#### Abstract

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

#### 1. Introduction

Graph theory is an extremely useful tool in solving the combinatorial problems in different areas. Point-to-point interconnection networks for parallel and distributed systems are usually modeled by *directed graphs* (or digraphs). A digraph is a graph whose edges have direction and are called *arcs* (edges). Arrows on the arcs are used to encode the directional information: an arc from vertex (node) to vertex indicates that one may move from to but not from to . The Cayley graph was first considered for finite groups by Cayley in 1878. Max Dehn in his unpublished lectures on group theory from 1909 to 1910 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point [1].

The notion of fuzzy sets was introduced by Zadeh [2] as a method of representing uncertainty and vagueness. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines. In 1994, Zhang [3] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets [4]. A bipolar fuzzy set is an extension of fuzzy sets whose membership degree range is . In a bipolar fuzzy set, the membership degree of an element means that the element is irrelevant to the corresponding property, the membership degree of an element indicates that the element somewhat satisfies the property, and the membership degree of an element indicates that the element somewhat satisfies the implicit counterproperty.

Kaufmann's initial definition of a fuzzy graph [5] was based on Zadeh's fuzzy relations [2]. Mordeson and Nair [6] introduced the fuzzy analogue of several basic graph-theoretic concepts. Kauffman [5] defined the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. Akram and Dudek [7–9] introduced many new concepts, including bipolar fuzzy graphs, complete bipolar fuzzy graphs, regular bipolar fuzzy graphs, and irregular bipolar fuzzy graphs. Wu [4] discussed fuzzy digraphs. Shahzamanian et al. [10] introduced the notion of roughness in Cayley graphs and investigated several properties. Namboothiri et al. [11] discussed Cayley fuzzy graphs. In this paper, we introduce the concept of Cayley bipolar fuzzy graphs and investigate some of their properties. We present some interesting properties of bipolar fuzzy graphs in terms of algebraic structures. We also discuss connectedness in Cayley bipolar fuzzy graphs.

#### 2. Preliminaries

A digraph is a pair , where is a finite set and . Let and be two digraphs. The *Cartesian product* of and gives a digraph with and
In this paper, we will write to mean , and if , we say and are *adjacent* such that is a starting node and is an ending node.

The study of vertex transitive graphs has a long and rich history in discrete mathematics. Prominent examples of vertex transitive graphs are Cayley graphs which are important in both theory and applications; for example, Cayley graphs are good models for interconnection networks.

*Definition 1. *Let be a finite group and let be a minimal generating set of . A Cayley graph has elements of as its vertices; the edge set is given by . Two vertices and are adjacent if , where . Note that a generating set is minimal if generates but no proper subset of does.

Theorem 2. *All Cayley graphs are vertex transitive.*

*Definition 3. *Let be a group and let be any subset of . Then the Cayley graph induced by is the graph , where .

*Definition 4 (see [2]). *A fuzzy subset on a set is a map . A *fuzzy binary relation* on is a fuzzy subset on . By a fuzzy relation we mean a fuzzy binary relation given by .

*Definition 5 (see [11]). *Let be a group and let be a fuzzy subset of . Then the fuzzy relation on defined by
induces a fuzzy graph , called the *Cayley fuzzy graph* induced by the .

*Definition 6 (see [3]). *Let be a nonempty set. A *bipolar fuzzy set * in is an object having the form
where and are mappings.

We use the positive membership degree to denote the satisfaction degree of an element to the property corresponding to a bipolar fuzzy set and the negative membership degree to denote the satisfaction degree of an element to some implicit counterproperty corresponding to a bipolar fuzzy set . If and , it is the situation that is regarded as having only positive satisfaction for . If and , it is the situation that does not satisfy the property of but somewhat satisfies the counterproperty of . It is possible for an element to be such that and when the membership function of the property overlaps that of its counterpro perty over some portion of .

For the sake of simplicity, we shall use the symbol for the bipolar fuzzy set A nice application of bipolar fuzzy concept is a political acceptation (map to [0, 1]) and nonacceptation (map to ).

*Definition 7 (see [3]). *A bipolar fuzzy relation in a universe (, for short) is a bipolar fuzzy set of the form
where and .

*Definition 8. *Let be a bipolar fuzzy relation on universe . Then is called *a bipolar fuzzy equivalence relation* on if it satisfies the following conditions: (a)is bipolar fuzzy reflexive; that is, = (1, −1) for each ;(b) is bipolar fuzzy symmetric; that is, for any , ;(c) is bipolar fuzzy transitive; that is, .

*Definition 9. *Let be a bipolar fuzzy relation on universe . Then is called *a bipolar fuzzy partial order relation* on if it satisfies the following conditions:(a) is bipolar fuzzy reflexive; that is, for each ;(b) is bipolar fuzzy antisymmetric; that is, for any , ;(c) is bipolar fuzzy transitive; that is, .

*Definition 10. *Let be a bipolar fuzzy relation on universe . Then is called *a bipolar fuzzy linear order relation* on if it satisfies the following conditions:(a) is bipolar fuzzy partial relation;(b), for all , .

#### 3. Cayley Bipolar Fuzzy Graphs

*Definition 11. *A *bipolar fuzzy digraph* of a digraph is a pair where is a bipolar fuzzy set in and is a bipolar fuzzy relation on such that
for all . We note that need not to be symmetric.

*Definition 12. *Let be a bipolar fuzzy digraph. The indegree of a vertex in is defined by , where and . Similarly, the outdegree of a vertex in is defined by , where and . A bipolar fuzzy digraph in which each vertex has the same outdegree is called *an outregular digraph* with index of outregularity . In-regular digraphs are defined similarly.

*Definition 13. *Let be a group and let be the bipolar fuzzy subset of . Then the bipolar fuzzy relation defined on by
induces a bipolar fuzzy graph called the Cayley bipolar fuzzy graph induced by the .

We now introduce Cayley bipolar fuzzy graphs and prove that all Cayley bipolar fuzzy graphs are regular.

*Definition 14. *Let be a group and let be a bipolar fuzzy subset of . Then the bipolar fuzzy relation on defined by
induces a bipolar fuzzy graph , called the *Cayley bipolar fuzzy graph* induced by the .

*Example 15. *Consider the group and take . Define and by , . Then the Cayley bipolar fuzzy graph induced by is given by Table 1 and Figure 1.

We see that Cayley bipolar fuzzy graphs are actually bipolar fuzzy digraphs. Furthermore, the relation in the above definition describes the strength of each directed edge. Let denote a bipolar fuzzy graph induced by the triple .

Theorem 16. *The Cayley bipolar fuzzy graph is vertex transitive.*

*Proof. *Let ,. Define by for all . Clearly, is a bijective map. For each , ,
Therefore, . Hence is an automorphism on . Also . Hence is vertex transitive.

Theorem 17. *Every vertex transitive bipolar fuzzy graph is regular.*

*Proof. *Let be any vertex transitive bipolar fuzzy graph. Let . Then there is an automorphism on such that . Note that
Hence is regular.

Theorem 18. *Cayley bipolar fuzzy graphs are regular.*

*Proof. *Proof follows from Theorems 16 and 17.

Theorem 19. *Let denote bipolar fuzzy graph. Then bipolar fuzzy relation is reflexive if and only if and .*

*Proof. * is reflexive if and only if for all . Now
Hence is reflexive if and only if and .

Theorem 20. *Let denote bipolar fuzzy graph. Then bipolar fuzzy relation is symmetric if and only if for all .*

*Proof. *Suppose that is symmetric. Then for any ,
Conversely, suppose that for all . Then for all ,
Hence is symmetric.

Theorem 21. *A bipolar fuzzy relation is antisymmetric if and only if .*

*Definition 22. *Let be a semigroup. Let be a bipolar fuzzy subset of . Then is said to be a bipolar fuzzy subsemigroup of if for all , and .

Theorem 23. *A bipolar fuzzy relation is transitive if and only if is a bipolar fuzzy subsemigroup of .*

*Proof. *Suppose that is transitive and let . Then . Now for any , we have . This implies that . That is and . Hence and . Hence is a bipolar fuzzy subsemigroup of .

Conversely, suppose that is a bipolar fuzzy subsemigroup of . That is, for all and . Then for any ,
Hence and . Hence is transitive.

We conclude that.

Theorem 24. *A bipolar fuzzy relation is a partial order if and only if is a bipolar fuzzy subsemigroup of satisfying*(i)* and ,*(ii)*.*

Theorem 25. *A bipolar fuzzy relation is a linear order if and only if is a bipolar fuzzy subsemigroup of satisfying*(i)* and ,*(ii)*,*(iii)*. *

* Proof. *Suppose is a linear order. Then by Theorem 24, conditions (i), (ii), and (iii) are satisfied. For any , . This implies that . Hence .

Conversely, suppose that conditions (i), (ii), and (iii) hold. Then by Theorem 24, is partial order. Now for any , we have , . Then by condition (iv), . Therefore is linear order.

Theorem 26. *A bipolar fuzzy relation is a equivalence relation if and only if is a bipolar fuzzy subsemigroup of satisfying*(i)* and ,*(ii)* for all .*

Theorem 27. * is a Hasse diagram if and only if for any collection of vertices in with and , for , we have, and .*

*Proof. *Suppose is a Hasse diagram and let be vertices in with and , for . Then it is obvious that , for , where . Therefore is a path from to . Since is a Hasse diagram, we have . This implies that and . Conversely, suppose that for any collection of vertices in with and , for , we have, and . Let be a path in from to with . Then , for . Therefore , for . Now consider the elements in . Then by assumption and . That is, and . Hence, . Thus is a Hasse diagram.

Let be any bipolar fuzzy graph; then is connected (weakly connected, semiconnected, locally connected, or quasi-connected) if and only if the induce fuzzy graph is connected (weakly connected, semiconnected, locally connected, or quasi-connected).

*Definition 28. *Let be a semigroup and let be a bipolar fuzzy subset of . Then the subsemigroup generated by is the meeting of all bipolar fuzzy subsemigroups of which contains . It is denoted by .

Lemma 29. *Let be a semigroup and be a bipolar fuzzy subset of . Then bipolar fuzzy subset is precisely given by with for , with for any .*

*Proof. *Let be a bipolar fuzzy subset of defined by with for , with for , for any . Let . If or , then and or , and then . Therefore, and . Again, if , then by definition of and , we have and . Hence is a bipolar fuzzy subsemigroup of containing . Now let be any bipolar fuzzy subsemigroup of containing . Then for any with with , for , we have and . Thus with for , with for , for any . Hence , for all . Thus , . Thus is the meeting of all bipolar fuzzy subsemigroups containing .

Theorem 30. *Let be a semigroup and be a bipolar fuzzy subset of . Then for any , and , where denotes the subsemigroup generated by and denotes bipolar fuzzy subsemigroup generated by .*

*Proof. *
Therefore . Similarly, we have .

*Remark 31. *Let be a semigroup and be a bipolar fuzzy subset of . Then by Theorem 30, we have .

Let denote the Cayley bipolar fuzzy graphs induced by . Then we have the following results.

Theorem 32. *Let be any subset of and be the Cayley graph induced by . Then is connected if and only if .*

Theorem 33. * is connected if and only if.*

Theorem 34. *Let be any subset of a set and let be the Cayley graph induced by the triplet . Then is weakly connected if and only if , where .*

*Definition 35. *Let be a group and let be a bipolar fuzzy subset of . Then we define as bipolar fuzzy subset of given by for all .

Theorem 36. * is weakly connected if and only if .*

*Proof. *

Theorem 37. *Let be any subset of a set and let be the Cayley graph induced by the triplet . Then is semiconnected if and only if , where .*

Theorem 38. * is semi-connected if and only if .*

*Proof. *

Theorem 39. *Let be the Cayley graph induced by the triplet . Then is locally connected if and only if , where .*

Theorem 40. *Let is locally connected if and only if .*

*Proof. *

Theorem 41. *Let be the Cayley graph induced by the triplet , where is finite. Then is quasi-connected if and only if it is connected.*

Theorem 42. *A finite Cayley bipolar fuzzy graph is quasi-connected if and only if it is connected.*

*Proof. *

*Definition 43. *The strength of a path is defined as for all and and is denoted by . The strength of a path is defined as for all and and is denoted by .

*Definition 44. *Let be a bipolar fuzzy graph. Then is said to be(1)*-connected* if for every pair of vertices , there is a path from to such that strength ,(2)*weakly **-connected* if a bipolar fuzzy graph is -connected,(3)*semi*-*-connected* if for every , there is a path of strength greater than or equal to from to or from to in ,(4)*locally **-connected* if for every pair of vertices and , there is a path of strength greater than or equal to from to whenever there is a path of strength greater than or equal to from to ,(5)*quasi*-*-connected* if for every pair , there is some such that there is directed path from to of strength greater than or equal to and there is a directed path from to of strength greater than or equal to .

*Remark 45. *Let be any bipolar fuzzy graph; then is -connected (weakly -connected, semi -connected, locally -connected or quasi -connected) if and only if the induce fuzzy graph is connected (weakly connected, semiconnected, locally connected, or quasi-connected).

Let denote the Cayley bipolar fuzzy graphs induced by . Also for any , we have the following results.

Theorem 46. * is -connected if and only if .*

*Proof. *

Theorem 47. * is weakly -connected if and only if .*

*Proof. *

Theorem 48. * is semi--connected if and only if .*

Theorem 49. *Let be locally -connected if and only if .*

Theorem 50. *A finite Cayley bipolar fuzzy graph is quasi--connected if and only if it is -connected.*

#### 4. Conclusions

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

#### Acknowledgments

This Project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 363-014-D1434. The authors, therefore, acknowledge with thanks DSR technical and financial support.