26, 27, 29]. 2 Preliminaries By graph, we mean a pair G* = (V, E), where V is the set and E is a relation on V. The elements of V are vertices of G* and the elements of E are edges of G*. We write xy 2 E
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to mean fx; yg 2 E; and if e ¼ xy 2 E; we say x and y are adjacent. Formally, given a graph G* = (V, E), two vertices x; y 2 V are said to be neighbors, or adjacent nodes, if xy 2 E: The neighborhood of a vertex v in a graph G* is the induced subgraph of G* consisting of all vertices adjacent to v and all edges connecting two such vertices. The neighborhood is often denoted N(v). The degree deg(v) of vertex v is the number of edges incident on v or equivalently, deg(v) = |N(v)|. The set of neighbors, called a (open) neighborhood N(v) for a vertex v in a graph G*, consists of all vertices adjacent to v but not including v, that is, NðvÞ ¼ fu 2 V j vu 2 Eg: When v is also included, it is called a closed neighborhood N[v], that is, N[v] = N(v) [ {v}. A regular graph is a graph where each vertex has the same number of neighbors, i.e., all the vertices have the same open neighborhood degree. A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. An isomorphism of graphs G*1 and G*2 is a bijection between the vertex sets of G*1 and G*2 such that any two 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. Isomorphic graphs are denoted by G*1 ^ G*2. In graph theory, the line graph L(G*) of a simple graph * G is another graph L(G*) that represents the adjacencies between edges of G*. Given a graph G*, its line graph L(G*) is a graph such that: • •
each vertex of L(G*) represents an edge of G*; and two vertices of L(G*) are adjacent if and only if their corresponding edges share a common endpoint (‘‘are adjacent’’) in G*.
Let G* = (V, E) be an undirected graph, where V ¼ fv1 ; v2 ; . . .; vn g: Let Si ¼ fvi ; xi1 ; . . .; xiqi g where xij 2 E has vertex vi ; i ¼ 1; 2; . . .; n; j ¼ 1; 2; . . .; qi : Let S ¼ fS1 ; S2 ; . . .; Sn g: Let T ¼ fSi Sj jSi ; Sj 2 S; Si \ Sj 6¼ ;; i 6¼ jg: Then P(S) = (S, T) is an intersection graph and P(S) = G*. The line graph L(G*) is by definition the intersection graph P(E). That is, L(G*) = (Z, W) where Z ¼ ffxg [ fux ; vx gjx 2 E; ux ; vx 2 V; x ¼ ux vx g and W ¼ fSx Sy jSx \ Sy 6¼ ;; x; y 2 E; x 6¼ yg; and Sx ¼ fxg[ fux ; vx g; x 2 E: Proposition 2.1 If G is regular of degree k, graph L(G) is regular of degree 2k - 2.
then line
Definition 2.2 [39, 40] A fuzzy subset l on a set X is a map l: X? [0, 1]. A map m: X 9 X? [0, 1] is called a fuzzy relation on X if m(x, y) B min(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 2.3 Let X be a non-empty set. A bipolar fuzzy set B in X is an object having the form
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B ¼ fðx; lPB ðxÞ; lNB ðxÞÞjx 2 Xg where lPB : X ! ½0; 1 and lNB : X ! ½1; 0 are mappings. We use the positive membership degree lPB (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 lNB (x) to denote the satisfaction degree of an element x to some implicit counter-property corresponding to a bipolar fuzzy set B. If lPB ðxÞ 6¼ 0 and lNB (x) = 0, it is the situation that x is regarded as having only positive satisfaction for B. If lPB (x) = 0 and lNB ðxÞ 6¼ 0; it is the situation that x does not satisfy the property of B, but somewhat satisfies the counter-property of B. It is possible for an element x to be such that lPB ðxÞ 6¼ 0 and lNB ðxÞ 6¼ 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 = (lPB, lNB ) for the bipolar fuzzy set
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B is a symmetric 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 lPB ðxyÞ minðlPA ðxÞ; lPA ðyÞÞ and lNB ðxyÞ maxðlNA ðxÞ; lNA ðyÞÞ for all xy 2 E: Example 3.2
Consider the bipolar fuzzy graph.
B ¼ fðx; lPB ðxÞ; lNB ðxÞÞjx 2 Xg: A nice application of bipolar fuzzy concept is a political acceptation (map to [0, 1]) and non-acceptation (map to [-1, 0]). Definition 2.4 [43] Let X be a non-empty set. Then we call a mapping A = (lPA, lNA ): X 9 X ? [-1, 1] 9 [-1, 1] a bipolar fuzzy relation on X such that lPA ðx; yÞ 2 ½0; 1 and lNA ðx; yÞ 2 ½1; 0: Definition 2.5 [43] Let A = (lPA, lNA) and B = (lPB, lNB) be bipolar fuzzy sets on a set X. If A = (lPA, lNA) is a bipolar fuzzy relation on a set X, then A = (lPA, lNA ) is called a bipolar fuzzy relation on B = (lPB, lNB) if lPA(x, y) B min (lPB(x), lPB(y)) and lNA (x, y) C max(lNB (x), lNB(y)) for all x; y 2 X: A bipolar fuzzy relation A on X is called symmetric if lPA(x, y) = lPA(y, x) and lNA(x, y) = lNA(y, x) for all x; y 2 X: 3 Regular bipolar fuzzy graphs Throughout this paper, G* will be a crisp graph and G a bipolar fuzzy graph. Definition 3.1 [3] By a bipolar fuzzy graph, we mean a pair G = (A, B) where A = (lPA, lNA ) is a bipolar fuzzy set in V and B = (lPB, lNB ) is bipolar relation on V such that lPB ðfx; ygÞ minðlPA ðxÞ; lPA ðyÞÞ maxðlNA ðxÞ; lNA ðyÞÞ
and
lNB ðfx; ygÞ
for all fx; yg 2 E: We call A the bipolar fuzzy vertex set of V, B the bipolar fuzzy edge set of E, respectively. Note that
Definition 3.3 Let G = (A, B) be a bipolar fuzzy graph on G*. If all the vertices have the same open neighborhood degree n, then G is called an n-regular bipolar fuzzy graph. The open neighborhood degree of a vertex x in G is defined by deg(x) = (degP (x), degN (x)), where degP ðxÞ ¼ P P N P N x2V lA ðxÞ and deg ðxÞ ¼ x2V lA ðxÞ: Definition 3.4 Let G = (A, B) be a regular bipolar fuzzy graph. The order of a regular bipolar fuzzy graph G is P P OðGÞ ¼ ð x2V lPA ðxÞ; x2V lNA ðxÞÞ: The size of a regular P bipolar fuzzy graph G is SðGÞ ¼ ð xy2V lPA ðxyÞ; P N xy2E lA ðxyÞÞ: Definition 3.5 Let G = (A, B) be a bipolar fuzzy graph. If each vertex of G has same closed neighborhood degree m, then G is called a totally regular bipolar fuzzy graph. The closed neighborhood degree of a vertex x is defined by deg[x] = (degP [x], degN [x]), where degP ½x ¼ degP ðxÞ þ lPA ðxÞ; degN ½x ¼ degN ðxÞ þ lNA ðxÞ: We show with the following examples that there is no relationship between n-regular bipolar fuzzy graph and mtotally regular bipolar fuzzy graph. Example 3.6 Consider a graph G* such that V = {a, b, c, d}, E = {ab, bc, cd, ad}. Let A be a bipolar
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fuzzy subset of V and let B be a bipolar fuzzy subset of E defined by
Routine computations show that bipolar fuzzy graph G is both regular and totally regular. The following propositions are obvious.
a lPA lNA
b
c
d
0.5
0.5
0.5
0.5
-0.3
-0.3
-0.3
-0.3
ab
bc
cd
da
lPB
0.2
0.4
0.2
0.4
lNB
-0.1
-0.1
-0.1
-0.1
Proposition 3.9 The size of a n-regular bipolar fuzzy graph G is nk2 ; where |V| = k. Proposition 3.10 If G is a m-totally regular bipolar fuzzy graph G, then 2S(G) ? O(G) = mk, where |V| = k. Proposition 3.11 If G is a both n-regular and m-totally regular bipolar fuzzy graph G, then O(G) = k(m n), where |V| = k. Definition 3.12 A bipolar fuzzy graph G = (A, B) is called complete if lPB ðxyÞ ¼ minðlPA ðxÞ; lPA ðyÞÞ and lNB ðxyÞ ¼ maxðlNA ðxÞ; lNA ðyÞÞ for all x; y 2 V: Example 3.13 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 bipolar fuzzy subset of E defined by lPA ðxÞ ¼ 0:5; lPA ðyÞ ¼ 0:7; lPA ðzÞ ¼ 0:6; lNA ðxÞ ¼ 0:3; lNA ðyÞ ¼ 0:4; lNA ðzÞ ¼ 0:5; lPA ðxyÞ ¼ 0:5; lPA ðyzÞ ¼ 0:6; lPA ðzxÞ ¼ 0:5; lNA ðxyÞ ¼ 0:3; lNA ðyzÞ ¼ 0:4; lNA ðzxÞ ¼ 0:3:
Routine computations show that a bipolar fuzzy graph G is both regular and totally regular. Example 3.7 Consider a graph G* such that V = {v1, v2, v3}, E = {v1v2, v1v3}. Let A be a bipolar fuzzy subset of V and let B be a bipolar fuzzy subset of E defined by lPA ðv1 Þ ¼ 0:4; lPA ðv2 Þ ¼ 0:8; lPA ðv3 Þ ¼ 0:7; lNA ðv1 Þ ¼ 0:4; lNA ðv2 Þ ¼ 0:7; lNA ðv3 Þ ¼ 0:6; lPB ðv1 v2 Þ ¼ 0:3; lPB ðv1 v3 Þ ¼ 0:4; lNB ðv1 v2 Þ ¼ 0:2; lNB ðv1 v3 Þ ¼ 0:2: Routine computations show that a bipolar fuzzy graph G is neither totally regular nor regular. Example 3.8 Consider a graph G* such that V = {v1, v2, v3}, E = {v1v2, v2v3, v3v1}. Let A be a bipolar fuzzy subset of V and let B be a bipolar fuzzy subset of E defined by lPA ðv1 Þ ¼ 0:4; lPA ðv2 Þ ¼ 0:4; lPA ðv3 Þ ¼ 0:4; lNA ðv1 Þ ¼ 0:4; lNA ðv2 Þ ¼ 0:4; lNA ðv3 Þ ¼ 0:4; lPB ðv1 v2 Þ ¼ 0:3; lPB ðv2 v3 Þ ¼ 0:3; lPB ðv3 v1 Þ ¼ 0:3; lNB ðv1 v2 Þ ¼ 0:2; lNB ðv2 v3 Þ ¼ 0:2; lNB ðv3 v1 Þ ¼ 0:2:
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Routine computations show that G is a both complete and totally regular bipolar fuzzy graph, but G is not regular since deg(x) = deg(z) = deg(y). Theorem 3.14 Every complete bipolar fuzzy graph is a totally regular bipolar fuzzy graph. Theorem 3.15 Let G = (A, B) be a bipolar fuzzy graph of a graph G*. Then A = (lPA, lNA ) is a constant function if and only if the following are equivalent: (a) G is a regular bipolar fuzzy graph, (b) G is a totally regular bipolar fuzzy graph.
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Proof Suppose that A = (lPA, lNA ) is constant function. Let lPA (x) = c1 and lNA (x) = c2 for all x 2 V: (a))(b): assume that G is n-regular bipolar fuzzy graph, then degP (x) = n1 and degN (x) = n2 for all x 2 V: So degP ½x ¼ degP ðxÞ þ lPA ðxÞ; degN ½x ¼ degN ðxÞ þ lNA ðxÞ for all x 2 V; Thus degP ½x ¼ n1 þ c1 ; degN ½x ¼ n2 þ c2
for all x 2 V:
Hence, G is a totally regular bipolar fuzzy graph. (b) ) (a): suppose that G is a totally regular bipolar fuzzy graph, then degP ½x ¼ k1 ; degN ½x ¼ k2
for all x 2 V
4 Bipolar fuzzy line graphs Definition 4.1 Let P(S) = (S, T) be an intersection graph of a simple graph G* = (V, E). Let G = (A1, B1) be a bipolar fuzzy graph of G*. We define a bipolar fuzzy intersection graph P(G) = (A2, B2) of P(S) as follows: (1) (2) (3)
A2, B2 are bipolar fuzzy sets of S and T, respectively, lPA2 ðSi Þ ¼ lPB1 ðvi Þ; lNA2 ðSi Þ ¼ lNB1 ðvi Þ; lPB2 ðSi Sj Þ ¼ lPB1 ðvi vj Þ; lNB2 ðSi Sj Þ ¼ lNB1 ðvi vj Þ
for all Si ; Sj 2 S; Si Sj 2 T: That is, any bipolar fuzzy graph of P(S) is called a bipolar fuzzy intersection graph. The following Proposition is obvious.
or
Proposition 4.2 Let G = (A1, B1) be a bipolar fuzzy graph of G*, then
degP ðxÞ þ lPA ðxÞ ¼ k1 ; degN ðxÞ þ lNA ðxÞ ¼ k2 for all x 2 V
• •
or
This Proposition shows that any bipolar fuzzy graph is isomorphic to a bipolar fuzzy intersection graph.
degP ðxÞ þ c1 ¼ k1 ; degN ðxÞ þ c2 ¼ k2
for all x 2 V
or degP ðxÞ ¼ k1 c1 ; degN ðxÞ ¼ k2 c2
for all x 2 V:
Thus, G is a regular bipolar fuzzy graph. Hence, (a) and (b) are equivalent. The converse part is obvious. h Proposition 3.16 If a bipolar fuzzy graph G is both regular and totally regular, then A = (lPA, lNA) is constant function. Proof Let G be a regular and totally regular bipolar fuzzy graph, then degP ðxÞ ¼ n1 ; degN ðxÞ ¼ n2 P
N
deg ½x ¼ k1 ; deg ½x ¼ k2
for all x 2 V;
Definition 4.3 Let L(G*) = (Z, W) be a line graph of a simple graph G* = (V, E). Let G = (A1, B1) be a bipolar fuzzy graph of G*. We define a bipolar fuzzy line graph L(G) = (A2, B2) of G as follows: (4) (5) (6) (7) (8)
Example 4.4 Consider a graph G* = (V, E) such that V = {v1, v2, v3, v4} and E ¼ fx1 ¼ v1 v2 ; x2 ¼ v2 v3 ; x3 ¼ v3 v4 ; x4 ¼ v4 v1 g: Let A1 be a bipolar fuzzy subset of V and let B1 be a bipolar fuzzy subset of E defined by
Now degP ½x ¼ k1 , degP ðxÞ þ lPA ðxÞ ¼ k1 , n1 þ lPA ðxÞ ¼ k1
v1
for all x 2 V:
Likewise, lNA (x) = k2 - n2 for A = (lPA, lNA ) is constant function.
all
A2 and B2 are bipolar fuzzy sets of Z and W, respectively, lPA2 ðSx Þ ¼ lPB1 ðxÞ ¼ lPB1 ðux vx Þ; lNA2 ðSx Þ ¼ lNB1 ðxÞ ¼ lNB1 ðux vx Þ; lPB2 ðSx Sy Þ ¼ minðlPB1 ðxÞ; lPB1 ðyÞÞ; lNB2 ðSx Sy Þ ¼ maxðlNB1 ðxÞ; lNB1 ðyÞÞ
for all Sx ; Sy 2 Z; Sx Sy 2 W:
for all x 2 V:
, lPA ðxÞ ¼ k1 n1
P(G) = (A2, B2) is a bipolar fuzzy graph of P(S), G^ P(G).
x 2 V:
Hence, h
v2
v3
v4
lPA1
0.2
0.3
0.4
0.2
lNA1
-0.5
-0.4
-0.5
-0.3
x1
x2
x3
x4
Remark The converse of Proposition 3.16 may not be true, in general. We state the following Theorem without its proof. Theorem 3.17 Let G be a bipolar fuzzy graph where crisp graph G* is an odd cycle. Then G is regular bipolar fuzzy graph if and only if B is a constant function.
lPB1
0.1
0.2
0.1
0.1
lNB1
-0.2
-0.3
-0.2
-0.2
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By routine computations, it is clear that L(G) is a bipolar fuzzy line graph. It is neither regular bipolar fuzzy line graph nor totally regular bipolar fuzzy line graph. Proposition 4.5 L(G) is a bipolar fuzzy line graph corresponding to bipolar fuzzy graph G. Proof
Obvious.
h
Proposition 4.6 If L(G) is a bipolar fuzzy line graph of bipolar fuzzy graph G. Then L(G*) is the line graph of G*. Proof Since G = (A1, B1) is a bipolar fuzzy graph and L(G) is a bipolar fuzzy line graph, lPA1 ðSx Þ ¼ lPB1 ðxÞ; lNA1 ðSx Þ ¼ lNB1 ðxÞ for all x 2 E By routine computations, it is easy to see that G is a bipolar fuzzy graph. Consider a line graph L(G*) = (Z, W) such that
and so Sx 2 Z , x 2 E: Also lPB2 ðSx Sy Þ ¼ minðlPB1 ðxÞ; lPB1 ðyÞÞ;
Z ¼ fSx1 ; Sx2 ; Sx3 ; Sx4 g
lNB2 ðSx Sy Þ ¼ maxðlNB1 ðxÞ; lNB1 ðyÞÞ
and
for all Sx ; Sy 2 Z; and so
W ¼ fSx1 Sx2 ; Sx2 Sx3 ; Sx3 Sx4 ; Sx4 Sx1 g:
W ¼ fSx Sy jSx \ Sy 6¼ ;; x; y 2 E; x 6¼ yg:
Let A2 ¼ ðlPA2 ; lNA2 Þ and B2 ¼ ðlPB2 ; lNB2 Þ be bipolar fuzzy sets of Z and W, respectively. Then, by routine computations, we have
This completes the proof.
lPA2 ðSx1 Þ ¼ 0:1;lPA2 ðSx2 Þ ¼ 0:2;lPA2 ðSx3 Þ ¼ 0:1;lPA2 ðSx4 Þ ¼ 0:1;
lPB2 ðSx Sy Þ ¼ minðlPA2 ðSx Þ; lPA2 ðSy ÞÞ for all Sx Sy 2 W;
lNA2 ðSx1 Þ ¼ 0:2;lNA2 ðSx2 Þ ¼ 0:3;lNA2 ðSx3 Þ
lNB2 ðSx Sy Þ ¼ maxðlNA2 ðSx Þ; lNA2 ðSy ÞÞ for all Sx Sy 2 W:
¼ 0:2;lNA2 ðSx4 Þ ¼ 0:2: lPB2 ðSx1 Sx2 Þ ¼ 0:1;lPB2 ðSx2 Sx3 Þ ¼ 0:1;lPB2 ðSx3 Sx4 Þ ¼ 0:1;lPB2 ðSx4 Sx1 Þ ¼ 0:1; lNB2 ðSx1 Sx2 ¼ 0:2;lNB2 ðSx2 Sx3 Þ ¼ 0:2;lNB2 ðSx3 Sx4 Þ ¼ 0:2;lNB2 ðSx4 Sx1 Þ ¼ 0:2:
h
Proposition 4.7 L(G) is a bipolar fuzzy line graph of some bipolar fuzzy graph G if and only if
Proof Assume that lPB2 ðSx Sy Þ ¼ minðlPA2 ðSx Þ; lPA2 ðSy ÞÞ for all Sx Sy 2 W: We define lPA1 ðxÞ ¼ lPA2 ðSx Þ for all x 2 E: Then lPB2 ðSx Sy Þ ¼ minðlPA2 ðSx Þ; lPA2 ðSy ÞÞ ¼ minðlPA1 ðxÞ; lPA1 ðyÞÞ; lNB2 ðSx Sy Þ ¼ maxðlNA2 ðSx Þ; lNA2 ðSy ÞÞ ¼ maxðlNA1 ðxÞ; lNA1 ðyÞÞ: A bipolar fuzzy set A1 ¼ ðlPA1 ; lNA1 Þ that yields that the property lPB1 ðxyÞ minðlPA1 ðxÞ; lPA1 ðyÞÞ; lNB1 ðxyÞ maxðlNA1 ðxÞ; lNA1 ðyÞÞ will suffice. The converse part is obvious.
h
Proposition 4.8 L(G) is a bipolar fuzzy line graph if and only if L(G*) is a line graph and lPB2 ðuvÞ ¼ minðlPA2 ðuÞ; lPA2 ðvÞÞ for all uv 2 W; lNB2 ðuvÞ ¼ maxðlNA2 ðuÞ; lNA2 ðvÞÞ for all uv 2 W: Proof
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Follows from Propositions 4.6 and 4.7.
h
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Definition 4.9 Let G1 = (A1, B1) and G2 = (A2, B2) be two bipolar fuzzy graphs. A homomorphism f: G1 ? G2 is a mapping f: V1 ? V2 such that (a) lPA1 ðx1 Þ lPA2 ðf ðx1 ÞÞ; lNA1 ðx1 Þ lNA2 ðf ðx1 ÞÞ; (b) lPB1 ðx1 y1 Þ lPB2 ðf ðx1 Þf ðy1 ÞÞ; lNB1 ðx1 y1 Þ lNB2 ðf ðx1 Þf ðy1 ÞÞ
lPA1 ðx1 Þ ¼ lPA2 ðf ðx1 ÞÞ;
ri ri
Now W ¼ fSx1 Sx2 ; Sx2 Sx3 ; . . .; Sxn Sx1 g: Also for rn?1 = r1, lPA2 ðSxi Þ ¼ lPA1 ðxi Þ ¼ lPA1 ðvi viþ1 Þ ¼ ri ; lNA2 ðSxi Þ ¼ lNA1 ðxi Þ ¼ lNA1 ðvi viþ1 Þ ¼ ri ;
lNA1 ðx1 Þ ¼ lNA2 ðf ðx1 ÞÞ;
lPB2 ðSxi Sxiþ1 Þ ¼ minðlPB1 ðxi Þ; lPB1 ðxiþ1 ÞÞ
is called a (weak) vertex-isomorphism.
¼ minðlPB1 ðvi viþ1 Þ; lPB1 ðviþ1 viþ2 ÞÞ
A bijective homomorphism f: G1 ? G2 such that (d)
¼ minðri ; riþ1 Þ;
lPB1 ðx1 y1 Þ ¼ lPB2 ðf ðx1 Þf ðy1 ÞÞ; lNB1 ðx1 y1 Þ ¼ lNB2 ðf ðx1 Þf ðy1 ÞÞ
lNB2 ðSxi Sxiþ1 Þ
for all x1 y1 2 V1 is called a (weak) line-isomorphism. If f is a (weak) vertex-isomorphism and a (weak) lineisomorphism of G1 onto G2, then f is called a (weak) isomorphism of G1 and G2. Proposition 4.10 Let G1 and G2 be bipolar fuzzy graphs. If f is a weak isomorphism of G1 onto G2, then f is an isomorphism of G*1 onto G*2. Proof
h
Obvious.
minðsi ; siþ1 Þ; maxðsi ; siþ1 Þ; i ¼ 1; 2; . . .; n:
Z ¼ fSx1 ; Sx1 ; Sx2 ; . . .; Sxn g
for all x1 2 V1 ; x1 y1 2 E1 : A bijective homomorphism with the property (c)
ðaÞ
¼ maxðlNB1 ðxi Þ; lNB1 ðxiþ1 ÞÞ ¼ maxðlNB1 ðvi viþ1 Þ; lNB1 ðviþ1 viþ2 ÞÞ ¼ maxð ri ; riþ1 Þ
for i ¼ 1; 2; . . .; n; vnþ1 ¼ v1 ; vnþ2 ¼ v2 : Since f is an isomorphism of G* onto L(G*), f maps V one-to-one and onto Z. Also f preserves adjacency. Hence f induces a permutation p of f1; 2; . . .; ng such that f ðvi Þ ¼ SxpðiÞ ¼ SxpðiÞ Sxpðiþ1Þ and
Theorem 4.11 Let L(G) = (A2, B2) be the bipolar fuzzy line graph corresponding to bipolar fuzzy graph G = (A1, B1). Suppose that G* = (V, E) is connected. Then
xi ¼ vi viþ1 ! f ðvi Þf ðviþ1 Þ ¼ SvpðiÞ Svpðiþ1Þ Svpðiþ2Þ ; i ¼ 1; 2; . . .; n 1:
(1)
Now
(2)
there exists a week isomorphism of G onto L(G) if and only if G* is a cyclic and for all v 2 V; x 2 E; lPA1 ðvÞ ¼ lPB1 ðxÞ; lNA1 ðvÞ ¼ lNB1 ðxÞ; i.e., A1 ¼ ðlPA1 ; lNA1 Þ and B1 ¼ ðlPB1 ; lNB1 Þ are constant functions on V and E, respectively, taking on the same value. If f is a weak isomorphism of G onto L(G), then f is an isomorphism.
si ¼ lPA1 ðvi Þ lPA2 ðf ðvi ÞÞ ¼ lPA2 ðSvpðiÞ vpðiþ1Þ Þ ¼ rpðiÞ ; si ¼ lNA1 ðvi Þ lNA2 ðf ðvi ÞÞ ¼ lNA2 ðSvpðiÞ vpðiþ1Þ Þ ¼ rpðiÞ ; ri ¼ lPB1 ðvi viþ1 ÞlPB2 ðf ðvi Þf ðviþ1 ÞÞ ¼ lPB2 ðSvpðiÞ SvpðiÞþ1 Svpðiþ1Þþ1 Þ ¼ minðlPB1 ðvpðiÞ vpðiÞþ1 Þ;lPB1 ðvpðiÞþ1 vpðiþ1Þþ1 ÞÞ
Proof Assume that f is a weak isomorphism of L(G) onto G. From Proposition 4.10, it follows that G* = (V, E) is a cycle. Let V ¼ fv1 ; v2 ; . . .; vn g and E ¼ fx1 ¼ v1 v2 ; x2 ¼ v2 v3 ; . . .; xn ¼ vn v1 g; where v1 v2 v3 . . .vn v1 is a cyclic. Define bipolar fuzzy sets
¼ minðrpðiÞ ;rpðiþ1Þ Þ; ri ¼ lNB1 ðvi viþ1 ÞlNB2 ðf ðvi Þf ðviþ1 ÞÞ ¼ lNB2 ðSvpðiÞ SvpðiÞþ1 Svpðiþ1Þþ1 Þ ¼ maxðlNB1 ðvpðiÞ vpðiÞþ1 Þ;lNB1 ðvpðiÞþ1 vpðiþ1Þþ1 ÞÞ ¼ maxð rpðiÞ ; rpðiþ1Þ Þ
lPA1 ðvi Þ ¼ si ; lNA1 ðvi Þ ¼ si and lPB1 ðvi viþ1 Þ
for i ¼ 1;2;...;n: That is, ¼
ri ; lNB1 ðvi viþ1 Þ
¼ ri ; i ¼ 1; 2; . . .; n; vnþ1 ¼ v1 :
Then for snþ1 ¼ s1 ; snþ1 ¼ s1 ;
si rpðiÞ ; si rp ðiÞ and
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ri minðrpðiÞ ; rpðiþ1Þ Þ; ri maxð rpðiÞ ; rpðiþ1Þ Þ:
By (b), we have ri rpðiÞ ; ri rpðiÞ for i ¼ 1; 2; . . .; n and so rp ðiÞ rpðpðiÞÞ ; rp ðiÞ rpðpðiÞÞ for i ¼ 1; 2; . . .; n: Continuing, we have ri rpðiÞ rpj ðiÞ ri ;
Rough graphs, (3) Soft fuzzy graphs, (4) Soft intuitionistic fuzzy graphs, and (5) Soft fuzzy hypergraphs. Acknowledgments The authors are thankful to the referees for their valuable comments and suggestions for improving the paper.
References
ri rpðiÞ rpj ðiÞ ri and so ri ¼ rpðiÞ ; ri ¼ rpðiÞ ; i ¼ 1; 2; . . .; n; where pj?1 is the identity map. Again, by (b), we have ri rpðiþ1Þ ¼ riþ1 ; i ¼ 1; 2; . . .; rnþ1 ¼ r1 ; ri rpðiþ1Þ ¼ riþ1 ; i ¼ 1; 2; . . .; rnþ1 ¼ r1 : Hence by (a) and (b), r1 ¼ ¼ rn ¼ s1 ¼ ¼ sn ; r1 ¼ ¼ rn ¼ s1 ¼ . . . ¼ sn : Thus, we have not only proved the conclusion about A1 and B1 being constant function but we have also shown that (2) holds. The converse part is obvious. h We state the following Theorem without proof. Theorem 4.12 Let G and H be bipolar fuzzy graphs of G* and H*, respectively, such that G* and H* are connected. Let L(G) and L(H) be the bipolar fuzzy line graphs corresponding to G and H, respectively. Suppose that it is not the case that one of G* and H* is complete graph K3 and other is bipartite complete graph K1,3. If L(G) and L(H) are isomorphic, then G and H are line-isomorphic.
5 Conclusions Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science, and technology. In computer science, graphs are used to represent networks of communication, data organization, computational devices, and the flow of computation. The bipolar fuzzy sets constitute a generalization of Zadeh’s fuzzy set theory. The bipolar fuzzy models give more precision, flexibility, and compatibility to the system as compared to the classical and fuzzy models. We have introduced the concept of regular and totally regular bipolar fuzzy graphs in this paper. The concept of bipolar fuzzy graphs can be applied in various areas of engineering, computer science: database theory, expert systems, neural networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, and medical diagnosis. We plan to extend our research of fuzzification to (1) Soft graphs, (2)
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1. Abiyev RH (2011) Fuzzy wavelet neural network based on fuzzy clustering and gradient techniques for time series prediction. Neural Comput Appl 20(2):249–259 2. Akram M (2011) Interval-valued fuzzy line graphs. Neural Comput Appl. doi:10.1007/s00521-011-0733-0 3. Akram M (2011) Bipolar fuzzy graphs. Inf Sci 181:5548–5564 4. Akram M (2011) Cofuzzy graphs. J Fuzzy Math 19(4):877–888 5. Akram M, Dudek WA (2011) Interval-valued fuzzy graphs. Comput Math Appl 61:289–299 6. Akram M, Dudek WA (2008) Intuitionistic fuzzy left k-ideals of semirings. Soft Comput 12:881–890 7. Alaoui A (1999) On fuzzification of some concepts of graphs. Fuzzy Sets Syst 101:363–389 8. Atanassov KT (1999) Intuitionistic fuzzy fets: theory and applications, studies in fuzziness and soft computing. Physica-Verl, Heidelberg, New York 9. Atanassov KT (1994) Index matrix representation of the intuitionistic fuzzy graphs. Preprint MRL-MFAIS-10-94, Sofia, pp 36–41 10. Atanassov KT, Pasi G, Yager R, Atanassova V (2003) Intuitionistic fuzzy graph interpretations of multi-person multi-criteria decision making, EUSFLAT conference’, pp 177–182 11. Bhattacharya P (1987) remarks on fuzzy graphs. Pattern Recognit Lett 6:297–302 12. Bhutani KR (1989) On automorphism of fuzzy graphs. Pattern Recognit Lett 9:159–162 13. Bhutani KR, Rosenfeld A (2003) Strong arcs in fuzzy graphs. Inf Sci 152:319–322 14. Bhutani KR, Rosenfeld A (2003) Fuzzy end nodes in fuzzy graphs. Inf Sci 152:323–326 15. Bhutani KR, Battou A (2003) On M-strong fuzzy graphs. Inf Sci 155:103–109 16. Bloch I (2007) Dilation and erosion of spatial bipolar fuzzy sets. In: Proceedings of WILF, pp 385–393 17. Bloch I (2009) Geometry of spatial bipolar fuzzy sets based on bipolar fuzzy numbers and mathematical morphology. Fuzzy Log Appl Lect Note Comput Sci 5571:237–245 18. Dubois D, Kaci S, Prade H (2004) Bipolarity in reasoning and decision, an introduction. International conference on information processing and management of uncertainty IPMU’04, pp 959– 966 19. Harary F (1972) Graph theory. 3rd edn. Addison-Wesley, Reading, MA 20. Huber KP, Berthold MR, Application of fuzzy graphs for metamodeling. Proceedings of the 2002 IEEE conference, pp 640–644 21. Kauffman A (1973) Introduction a la Theorie des Sous-emsembles Flous. Masson et Cie, vol 1 22. Kiss A (1991) An application of fuzzy graphs in database theory. Pure Math Appl 1(3–4):337–342 23. Koczy LT (1992) Fuzzy graphs in the evaluation and optimization of networks. Fuzzy Sets Syst 46:307–319 24. Lee K-M (2000) Bipolar-valued fuzzy sets and their basic operations. Proceedings of international conference, Bangkok, Thailand, pp 307–317
Neural Comput & Applic (2012) 21 (Suppl 1):S197–S205 25. Lee K-M (2004) Comparison of bipolar fuzzy sets, intuitionistic fuzzy sets, and bipolar-valued fuzzy sets. J Fuzzy Logic Intell Syst 14:125–129 26. Mathew S, Sunitha MS (2009) Types of arcs in a fuzzy graph. Inf Sci 179(11):1760–1768 27. Mathew S, Sunitha MS (2010) Node connectivity and arc connectivity of a fuzzy graph. Inf Sci 180(4):519–531 28. Mordeson JN (1993) Fuzzy line graphs. Pattern Recognit Lett 14:381–384 29. Mordeson JN, Nair PS (1998) Fuzzy graphs and fuzzy hypergraphs, 2nd edn, 2001. Physica Verlag, Heidelberg 30. Nagoor Gani A, Radha K (2008) On regular fuzzy graphs. J Phys Sci 12:33–40 31. Perchant A, Bloch I (2002) morphisms between graphs. Fuzzy Sets Syst 128:149–168 32. Parvathi R, Karunambigai MG, Atanassov KT (2009) Operations on intuitionistic fuzzy graphs. Fuzzy Systems, FUZZ-IEEE 2009. IEEE international conference, pp 1396–1401 33. Pasi G, Yager R, Atanassov KT (2004) Intuitionistic fuzzy graph interpretations of multi-person multi-criteria decision making: generalized net approach. Intelligent systems, proceedings, 2004 2nd international IEEE conference 2:434–439 34. Rosenfeld A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications, Academic Press, New York, pp 77–95 35. Shannon A, Atanassov KT (1994) A first step to a theory of the intuitionistic fuzzy graphs. In: Lakov D (ed) Proceeding of FUBEST, Sofia, Sept 28–30, pp 59–61
S205 36. Shannon A, Atanassov KT (1995) Intuitionistic fuzzy graphs from a-, b-, and (a, b)- levels. Note Intuitionistic Fuzzy Sets 1(1):32–35 37. Riff M, Zu´n¨iga M, Montero E (2010) A graph-based immuneinspired constraint satisfaction search. Neural Comput Appl 19(8):1133–1142 38. Yin Y, Zhan J, Corsini P (2011) Fuzzy roughness of n-ary hypergroups based on a complete residuated lattice. Neural Comput Appl 20(1):41–57 39. Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353 40. Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf Sci 3(2):177–200 41. Zadeh LA (2005) Toward a generalized theory of uncertainty (GTU)—an outline. Inf Sci 172:1–40 42. Zadeh LA (2005) From imprecise to granular probabilities. Fuzzy Sets Syst 154:370–374 43. Zhang W-R (1994) Bipolar fuzzy sets and relations: a computational framework forcognitive modeling and multiagent decision analysis, Proceedings of IEEE conference, pp 305–309 44. Zhang W-R (1998) Bipolar fuzzy sets. Proceedings of FUZZIEEE, pp 835–840 45. Zhang J, Yang X (2010) Some properties of fuzzy reasoning in propositional fuzzy logic systems. Inf Sci 180:4661–4671
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ORIGINAL ARTICLE
Regular bipolar fuzzy graphs Muhammad Akram • Wieslaw A. Dudek
Received: 14 July 2011 / Accepted: 30 November 2011 / Published online: 27 December 2011 Springer-Verlag London Limited 2011
Abstract We introduce the concepts of regular and totally regular bipolar fuzzy graphs. We prove necessary and sufficient condition under which regular bipolar fuzzy graph and totally bipolar fuzzy graph are equivalent. We introduce the notion of bipolar fuzzy line graphs and present some of their properties. We state a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We examine when an isomorphism between two bipolar fuzzy graphs follows from an isomorphism of their corresponding bipolar fuzzy line graphs. Keywords Bipolar fuzzy sets Bipolar fuzzy graphs Bipolar fuzzy line graphs Regular bipolar fuzzy graphs Totally regular bipolar fuzzy graphs
1 Introduction In 1736, Euler first introduced the notion of graph theory. In the history of mathematics, the solution given by Euler of the well-known Ko¨nigsberg bridge problem is considered to be the first theorem of graph theory. This has now become a subject generally regarded as a branch of combinatorics. The theory of graph is an extremely useful tool for solving combinatorial problems in different areas such
M. Akram (&) Punjab University College of Information Technology, University of the Punjab, Old Campus, Lahore 54000, Pakistan e-mail: [email protected]; [email protected] W. A. Dudek Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50-370 Wrocław, Poland e-mail: [email protected]
as geometry, algebra, number theory, topology, operations research, optimization, and computer science. In 1965, Zadeh [39] introduced the notion of a fuzzy subset of a set. 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, multiagent systems, pattern recognition, robotics, computer networks, expert systems, decision making, and automata theory. In 1994, Zhang [43, 44] 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 [25]. In many domains, it is important to be able to deal with bipolar information. 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. In particular, fuzzy and possibilistic formalisms for bipolar information have been proposed [18], because when we deal with spatial information in image processing or in spatial reasoning applications, this bipolarity also occurs. For instance, when we assess the position of an object in a space, we may have positive information expressed as a set of possible places and negative information expressed as a set of impossible
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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 that are considered neither to the right nor to the left of some reference object, thus leaving some room for indetermination. This corresponds to the idea that the union of positive and negative information does not cover the whole space. In 1975, Rosenfeld [34] discussed the concept of fuzzy graphs whose basic idea was introduced by Kauffmann [21] in 1973. The fuzzy relations between fuzzy sets were also considered by Rosenfeld, and he developed the structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts. Later on, Bhattacharya [11] gave some remarks on fuzzy graphs. The notion of fuzzy line graph was introduced by Mordeson in [28]. Bhutani and Rosenfeld introduced the concept of M-strong fuzzy graphs in [15] and studied some of their properties. The concept of strong arcs in fuzzy graphs was discussed in [13]. Shannon and Atanassov [35] introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs, and investigated some of their properties in [36]. Parvathi et al. defined operations on intuitionistic fuzzy graphs in [32]. Akram and Dudek [5] discussed intervalvalued fuzzy graphs. Recently, Akram has introduced the notions of bipolar fuzzy graphs in [3] and interval-valued fuzzy line graphs in [2]. In this paper, we introduce the concepts of regular and totally regular bipolar fuzzy graphs. We prove necessary and sufficient condition under which regular bipolar fuzzy graph and totally bipolar fuzzy graph are equivalent. We introduce the notion of bipolar fuzzy line graphs and present some of their properties. We give a necessary and sufficient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We examine when an isomorphism between two bipolar fuzzy graphs follows from an isomorphism of their corresponding bipolar fuzzy line graphs.ient condition for a bipolar fuzzy graph to be isomorphic to its corresponding bipolar fuzzy line graph. We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [1, 4, 7, 8, 20, 22, 23, 26, 27, 29]. 2 Preliminaries By graph, we mean a pair G* = (V, E), where V is the set and E is a relation on V. The elements of V are vertices of G* and the elements of E are edges of G*. We write xy 2 E
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to mean fx; yg 2 E; and if e ¼ xy 2 E; we say x and y are adjacent. Formally, given a graph G* = (V, E), two vertices x; y 2 V are said to be neighbors, or adjacent nodes, if xy 2 E: The neighborhood of a vertex v in a graph G* is the induced subgraph of G* consisting of all vertices adjacent to v and all edges connecting two such vertices. The neighborhood is often denoted N(v). The degree deg(v) of vertex v is the number of edges incident on v or equivalently, deg(v) = |N(v)|. The set of neighbors, called a (open) neighborhood N(v) for a vertex v in a graph G*, consists of all vertices adjacent to v but not including v, that is, NðvÞ ¼ fu 2 V j vu 2 Eg: When v is also included, it is called a closed neighborhood N[v], that is, N[v] = N(v) [ {v}. A regular graph is a graph where each vertex has the same number of neighbors, i.e., all the vertices have the same open neighborhood degree. A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. An isomorphism of graphs G*1 and G*2 is a bijection between the vertex sets of G*1 and G*2 such that any two 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. Isomorphic graphs are denoted by G*1 ^ G*2. In graph theory, the line graph L(G*) of a simple graph * G is another graph L(G*) that represents the adjacencies between edges of G*. Given a graph G*, its line graph L(G*) is a graph such that: • •
each vertex of L(G*) represents an edge of G*; and two vertices of L(G*) are adjacent if and only if their corresponding edges share a common endpoint (‘‘are adjacent’’) in G*.
Let G* = (V, E) be an undirected graph, where V ¼ fv1 ; v2 ; . . .; vn g: Let Si ¼ fvi ; xi1 ; . . .; xiqi g where xij 2 E has vertex vi ; i ¼ 1; 2; . . .; n; j ¼ 1; 2; . . .; qi : Let S ¼ fS1 ; S2 ; . . .; Sn g: Let T ¼ fSi Sj jSi ; Sj 2 S; Si \ Sj 6¼ ;; i 6¼ jg: Then P(S) = (S, T) is an intersection graph and P(S) = G*. The line graph L(G*) is by definition the intersection graph P(E). That is, L(G*) = (Z, W) where Z ¼ ffxg [ fux ; vx gjx 2 E; ux ; vx 2 V; x ¼ ux vx g and W ¼ fSx Sy jSx \ Sy 6¼ ;; x; y 2 E; x 6¼ yg; and Sx ¼ fxg[ fux ; vx g; x 2 E: Proposition 2.1 If G is regular of degree k, graph L(G) is regular of degree 2k - 2.
then line
Definition 2.2 [39, 40] A fuzzy subset l on a set X is a map l: X? [0, 1]. A map m: X 9 X? [0, 1] is called a fuzzy relation on X if m(x, y) B min(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 2.3 Let X be a non-empty set. A bipolar fuzzy set B in X is an object having the form
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B ¼ fðx; lPB ðxÞ; lNB ðxÞÞjx 2 Xg where lPB : X ! ½0; 1 and lNB : X ! ½1; 0 are mappings. We use the positive membership degree lPB (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 lNB (x) to denote the satisfaction degree of an element x to some implicit counter-property corresponding to a bipolar fuzzy set B. If lPB ðxÞ 6¼ 0 and lNB (x) = 0, it is the situation that x is regarded as having only positive satisfaction for B. If lPB (x) = 0 and lNB ðxÞ 6¼ 0; it is the situation that x does not satisfy the property of B, but somewhat satisfies the counter-property of B. It is possible for an element x to be such that lPB ðxÞ 6¼ 0 and lNB ðxÞ 6¼ 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 = (lPB, lNB ) for the bipolar fuzzy set
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B is a symmetric 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 lPB ðxyÞ minðlPA ðxÞ; lPA ðyÞÞ and lNB ðxyÞ maxðlNA ðxÞ; lNA ðyÞÞ for all xy 2 E: Example 3.2
Consider the bipolar fuzzy graph.
B ¼ fðx; lPB ðxÞ; lNB ðxÞÞjx 2 Xg: A nice application of bipolar fuzzy concept is a political acceptation (map to [0, 1]) and non-acceptation (map to [-1, 0]). Definition 2.4 [43] Let X be a non-empty set. Then we call a mapping A = (lPA, lNA ): X 9 X ? [-1, 1] 9 [-1, 1] a bipolar fuzzy relation on X such that lPA ðx; yÞ 2 ½0; 1 and lNA ðx; yÞ 2 ½1; 0: Definition 2.5 [43] Let A = (lPA, lNA) and B = (lPB, lNB) be bipolar fuzzy sets on a set X. If A = (lPA, lNA) is a bipolar fuzzy relation on a set X, then A = (lPA, lNA ) is called a bipolar fuzzy relation on B = (lPB, lNB) if lPA(x, y) B min (lPB(x), lPB(y)) and lNA (x, y) C max(lNB (x), lNB(y)) for all x; y 2 X: A bipolar fuzzy relation A on X is called symmetric if lPA(x, y) = lPA(y, x) and lNA(x, y) = lNA(y, x) for all x; y 2 X: 3 Regular bipolar fuzzy graphs Throughout this paper, G* will be a crisp graph and G a bipolar fuzzy graph. Definition 3.1 [3] By a bipolar fuzzy graph, we mean a pair G = (A, B) where A = (lPA, lNA ) is a bipolar fuzzy set in V and B = (lPB, lNB ) is bipolar relation on V such that lPB ðfx; ygÞ minðlPA ðxÞ; lPA ðyÞÞ maxðlNA ðxÞ; lNA ðyÞÞ
and
lNB ðfx; ygÞ
for all fx; yg 2 E: We call A the bipolar fuzzy vertex set of V, B the bipolar fuzzy edge set of E, respectively. Note that
Definition 3.3 Let G = (A, B) be a bipolar fuzzy graph on G*. If all the vertices have the same open neighborhood degree n, then G is called an n-regular bipolar fuzzy graph. The open neighborhood degree of a vertex x in G is defined by deg(x) = (degP (x), degN (x)), where degP ðxÞ ¼ P P N P N x2V lA ðxÞ and deg ðxÞ ¼ x2V lA ðxÞ: Definition 3.4 Let G = (A, B) be a regular bipolar fuzzy graph. The order of a regular bipolar fuzzy graph G is P P OðGÞ ¼ ð x2V lPA ðxÞ; x2V lNA ðxÞÞ: The size of a regular P bipolar fuzzy graph G is SðGÞ ¼ ð xy2V lPA ðxyÞ; P N xy2E lA ðxyÞÞ: Definition 3.5 Let G = (A, B) be a bipolar fuzzy graph. If each vertex of G has same closed neighborhood degree m, then G is called a totally regular bipolar fuzzy graph. The closed neighborhood degree of a vertex x is defined by deg[x] = (degP [x], degN [x]), where degP ½x ¼ degP ðxÞ þ lPA ðxÞ; degN ½x ¼ degN ðxÞ þ lNA ðxÞ: We show with the following examples that there is no relationship between n-regular bipolar fuzzy graph and mtotally regular bipolar fuzzy graph. Example 3.6 Consider a graph G* such that V = {a, b, c, d}, E = {ab, bc, cd, ad}. Let A be a bipolar
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fuzzy subset of V and let B be a bipolar fuzzy subset of E defined by
Routine computations show that bipolar fuzzy graph G is both regular and totally regular. The following propositions are obvious.
a lPA lNA
b
c
d
0.5
0.5
0.5
0.5
-0.3
-0.3
-0.3
-0.3
ab
bc
cd
da
lPB
0.2
0.4
0.2
0.4
lNB
-0.1
-0.1
-0.1
-0.1
Proposition 3.9 The size of a n-regular bipolar fuzzy graph G is nk2 ; where |V| = k. Proposition 3.10 If G is a m-totally regular bipolar fuzzy graph G, then 2S(G) ? O(G) = mk, where |V| = k. Proposition 3.11 If G is a both n-regular and m-totally regular bipolar fuzzy graph G, then O(G) = k(m n), where |V| = k. Definition 3.12 A bipolar fuzzy graph G = (A, B) is called complete if lPB ðxyÞ ¼ minðlPA ðxÞ; lPA ðyÞÞ and lNB ðxyÞ ¼ maxðlNA ðxÞ; lNA ðyÞÞ for all x; y 2 V: Example 3.13 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 bipolar fuzzy subset of E defined by lPA ðxÞ ¼ 0:5; lPA ðyÞ ¼ 0:7; lPA ðzÞ ¼ 0:6; lNA ðxÞ ¼ 0:3; lNA ðyÞ ¼ 0:4; lNA ðzÞ ¼ 0:5; lPA ðxyÞ ¼ 0:5; lPA ðyzÞ ¼ 0:6; lPA ðzxÞ ¼ 0:5; lNA ðxyÞ ¼ 0:3; lNA ðyzÞ ¼ 0:4; lNA ðzxÞ ¼ 0:3:
Routine computations show that a bipolar fuzzy graph G is both regular and totally regular. Example 3.7 Consider a graph G* such that V = {v1, v2, v3}, E = {v1v2, v1v3}. Let A be a bipolar fuzzy subset of V and let B be a bipolar fuzzy subset of E defined by lPA ðv1 Þ ¼ 0:4; lPA ðv2 Þ ¼ 0:8; lPA ðv3 Þ ¼ 0:7; lNA ðv1 Þ ¼ 0:4; lNA ðv2 Þ ¼ 0:7; lNA ðv3 Þ ¼ 0:6; lPB ðv1 v2 Þ ¼ 0:3; lPB ðv1 v3 Þ ¼ 0:4; lNB ðv1 v2 Þ ¼ 0:2; lNB ðv1 v3 Þ ¼ 0:2: Routine computations show that a bipolar fuzzy graph G is neither totally regular nor regular. Example 3.8 Consider a graph G* such that V = {v1, v2, v3}, E = {v1v2, v2v3, v3v1}. Let A be a bipolar fuzzy subset of V and let B be a bipolar fuzzy subset of E defined by lPA ðv1 Þ ¼ 0:4; lPA ðv2 Þ ¼ 0:4; lPA ðv3 Þ ¼ 0:4; lNA ðv1 Þ ¼ 0:4; lNA ðv2 Þ ¼ 0:4; lNA ðv3 Þ ¼ 0:4; lPB ðv1 v2 Þ ¼ 0:3; lPB ðv2 v3 Þ ¼ 0:3; lPB ðv3 v1 Þ ¼ 0:3; lNB ðv1 v2 Þ ¼ 0:2; lNB ðv2 v3 Þ ¼ 0:2; lNB ðv3 v1 Þ ¼ 0:2:
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Routine computations show that G is a both complete and totally regular bipolar fuzzy graph, but G is not regular since deg(x) = deg(z) = deg(y). Theorem 3.14 Every complete bipolar fuzzy graph is a totally regular bipolar fuzzy graph. Theorem 3.15 Let G = (A, B) be a bipolar fuzzy graph of a graph G*. Then A = (lPA, lNA ) is a constant function if and only if the following are equivalent: (a) G is a regular bipolar fuzzy graph, (b) G is a totally regular bipolar fuzzy graph.
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Proof Suppose that A = (lPA, lNA ) is constant function. Let lPA (x) = c1 and lNA (x) = c2 for all x 2 V: (a))(b): assume that G is n-regular bipolar fuzzy graph, then degP (x) = n1 and degN (x) = n2 for all x 2 V: So degP ½x ¼ degP ðxÞ þ lPA ðxÞ; degN ½x ¼ degN ðxÞ þ lNA ðxÞ for all x 2 V; Thus degP ½x ¼ n1 þ c1 ; degN ½x ¼ n2 þ c2
for all x 2 V:
Hence, G is a totally regular bipolar fuzzy graph. (b) ) (a): suppose that G is a totally regular bipolar fuzzy graph, then degP ½x ¼ k1 ; degN ½x ¼ k2
for all x 2 V
4 Bipolar fuzzy line graphs Definition 4.1 Let P(S) = (S, T) be an intersection graph of a simple graph G* = (V, E). Let G = (A1, B1) be a bipolar fuzzy graph of G*. We define a bipolar fuzzy intersection graph P(G) = (A2, B2) of P(S) as follows: (1) (2) (3)
A2, B2 are bipolar fuzzy sets of S and T, respectively, lPA2 ðSi Þ ¼ lPB1 ðvi Þ; lNA2 ðSi Þ ¼ lNB1 ðvi Þ; lPB2 ðSi Sj Þ ¼ lPB1 ðvi vj Þ; lNB2 ðSi Sj Þ ¼ lNB1 ðvi vj Þ
for all Si ; Sj 2 S; Si Sj 2 T: That is, any bipolar fuzzy graph of P(S) is called a bipolar fuzzy intersection graph. The following Proposition is obvious.
or
Proposition 4.2 Let G = (A1, B1) be a bipolar fuzzy graph of G*, then
degP ðxÞ þ lPA ðxÞ ¼ k1 ; degN ðxÞ þ lNA ðxÞ ¼ k2 for all x 2 V
• •
or
This Proposition shows that any bipolar fuzzy graph is isomorphic to a bipolar fuzzy intersection graph.
degP ðxÞ þ c1 ¼ k1 ; degN ðxÞ þ c2 ¼ k2
for all x 2 V
or degP ðxÞ ¼ k1 c1 ; degN ðxÞ ¼ k2 c2
for all x 2 V:
Thus, G is a regular bipolar fuzzy graph. Hence, (a) and (b) are equivalent. The converse part is obvious. h Proposition 3.16 If a bipolar fuzzy graph G is both regular and totally regular, then A = (lPA, lNA) is constant function. Proof Let G be a regular and totally regular bipolar fuzzy graph, then degP ðxÞ ¼ n1 ; degN ðxÞ ¼ n2 P
N
deg ½x ¼ k1 ; deg ½x ¼ k2
for all x 2 V;
Definition 4.3 Let L(G*) = (Z, W) be a line graph of a simple graph G* = (V, E). Let G = (A1, B1) be a bipolar fuzzy graph of G*. We define a bipolar fuzzy line graph L(G) = (A2, B2) of G as follows: (4) (5) (6) (7) (8)
Example 4.4 Consider a graph G* = (V, E) such that V = {v1, v2, v3, v4} and E ¼ fx1 ¼ v1 v2 ; x2 ¼ v2 v3 ; x3 ¼ v3 v4 ; x4 ¼ v4 v1 g: Let A1 be a bipolar fuzzy subset of V and let B1 be a bipolar fuzzy subset of E defined by
Now degP ½x ¼ k1 , degP ðxÞ þ lPA ðxÞ ¼ k1 , n1 þ lPA ðxÞ ¼ k1
v1
for all x 2 V:
Likewise, lNA (x) = k2 - n2 for A = (lPA, lNA ) is constant function.
all
A2 and B2 are bipolar fuzzy sets of Z and W, respectively, lPA2 ðSx Þ ¼ lPB1 ðxÞ ¼ lPB1 ðux vx Þ; lNA2 ðSx Þ ¼ lNB1 ðxÞ ¼ lNB1 ðux vx Þ; lPB2 ðSx Sy Þ ¼ minðlPB1 ðxÞ; lPB1 ðyÞÞ; lNB2 ðSx Sy Þ ¼ maxðlNB1 ðxÞ; lNB1 ðyÞÞ
for all Sx ; Sy 2 Z; Sx Sy 2 W:
for all x 2 V:
, lPA ðxÞ ¼ k1 n1
P(G) = (A2, B2) is a bipolar fuzzy graph of P(S), G^ P(G).
x 2 V:
Hence, h
v2
v3
v4
lPA1
0.2
0.3
0.4
0.2
lNA1
-0.5
-0.4
-0.5
-0.3
x1
x2
x3
x4
Remark The converse of Proposition 3.16 may not be true, in general. We state the following Theorem without its proof. Theorem 3.17 Let G be a bipolar fuzzy graph where crisp graph G* is an odd cycle. Then G is regular bipolar fuzzy graph if and only if B is a constant function.
lPB1
0.1
0.2
0.1
0.1
lNB1
-0.2
-0.3
-0.2
-0.2
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By routine computations, it is clear that L(G) is a bipolar fuzzy line graph. It is neither regular bipolar fuzzy line graph nor totally regular bipolar fuzzy line graph. Proposition 4.5 L(G) is a bipolar fuzzy line graph corresponding to bipolar fuzzy graph G. Proof
Obvious.
h
Proposition 4.6 If L(G) is a bipolar fuzzy line graph of bipolar fuzzy graph G. Then L(G*) is the line graph of G*. Proof Since G = (A1, B1) is a bipolar fuzzy graph and L(G) is a bipolar fuzzy line graph, lPA1 ðSx Þ ¼ lPB1 ðxÞ; lNA1 ðSx Þ ¼ lNB1 ðxÞ for all x 2 E By routine computations, it is easy to see that G is a bipolar fuzzy graph. Consider a line graph L(G*) = (Z, W) such that
and so Sx 2 Z , x 2 E: Also lPB2 ðSx Sy Þ ¼ minðlPB1 ðxÞ; lPB1 ðyÞÞ;
Z ¼ fSx1 ; Sx2 ; Sx3 ; Sx4 g
lNB2 ðSx Sy Þ ¼ maxðlNB1 ðxÞ; lNB1 ðyÞÞ
and
for all Sx ; Sy 2 Z; and so
W ¼ fSx1 Sx2 ; Sx2 Sx3 ; Sx3 Sx4 ; Sx4 Sx1 g:
W ¼ fSx Sy jSx \ Sy 6¼ ;; x; y 2 E; x 6¼ yg:
Let A2 ¼ ðlPA2 ; lNA2 Þ and B2 ¼ ðlPB2 ; lNB2 Þ be bipolar fuzzy sets of Z and W, respectively. Then, by routine computations, we have
This completes the proof.
lPA2 ðSx1 Þ ¼ 0:1;lPA2 ðSx2 Þ ¼ 0:2;lPA2 ðSx3 Þ ¼ 0:1;lPA2 ðSx4 Þ ¼ 0:1;
lPB2 ðSx Sy Þ ¼ minðlPA2 ðSx Þ; lPA2 ðSy ÞÞ for all Sx Sy 2 W;
lNA2 ðSx1 Þ ¼ 0:2;lNA2 ðSx2 Þ ¼ 0:3;lNA2 ðSx3 Þ
lNB2 ðSx Sy Þ ¼ maxðlNA2 ðSx Þ; lNA2 ðSy ÞÞ for all Sx Sy 2 W:
¼ 0:2;lNA2 ðSx4 Þ ¼ 0:2: lPB2 ðSx1 Sx2 Þ ¼ 0:1;lPB2 ðSx2 Sx3 Þ ¼ 0:1;lPB2 ðSx3 Sx4 Þ ¼ 0:1;lPB2 ðSx4 Sx1 Þ ¼ 0:1; lNB2 ðSx1 Sx2 ¼ 0:2;lNB2 ðSx2 Sx3 Þ ¼ 0:2;lNB2 ðSx3 Sx4 Þ ¼ 0:2;lNB2 ðSx4 Sx1 Þ ¼ 0:2:
h
Proposition 4.7 L(G) is a bipolar fuzzy line graph of some bipolar fuzzy graph G if and only if
Proof Assume that lPB2 ðSx Sy Þ ¼ minðlPA2 ðSx Þ; lPA2 ðSy ÞÞ for all Sx Sy 2 W: We define lPA1 ðxÞ ¼ lPA2 ðSx Þ for all x 2 E: Then lPB2 ðSx Sy Þ ¼ minðlPA2 ðSx Þ; lPA2 ðSy ÞÞ ¼ minðlPA1 ðxÞ; lPA1 ðyÞÞ; lNB2 ðSx Sy Þ ¼ maxðlNA2 ðSx Þ; lNA2 ðSy ÞÞ ¼ maxðlNA1 ðxÞ; lNA1 ðyÞÞ: A bipolar fuzzy set A1 ¼ ðlPA1 ; lNA1 Þ that yields that the property lPB1 ðxyÞ minðlPA1 ðxÞ; lPA1 ðyÞÞ; lNB1 ðxyÞ maxðlNA1 ðxÞ; lNA1 ðyÞÞ will suffice. The converse part is obvious.
h
Proposition 4.8 L(G) is a bipolar fuzzy line graph if and only if L(G*) is a line graph and lPB2 ðuvÞ ¼ minðlPA2 ðuÞ; lPA2 ðvÞÞ for all uv 2 W; lNB2 ðuvÞ ¼ maxðlNA2 ðuÞ; lNA2 ðvÞÞ for all uv 2 W: Proof
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Follows from Propositions 4.6 and 4.7.
h
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Definition 4.9 Let G1 = (A1, B1) and G2 = (A2, B2) be two bipolar fuzzy graphs. A homomorphism f: G1 ? G2 is a mapping f: V1 ? V2 such that (a) lPA1 ðx1 Þ lPA2 ðf ðx1 ÞÞ; lNA1 ðx1 Þ lNA2 ðf ðx1 ÞÞ; (b) lPB1 ðx1 y1 Þ lPB2 ðf ðx1 Þf ðy1 ÞÞ; lNB1 ðx1 y1 Þ lNB2 ðf ðx1 Þf ðy1 ÞÞ
lPA1 ðx1 Þ ¼ lPA2 ðf ðx1 ÞÞ;
ri ri
Now W ¼ fSx1 Sx2 ; Sx2 Sx3 ; . . .; Sxn Sx1 g: Also for rn?1 = r1, lPA2 ðSxi Þ ¼ lPA1 ðxi Þ ¼ lPA1 ðvi viþ1 Þ ¼ ri ; lNA2 ðSxi Þ ¼ lNA1 ðxi Þ ¼ lNA1 ðvi viþ1 Þ ¼ ri ;
lNA1 ðx1 Þ ¼ lNA2 ðf ðx1 ÞÞ;
lPB2 ðSxi Sxiþ1 Þ ¼ minðlPB1 ðxi Þ; lPB1 ðxiþ1 ÞÞ
is called a (weak) vertex-isomorphism.
¼ minðlPB1 ðvi viþ1 Þ; lPB1 ðviþ1 viþ2 ÞÞ
A bijective homomorphism f: G1 ? G2 such that (d)
¼ minðri ; riþ1 Þ;
lPB1 ðx1 y1 Þ ¼ lPB2 ðf ðx1 Þf ðy1 ÞÞ; lNB1 ðx1 y1 Þ ¼ lNB2 ðf ðx1 Þf ðy1 ÞÞ
lNB2 ðSxi Sxiþ1 Þ
for all x1 y1 2 V1 is called a (weak) line-isomorphism. If f is a (weak) vertex-isomorphism and a (weak) lineisomorphism of G1 onto G2, then f is called a (weak) isomorphism of G1 and G2. Proposition 4.10 Let G1 and G2 be bipolar fuzzy graphs. If f is a weak isomorphism of G1 onto G2, then f is an isomorphism of G*1 onto G*2. Proof
h
Obvious.
minðsi ; siþ1 Þ; maxðsi ; siþ1 Þ; i ¼ 1; 2; . . .; n:
Z ¼ fSx1 ; Sx1 ; Sx2 ; . . .; Sxn g
for all x1 2 V1 ; x1 y1 2 E1 : A bijective homomorphism with the property (c)
ðaÞ
¼ maxðlNB1 ðxi Þ; lNB1 ðxiþ1 ÞÞ ¼ maxðlNB1 ðvi viþ1 Þ; lNB1 ðviþ1 viþ2 ÞÞ ¼ maxð ri ; riþ1 Þ
for i ¼ 1; 2; . . .; n; vnþ1 ¼ v1 ; vnþ2 ¼ v2 : Since f is an isomorphism of G* onto L(G*), f maps V one-to-one and onto Z. Also f preserves adjacency. Hence f induces a permutation p of f1; 2; . . .; ng such that f ðvi Þ ¼ SxpðiÞ ¼ SxpðiÞ Sxpðiþ1Þ and
Theorem 4.11 Let L(G) = (A2, B2) be the bipolar fuzzy line graph corresponding to bipolar fuzzy graph G = (A1, B1). Suppose that G* = (V, E) is connected. Then
xi ¼ vi viþ1 ! f ðvi Þf ðviþ1 Þ ¼ SvpðiÞ Svpðiþ1Þ Svpðiþ2Þ ; i ¼ 1; 2; . . .; n 1:
(1)
Now
(2)
there exists a week isomorphism of G onto L(G) if and only if G* is a cyclic and for all v 2 V; x 2 E; lPA1 ðvÞ ¼ lPB1 ðxÞ; lNA1 ðvÞ ¼ lNB1 ðxÞ; i.e., A1 ¼ ðlPA1 ; lNA1 Þ and B1 ¼ ðlPB1 ; lNB1 Þ are constant functions on V and E, respectively, taking on the same value. If f is a weak isomorphism of G onto L(G), then f is an isomorphism.
si ¼ lPA1 ðvi Þ lPA2 ðf ðvi ÞÞ ¼ lPA2 ðSvpðiÞ vpðiþ1Þ Þ ¼ rpðiÞ ; si ¼ lNA1 ðvi Þ lNA2 ðf ðvi ÞÞ ¼ lNA2 ðSvpðiÞ vpðiþ1Þ Þ ¼ rpðiÞ ; ri ¼ lPB1 ðvi viþ1 ÞlPB2 ðf ðvi Þf ðviþ1 ÞÞ ¼ lPB2 ðSvpðiÞ SvpðiÞþ1 Svpðiþ1Þþ1 Þ ¼ minðlPB1 ðvpðiÞ vpðiÞþ1 Þ;lPB1 ðvpðiÞþ1 vpðiþ1Þþ1 ÞÞ
Proof Assume that f is a weak isomorphism of L(G) onto G. From Proposition 4.10, it follows that G* = (V, E) is a cycle. Let V ¼ fv1 ; v2 ; . . .; vn g and E ¼ fx1 ¼ v1 v2 ; x2 ¼ v2 v3 ; . . .; xn ¼ vn v1 g; where v1 v2 v3 . . .vn v1 is a cyclic. Define bipolar fuzzy sets
¼ minðrpðiÞ ;rpðiþ1Þ Þ; ri ¼ lNB1 ðvi viþ1 ÞlNB2 ðf ðvi Þf ðviþ1 ÞÞ ¼ lNB2 ðSvpðiÞ SvpðiÞþ1 Svpðiþ1Þþ1 Þ ¼ maxðlNB1 ðvpðiÞ vpðiÞþ1 Þ;lNB1 ðvpðiÞþ1 vpðiþ1Þþ1 ÞÞ ¼ maxð rpðiÞ ; rpðiþ1Þ Þ
lPA1 ðvi Þ ¼ si ; lNA1 ðvi Þ ¼ si and lPB1 ðvi viþ1 Þ
for i ¼ 1;2;...;n: That is, ¼
ri ; lNB1 ðvi viþ1 Þ
¼ ri ; i ¼ 1; 2; . . .; n; vnþ1 ¼ v1 :
Then for snþ1 ¼ s1 ; snþ1 ¼ s1 ;
si rpðiÞ ; si rp ðiÞ and
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ri minðrpðiÞ ; rpðiþ1Þ Þ; ri maxð rpðiÞ ; rpðiþ1Þ Þ:
By (b), we have ri rpðiÞ ; ri rpðiÞ for i ¼ 1; 2; . . .; n and so rp ðiÞ rpðpðiÞÞ ; rp ðiÞ rpðpðiÞÞ for i ¼ 1; 2; . . .; n: Continuing, we have ri rpðiÞ rpj ðiÞ ri ;
Rough graphs, (3) Soft fuzzy graphs, (4) Soft intuitionistic fuzzy graphs, and (5) Soft fuzzy hypergraphs. Acknowledgments The authors are thankful to the referees for their valuable comments and suggestions for improving the paper.
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5 Conclusions Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science, and technology. In computer science, graphs are used to represent networks of communication, data organization, computational devices, and the flow of computation. The bipolar fuzzy sets constitute a generalization of Zadeh’s fuzzy set theory. The bipolar fuzzy models give more precision, flexibility, and compatibility to the system as compared to the classical and fuzzy models. We have introduced the concept of regular and totally regular bipolar fuzzy graphs in this paper. The concept of bipolar fuzzy graphs can be applied in various areas of engineering, computer science: database theory, expert systems, neural networks, artificial intelligence, signal processing, pattern recognition, robotics, computer networks, and medical diagnosis. We plan to extend our research of fuzzification to (1) Soft graphs, (2)
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