Cayley Bipolar Fuzzy Graphs

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) 𝑥 to vertex 𝑦 indicates that one may move from 𝑥 to 𝑦 but not from 𝑦 to 𝑥. The Cayley graph was first considered for finite groups by Cayley in 1878. Max Dehn in his unpublished lectures on group theory from 1909 to 1910 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point [1].

The notion of fuzzy sets was introduced by Zadeh [2] as a method of representing uncertainty and vagueness. Since then, the theory of fuzzy sets has become a vigorous area of research in different disciplines. In 1994, Zhang [3] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets [4]. A bipolar fuzzy set is an extension of fuzzy sets whose membership degree range is [−1,1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0,1] of an element indicates that the element somewhat satisfies the property, and the membership degree [−1,0) of an element indicates that the element somewhat satisfies the implicit counterproperty.

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

2. Preliminaries

A digraph is a pair 𝐷^∗=(𝑉,𝐸), where 𝑉 is a finite set and 𝐸⊆𝑉×𝑉. Let 𝐷^∗₁=(𝑉₁,𝐸₁) and 𝐷^∗₂=(𝑉₂,𝐸₂) be two digraphs. The Cartesian product of 𝐷^∗₁ and 𝐷^∗₂ gives a digraph 𝐷^∗₁×𝐷^∗₂=(𝑉,𝐸) with 𝑉=𝑉₁×𝑉₂ and 𝐸=𝑥,𝑥₂⟶𝑥,𝑦₂∣𝑥∈𝑉₁,𝑥₂⟶𝑦₂∈𝐸₂∪𝑥₁,𝑧⟶𝑦₁,𝑧∣𝑥₁⟶𝑦₁∈𝐸₁,𝑧∈𝑉₂. (1) In this paper, we will write 𝑥𝑦∈𝐸 to mean 𝑥→𝑦∈𝐸, and if 𝑒=𝑥𝑦∈𝐸, we say 𝑥 and 𝑦 are adjacent such that 𝑥 is a starting node and 𝑦 is an ending node.

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

Definition 1. Let 𝐺 be a finite group and let 𝑆 be a minimal generating set of 𝐺. A Cayley graph (𝐺,𝑆) has elements of 𝐺 as its vertices; the edge set is given by {(𝑔,𝑔_𝑠)∶𝑔∈𝐺,𝑠∈𝑆}. Two vertices 𝑔₁ and 𝑔₂ are adjacent if 𝑔₂=𝑔₁⋅𝑠, where 𝑠∈𝑆. Note that a generating set 𝑆 is minimal if 𝑆 generates 𝐺 but no proper subset of 𝑆 does.

Theorem 2. All Cayley graphs are vertex transitive.

Definition 3. Let (𝑉,∗) be a group and let 𝐴 be any subset of 𝑉. Then the Cayley graph induced by (𝑉,∗,𝐴) is the graph 𝐺=(𝑉,𝑅), where 𝑅={(𝑥,𝑦)∶𝑥⁻¹𝑦∈𝐴}.

Definition 4 (see [2]). A fuzzy subset 𝜇 on a set 𝑋 is a map 𝜇∶𝑋→[0,1]. A fuzzy binary relation on 𝑋 is a fuzzy subset 𝜇 on 𝑋×𝑋. By a fuzzy relation we mean a fuzzy binary relation given by 𝜇∶𝑋×𝑋→[0,1].

Definition 5 (see [11]). Let (𝑉,∗) be a group and let 𝜇 be a fuzzy subset of 𝑉. Then the fuzzy relation 𝑅 on 𝑉 defined by 𝑅𝑥,𝑦=𝜇𝑥⁻¹∗𝑦∀𝑥,𝑦∈𝑉 (2) induces a fuzzy graph 𝐺=(𝑉,𝑅), called the Cayley fuzzy graph induced by the (𝑉,∗,𝜇).

Definition 6 (see [3]). Let 𝑋 be a nonempty set. A bipolar fuzzy set 𝐵 in 𝑋 is an object having the form 𝐵=𝑥,𝜇^𝑃_𝐵(𝑥),𝜇^𝑁_𝐵(𝑥)∣𝑥∈𝑋, (3) where 𝜇^𝑃_𝐵∶𝑋→[0,1] and 𝜇^𝑁_𝐵∶𝑋→[−1,0] are mappings.

We use the positive membership degree 𝜇^𝑃_𝐵(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 𝜇^𝑁_𝐵(𝑥) to denote the satisfaction degree of an element 𝑥 to some implicit counterproperty corresponding to a bipolar fuzzy set 𝐵. If 𝜇^𝑃_𝐵(𝑥)≠0 and 𝜇^𝑁_𝐵(𝑥)=0, it is the situation that 𝑥 is regarded as having only positive satisfaction for 𝐵. If 𝜇^𝑃_𝐵(𝑥)=0 and 𝜇^𝑁_𝐵(𝑥)≠0, it is the situation that 𝑥 does not satisfy the property of 𝐵 but somewhat satisfies the counterproperty of 𝐵. It is possible for an element 𝑥 to be such that 𝜇^𝑃_𝐵(𝑥)≠0 and 𝜇^𝑁_𝐵(𝑥)≠0 when the membership function of the property overlaps that of its counterpro perty over some portion of 𝑋.

For the sake of simplicity, we shall use the symbol 𝐵=(𝜇^𝑃_𝐵,𝜇^𝑁_𝐵) for the bipolar fuzzy set 𝐵=𝑥,𝜇^𝑃_𝐵(𝑥),𝜇^𝑁_𝐵(𝑥)∣𝑥∈𝑋. (4) A nice application of bipolar fuzzy concept is a political acceptation (map to [0, 1]) and nonacceptation (map to [−1,0]).

Definition 7 (see [3]). A bipolar fuzzy relation 𝑅=(𝜇^𝑃_𝑅(𝑥,𝑦),𝜇^𝑁_𝑅(𝑥,𝑦)) in a universe 𝑋×𝑌 (𝑅(𝑋→𝑌), for short) is a bipolar fuzzy set of the form 𝑅=𝑥,𝑦,𝜇^𝑃_𝑅𝑥,𝑦,𝜇^𝑁_𝑅𝑥,𝑦∣𝑥,𝑦∈𝑋×𝑌, (5) where 𝜇^𝑃_𝑅∶𝑋×𝑌→[0,1] and 𝜇^𝑁_𝑅∶𝑋×𝑌→[−1,0].

Definition 8. Let 𝑅 be a bipolar fuzzy relation on universe 𝑋. Then 𝑅 is called a bipolar fuzzy equivalence relation on 𝑋 if it satisfies the following conditions: (a)𝑅is bipolar fuzzy reflexive; that is, 𝑅(𝑥,𝑥) = (1, −1) for each 𝑥∈𝑋;(b)𝑅 is bipolar fuzzy symmetric; that is, 𝑅(𝑥,𝑦)=𝑅(𝑦,𝑥) for any 𝑥, 𝑦∈𝑋;(c)𝑅 is bipolar fuzzy transitive; that is, 𝑅(𝑥,𝑧)≥⋁_𝑦(𝑅(𝑥,𝑦)∧𝑅(𝑦,𝑧)).

Definition 9. Let 𝑅 be a bipolar fuzzy relation on universe 𝑋. Then 𝑅 is called a bipolar fuzzy partial order relation on 𝑋 if it satisfies the following conditions:(a)𝑅 is bipolar fuzzy reflexive; that is, 𝑅(𝑥,𝑥)=(1,−1) for each 𝑥∈𝑋;(b)𝑅 is bipolar fuzzy antisymmetric; that is, 𝑅(𝑥,𝑦)≠𝑅(𝑦,𝑥) for any 𝑥, 𝑦∈𝑋;(c)𝑅 is bipolar fuzzy transitive; that is, 𝑅(𝑥,𝑧)≥⋁_𝑦(𝑅(𝑥,𝑦)∧𝑅(𝑦,𝑧)).

Definition 10. Let 𝑅 be a bipolar fuzzy relation on universe 𝑋. Then 𝑅 is called a bipolar fuzzy linear order relation on 𝑋 if it satisfies the following conditions:(a)𝑅 is bipolar fuzzy partial relation;(b)(𝜇^𝑃_𝑅∨𝜇^𝑃_𝑅−1)(𝑥,𝑦)>0, (𝜇^𝑁_𝑅∧𝜇^𝑁_𝑅−1)(𝑥,𝑦)<0 for all 𝑥, 𝑦∈𝑋.

3. Cayley Bipolar Fuzzy Graphs

Definition 11. A bipolar fuzzy digraph of a digraph 𝐷^∗ is a pair 𝐷=(𝐴,𝐵) where 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy set in 𝑉 and 𝐵=(𝜇^𝑃_𝐵,𝜇^𝑁_𝐵) is a bipolar fuzzy relation on 𝐸 such that 𝜇^𝑃_𝐵𝑥𝑦≤min𝜇^𝑃_𝐴(𝑥),𝜇^𝑃_𝐴𝑦,𝜇^𝑁_𝐵𝑥𝑦≥max𝜇^𝑁_𝐴(𝑥),𝜇^𝑁_𝐴𝑦 (6) for all 𝑥𝑦∈𝐸. We note that 𝐵 need not to be symmetric.

Definition 12. Let 𝐷 be a bipolar fuzzy digraph. The indegree of a vertex 𝑥 in 𝐷 is defined by ind(𝑥)=(ind^𝑃_𝜇(𝑥),ind^𝑁_𝜇(𝑥)), where ind^𝑃_𝜇(𝑥)=∑_𝑦≠𝑥𝜇^𝑃_𝐴(𝑥𝑦) and ind^𝑁_𝜇(𝑥)=∑_𝑦≠𝑥𝜇^𝑁_𝐴(𝑥𝑦). Similarly, the outdegree of a vertex 𝑥 in 𝐷 is defined by outd(𝑥)=(outd^𝑃_𝜇(𝑥),outd^𝑁_𝜇(𝑥)), where outd^𝑃_𝜇(𝑥)=∑_𝑦≠𝑥𝜇^𝑃_𝐴(𝑥𝑦) and outd^𝑁_𝜇(𝑥)=∑_𝑦≠𝑥𝜇^𝑁_𝐴(𝑥𝑦). A bipolar fuzzy digraph in which each vertex has the same outdegree 𝑟 is called an outregular digraph with index of outregularity 𝑟. In-regular digraphs are defined similarly.

Definition 13. Let (𝑉,∗) be a group and let 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be the bipolar fuzzy subset of 𝑉. Then the bipolar fuzzy relation 𝑅 defined on 𝑉 by 𝑅𝑥,𝑦=𝜇^𝑃_𝑅𝑥⁻¹𝑦,𝜇^𝑁_𝑅𝑥⁻¹𝑦,∀𝑥,𝑦∈𝑉, (7) induces a bipolar fuzzy graph 𝐺=(𝑉,𝑅) called the Cayley bipolar fuzzy graph induced by the (𝑉,∗,𝜇^𝑃_𝑅,𝜇^𝑁_𝑅).

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

Definition 14. Let (𝑉,∗) be a group and let 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be a bipolar fuzzy subset of 𝑉. Then the bipolar fuzzy relation 𝑅 on 𝑉 defined by 𝑅𝑥,𝑦=𝜇^𝑃_𝑅𝑥⁻¹𝑦,𝜇^𝑁_𝑅𝑥⁻¹𝑦∀𝑥,𝑦∈𝑉 (8) induces a bipolar fuzzy graph 𝐺=(𝑉,𝑅), called the Cayley bipolar fuzzy graph induced by the (𝑉,∗,𝐴).

Example 15. Consider the group 𝑍₃ and take 𝑉={0,1,2}. Define 𝜇^𝑃_𝐴∶𝑉→[0,1] and 𝜇^𝑁_𝐴∶𝑉→[−1,0] by 𝜇^𝑃_𝐴(0)=𝜇^𝑃_𝐴(1)=𝜇^𝑃_𝐴(2)=0.5, 𝜇^𝑁_𝐴(0)=𝜇^𝑁_𝐴(1)=𝜇^𝑁_𝐴(2)=−0.4. Then the Cayley bipolar fuzzy graph 𝐺=(𝑉,𝑅) induced by (𝑍₃,+,𝐴) is given by Table 1 and Figure 1.

Table 1 

𝑅(𝑎,𝑏) for Cayley bipolar fuzzy graph.

Figure 1 

Cayley bipolar fuzzy graph.

We see that Cayley bipolar fuzzy graphs are actually bipolar fuzzy digraphs. Furthermore, the relation 𝑅 in the above definition describes the strength of each directed edge. Let 𝐺 denote a bipolar fuzzy graph 𝐺=(𝑉,𝑅) induced by the triple (𝑉,∗,𝐴).

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

Proof. Let 𝑎,𝑏∈𝑉. Define 𝜓∶𝑉→𝑉 by 𝜓(𝑥)=𝑏𝑎⁻¹𝑥 for all 𝑥∈𝑉. Clearly, 𝜓 is a bijective map. For each 𝑥, 𝑦∈𝑉,𝑅𝜓(𝑥),𝜓𝑦=𝑅_𝜇𝑃𝜓(𝑥)𝑦,𝑅,𝜓_𝜇𝑁𝜓(𝑥),𝜓𝑦.Now𝑅_𝜇𝑃𝜓(𝑥),𝜓𝑦=𝑅_𝜇𝑃𝑏𝑎⁻¹𝑥,𝑏𝑎⁻¹𝑦=𝜇^𝑃_𝐴𝑏𝑎⁻¹𝑥⁻¹𝑏𝑎⁻¹𝑥=𝜇^𝑃_𝐴𝑥⁻¹𝑦=𝑅_𝜇𝑃,𝑅𝑥,𝑦_𝜇𝑁𝜓(𝑥),𝜓𝑦=𝑅_𝜇𝑁𝑏𝑎⁻¹𝑥,𝑏𝑎⁻¹𝑦=𝜇^𝑁_𝐴𝑏𝑎⁻¹𝑥⁻¹𝑏𝑎⁻¹𝑥=𝜇^𝑁_𝐴𝑥⁻¹𝑦=𝑅_𝜇𝑁𝑥,𝑦. (9) Therefore, 𝑅(𝜓(𝑥),𝜓(𝑦))=𝑅(𝑥,𝑦). Hence 𝜓 is an automorphism on 𝐺. Also 𝜓(𝑎)=𝑏. Hence 𝐺 is vertex transitive.

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

Proof. Let 𝐺=(𝑉,𝑅) be any vertex transitive bipolar fuzzy graph. Let 𝑢,𝑣∈𝑉. Then there is an automorphism 𝑓 on 𝐺 such that 𝑓(𝑢)=𝑣. Note that ind(𝑢)=_𝑥∈𝑉𝑅(𝑥,𝑢)=_𝑥∈𝑉𝑅_𝜇𝑃(𝑥,𝑢),𝑅_𝜇𝑁(𝑥,𝑢)=_𝑥∈𝑉𝑅_𝜇𝑃𝑓(𝑥),𝑓(𝑢),𝑅_𝜇𝑁𝑓(𝑥),𝑓(𝑢)=_𝑥∈𝑉𝑅_𝜇𝑃𝑓(𝑥),𝑣,𝑅_𝜇𝑁𝑓(𝑥),𝑣=_𝑥∈𝑉𝑅_𝜇𝑃𝑦,𝑣,𝑅_𝜇𝑁(𝑣),(𝑢)𝑦,𝑣=indoutd=_𝑥∈𝑉𝑅(𝑥,𝑢)=_𝑥∈𝑉𝑅_𝜇𝑃(𝑢,𝑥),𝑅_𝜇𝑁(𝑢,𝑥)=_𝑥∈𝑉𝑅_𝜇𝑃𝑓(𝑢),𝑓(𝑥),𝑅_𝜇𝑁𝑓(𝑢),𝑓(𝑥)=_𝑥∈𝑉𝑅_𝜇𝑃𝑣,𝑓(𝑥),𝑅_𝜇𝑁𝑣,𝑓(𝑥)=_𝑥∈𝑉𝑅_𝜇𝑃𝑣,𝑦,𝑅_𝜇𝑁(𝑣).𝑣,𝑦=outd (10) Hence 𝐺 is regular.

Theorem 18. Cayley bipolar fuzzy graphs are regular.

Proof. Proof follows from Theorems 16 and 17.

Theorem 19. Let 𝐺=(𝑉,𝑅) denote bipolar fuzzy graph. Then bipolar fuzzy relation 𝑅 is reflexive if and only if 𝜇^𝑃_𝐴(1)=1 and 𝜇^𝑁_𝐴(1)=−1.

Proof. 𝑅 is reflexive if and only if 𝑅(𝑥,𝑥)=(1,−1) for all 𝑥∈𝑉. Now 𝑅(𝑥,𝑥)=𝜇^𝑃_𝐴𝑥⁻¹𝑥,𝜇^𝑁_𝐴𝑥⁻¹𝑥=𝜇^𝑃_𝐴(1),𝜇^𝑁_𝐴(1),∀𝑥∈𝑉. (11) Hence 𝑅 is reflexive if and only if 𝜇^𝑃_𝐴(1)=1 and 𝜇^𝑁_𝐴(1)=−1.

Theorem 20. Let 𝐺=(𝑉,𝑅) denote bipolar fuzzy graph. Then bipolar fuzzy relation 𝑅 is symmetric if and only if (𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥))=(𝜇^𝑃_𝐴(𝑥⁻¹),𝜇^𝑁_𝐴(𝑥⁻¹)) for all 𝑥∈𝑉.

Proof. Suppose that 𝑅 is symmetric. Then for any 𝑥∈𝑉, 𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥)=𝜇^𝑃_𝐴𝑥⁻¹𝑥²,𝜇^𝑁_𝐴𝑥⁻¹𝑥²=𝑅𝑥,𝑥²=𝑅𝑥²=𝜇,𝑥since𝑅issymmetric_𝑃𝐴𝑥²⁻¹𝑥,𝜇^𝑁_𝐴𝑥²⁻¹𝑥=𝜇^𝑃_𝐴𝑥⁻²𝑥,𝜇^𝑁_𝐴𝑥⁻²𝑥=𝜇^𝑃_𝐴𝑥⁻¹,𝜇^𝑁_𝐴𝑥⁻¹. (12) Conversely, suppose that (𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥))=(𝜇^𝑃_𝐴(𝑥⁻¹),𝜇^𝑁_𝐴(𝑥⁻¹)) for all 𝑥∈𝑉. Then for all 𝑥,𝑦∈𝑉, 𝑅𝑥,𝑦=𝜇^𝑃_𝐴𝑥⁻¹𝑦,𝜇^𝑁_𝐴𝑥⁻¹𝑦=𝜇^𝑃_𝐴𝑦⁻¹𝑥,𝜇^𝑁_𝐴𝑦⁻¹𝑥=𝑅𝑦,𝑥. (13) Hence 𝑅 is symmetric.

Theorem 21. A bipolar fuzzy relation 𝑅 is antisymmetric if and only if {𝑥∶(𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥))=(𝜇^𝑃_𝐴(𝑥⁻¹),𝜇^𝑁_𝐴(𝑥⁻¹))=(1,−1).

Definition 22. Let (𝑆,∗) be a semigroup. Let 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be a bipolar fuzzy subset of 𝑆. Then 𝐴 is said to be a bipolar fuzzy subsemigroup of 𝑆 if for all 𝑥,𝑦∈𝑆, 𝜇^𝑃_𝐵(𝑥𝑦)≥𝜇^𝑃_𝐴(𝑥)∧𝜇^𝑃_𝐴(𝑦) and 𝜇^𝑁_𝐵(𝑥𝑦)≤𝜇^𝑁_𝐴(𝑥)∨𝜇^𝑁_𝐴(𝑦).

Theorem 23. A bipolar fuzzy relation 𝑅 is transitive if and only if 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy subsemigroup of (𝑉,∗).

Proof. Suppose that 𝑅 is transitive and let 𝑥,𝑦∈𝑉. Then 𝑅²≤𝑅. Now for any 𝑥∈𝑉, we have 𝑅(1,𝑥)=(𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥)). This implies that {𝑅(1,𝑧)∧𝑅(𝑧,𝑥𝑦)∶𝑧∈𝑉}=𝑅²(1,𝑥𝑦)≤𝑅(1,𝑥𝑦). That is ∨{𝜇^𝑃_𝐴(𝑧)∧𝜇^𝑃_𝐴(𝑧⁻¹𝑥𝑦)∶𝑧∈𝑉}≤𝜇^𝑃_𝐴(𝑥𝑦) and ∧{𝜇^𝑁_𝐴(𝑧)∨𝜇^𝑁_𝐴(𝑧⁻¹𝑥𝑦)∶𝑧∈𝑉}≥𝜇^𝑁_𝐴(𝑥𝑦). Hence 𝜇^𝑃_𝐴(𝑥𝑦)≥𝜇^𝑃_𝐴(𝑥)∧𝜇^𝑃_𝐴(𝑦) and 𝜇^𝑁_𝐴(𝑥𝑦)≤𝜇^𝑃_𝐴(𝑥)∨𝜇^𝑁_𝐴(𝑦). Hence 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy subsemigroup of (𝑉,∗).
Conversely, suppose that 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy subsemigroup of (𝑉,∗). That is, for all 𝑥,𝑦∈𝑉𝜇^𝑃_𝐵(𝑥𝑦)≥𝜇^𝑃_𝐴(𝑥)∧𝜇^𝑃_𝐴(𝑦) and 𝜇^𝑁_𝐵(𝑥𝑦)≤𝜇^𝑁_𝐴(𝑥)∨𝜇^𝑁_𝐴(𝑦). Then for any 𝑥,𝑦∈𝑉, 𝑅²𝑥,𝑦=𝑅²_𝜇𝑃𝑥,𝑦,𝑅²_𝜇𝑁,𝑅𝑥,𝑦_2𝜇𝑃𝑥,𝑦=∨𝑅_𝜇𝑃(𝑥,𝑧)∧𝑅_𝜇𝑃𝜇𝑧,𝑦∶𝑧∈𝑉=∨^𝑃_𝐴𝑥⁻¹𝑧∧𝜇^𝑃_𝐴𝑧⁻¹𝑦∶𝑧∈𝑉≤𝜇^𝑃_𝐴𝑥⁻¹𝑦=𝑅_𝜇𝑃,𝑅𝑥,𝑦_2𝜇𝑁𝑥,𝑦=∧𝑅_𝜇𝑁(𝑥,𝑧)∨𝑅_𝜇𝑁𝜇𝑧,𝑦∶𝑧∈𝑉=∧^𝑁_𝐴𝑥⁻¹𝑧∨𝜇^𝑁_𝐴𝑧⁻¹𝑦∶𝑧∈𝑉≥𝜇^𝑁_𝐴𝑥⁻¹𝑦=𝑅_𝜇𝑁𝑥,𝑦. (14) Hence 𝑅²_𝜇𝑃(𝑥,𝑦)≤𝑅_𝜇𝑃(𝑥,𝑦) and 𝑅²_𝜇𝑁(𝑥,𝑦)≥𝑅_𝜇𝑁(𝑥,𝑦). Hence 𝑅 is transitive.

We conclude that.

Theorem 24. A bipolar fuzzy relation 𝑅 is a partial order if and only if 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy subsemigroup of (𝑉,∗) satisfying(i)𝜇^𝑃_𝐴(1)=1 and 𝜇^𝑁_𝐴(1)=−1,(ii){𝑥∶(𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥))=(𝜇^𝑃_𝐴(𝑥⁻¹),𝜇^𝑁_𝐴(𝑥⁻¹))={1,−1}}.

Theorem 25. A bipolar fuzzy relation 𝑅 is a linear order if and only if (𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy subsemigroup of (𝑉,∗) satisfying(i)𝜇^𝑃_𝐴(1)=1 and 𝜇^𝑁_𝐴(1)=−1,(ii){𝑥∶(𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥))=(𝜇^𝑃_𝐴(𝑥⁻¹),𝜇^𝑁_𝐴(𝑥⁻¹))={1,−1}},(iii){𝑥∶𝜇^𝑃_𝐴(𝑥)∨𝜇^𝑃_𝐴(𝑥⁻¹)>0,𝜇^𝑁_𝐴(𝑥)∧𝜇^𝑁_𝐴(𝑥⁻¹)<0}=𝑉.

Proof. Suppose 𝑅 is a linear order. Then by Theorem 24, conditions (i), (ii), and (iii) are satisfied. For any 𝑥∈𝑉, (𝑅∨𝑅⁻¹)(1,𝑥)>0. This implies that 𝑅(1,𝑥)∨𝑅(𝑥,1)>0. Hence {𝑥∶𝜇^𝑃_𝐴(𝑥)∨𝜇^𝑃_𝐴(𝑥⁻¹)>0,𝜇^𝑁_𝐴(𝑥)∧𝜇^𝑁_𝐴(𝑥⁻¹)<0}.
Conversely, suppose that conditions (i), (ii), and (iii) hold. Then by Theorem 24, 𝑅 is partial order. Now for any 𝑥,𝑦∈𝑉, we have (𝑥⁻¹𝑦), (𝑦⁻¹𝑥)∈𝑉. Then by condition (iv), {𝑥∶𝜇^𝑃_𝐴(𝑥)∨𝜇^𝑃_𝐴(𝑥⁻¹)>0,𝜇^𝑁_𝐴(𝑥)∧𝜇^𝑁_𝐴(𝑥⁻¹)<0}. Therefore 𝑅 is linear order.

Theorem 26. A bipolar fuzzy relation 𝑅 is a equivalence relation if and only if (𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) is a bipolar fuzzy subsemigroup of (𝑉,∗) satisfying(i)𝜇^𝑃_𝐴(1)=1 and 𝜇^𝑁_𝐴(1)=−1,(ii)(𝜇^𝑃_𝐴(𝑥),𝜇^𝑁_𝐴(𝑥))=(𝜇^𝑃_𝐴(𝑥⁻¹),𝜇^𝑁_𝐴(𝑥⁻¹)) for all 𝑥∈𝑉.

Theorem 27. 𝐺 is a Hasse diagram if and only if for any collection 𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑛 of vertices in 𝑉 with 𝑛≥2 and 𝜇^𝑃_𝐴(𝑥_𝑖)>0,𝜇^𝑁_𝐴(𝑥_𝑖)<0, for 𝑖=1,2,3,…,𝑛, we have, 𝜇^𝑃_𝐴(𝑥₁𝑥₂⋯𝑥_𝑛)=0 and 𝜇^𝑁_𝐴(𝑥₁𝑥₂⋯𝑥_𝑛)=0.

Proof. Suppose 𝐺 is a Hasse diagram and let 𝑥₁,𝑥₂,…,𝑥_𝑛 be vertices in 𝑉 with 𝑛≥2 and 𝜇^𝑃_𝐴(𝑥_𝑖)>0,𝜇^𝑁_𝐴(𝑥_𝑖)<0, for 𝑖=1,2,3,…,𝑛. Then it is obvious that 𝑅(𝑥₁𝑥₂⋯𝑥_𝑖−1,𝑥₁𝑥₂⋯𝑥_𝑖)=(𝜇^𝑃_𝐴(𝑥_𝑖),𝜇^𝑁_𝐴(𝑥_𝑖)), for 𝑖=1,2,…,𝑛, where 𝑥₀=1. Therefore (1,𝑥₁,𝑥₁𝑥₂,…,𝑥₁𝑥₂⋯𝑥_𝑛) is a path from 1 to 𝑥₁𝑥₂⋯𝑥_𝑛. Since 𝐺 is a Hasse diagram, we have 𝑅(1,𝑥₁𝑥₂⋯𝑥_𝑛)=0. This implies that 𝜇^𝑃_𝐴(𝑥₁𝑥₂⋯𝑥_𝑛)=0 and 𝜇^𝑁_𝐴(𝑥₁𝑥₂⋯𝑥_𝑛)=0. Conversely, suppose that for any collection 𝑥₁,𝑥₂,…𝑥_𝑛 of vertices in 𝑉 with 𝑛≥2 and 𝜇^𝑃_𝐴(𝑥_𝑖)>0,𝜇^𝑁_𝐴(𝑥_𝑖)<0, for 𝑖=1,2,3,…,𝑛, we have, 𝜇^𝑃_𝐴(𝑥₁𝑥₂⋯𝑥_𝑛)=0 and 𝜇^𝑁_𝐴(𝑥₁𝑥₂⋯𝑥_𝑛)=0. Let (𝑥₀,𝑥₁,𝑥₂,…𝑥_𝑛) be a path in 𝐺 from 𝑥₀ to 𝑥_𝑛 with 𝑛≥2. Then 𝑅(𝑥_𝑖−1,𝑥_𝑖)>0, for 𝑖=1,2,…,𝑛. Therefore 𝜇^𝑃_𝐴(𝑥⁻¹_𝑖−1𝑥_𝑖)>0,𝜇^𝑁_𝐴(𝑥⁻¹_𝑖−1𝑥_𝑖)<0, for 𝑖=1,2,…,𝑛. Now consider the elements 𝑥₀⁻¹𝑥₁,𝑥₁⁻¹𝑥₂,…,𝑥⁻¹_𝑛−1𝑥_𝑛 in 𝑉. Then by assumption 𝜇^𝑃_𝐴(𝑥₀⁻¹𝑥₁𝑥₁⁻¹𝑥₂⋯𝑥⁻¹_𝑛−1𝑥_𝑛)=0 and 𝜇^𝑁_𝐴(𝑥₀⁻¹𝑥₁𝑥₁⁻¹𝑥₂⋯𝑥⁻¹_𝑛−1𝑥_𝑛)=0. That is, 𝜇^𝑃_𝐴(𝑥₀⁻¹𝑥_𝑛)=0 and 𝜇^𝑁_𝐴(𝑥₀⁻¹𝑥_𝑛)=0. Hence, 𝑅(𝑥₀,𝑥_𝑛)=0. Thus 𝐺 is a Hasse diagram.

Let 𝐺=(𝑉,𝑅) be any bipolar fuzzy graph; then 𝐺 is connected (weakly connected, semiconnected, locally connected, or quasi-connected) if and only if the induce fuzzy graph (𝑉,𝑅⁺₀) is connected (weakly connected, semiconnected, locally connected, or quasi-connected).

Definition 28. Let (𝑆,∗) be a semigroup and let 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be a bipolar fuzzy subset of 𝑆. Then the subsemigroup generated by 𝐴 is the meeting of all bipolar fuzzy subsemigroups of 𝑆 which contains 𝐴. It is denoted by ⟨𝐴⟩.

Lemma 29. Let (𝑆,∗) be a semigroup and 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be a bipolar fuzzy subset of 𝑆. Then bipolar fuzzy subset ⟨𝐴⟩ is precisely given by ⟨𝜇^𝑃_𝐴⟩(𝑥)=∨{𝜇^𝑃_𝐴(𝑥₁)∧𝜇^𝑃_𝐴(𝑥₂)∧⋯∧𝜇^𝑃_𝐴(𝑥_𝑛)∶𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with𝜇^𝑃_𝐴(𝑥_𝑖)>0 for 𝑖=1,2,…,𝑛}, ⟨𝜇^𝑁_𝐴⟩(𝑥)=∧{𝜇^𝑁_𝐴(𝑥₁)∨𝜇^𝑁_𝐴(𝑥₂)∨⋯∨𝜇^𝑁_𝐴(𝑥_𝑛)∶𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with 𝜇^𝑁_𝐴(𝑥_𝑖)<0𝑓𝑜𝑟𝑖=1,2,…,𝑛} for any 𝑥∈𝑆.

Proof. Let 𝐴^′=(𝜇́^𝑃_𝐴,𝜇́^𝑁_𝐴) be a bipolar fuzzy subset of 𝑆 defined by 𝜇́^𝑃_𝐴(𝑥)=∨{𝜇^𝑃_𝐴(𝑥₁)∧𝜇^𝑃_𝐴(𝑥₂)∧⋯∧𝜇^𝑃_𝐴(𝑥_𝑛)∶𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with 𝜇^𝑃_𝐴(𝑥_𝑖)>0 for 𝑖=1,2,…,𝑛}, 𝜇́^𝑁_𝐴(𝑥)=∧{𝜇^𝑁_𝐴(𝑥₁)∨𝜇^𝑁_𝐴(𝑥₂)∨⋯∨𝜇^𝑁_𝐴(𝑥_𝑛)∶𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with 𝜇^𝑁_𝐴(𝑥_𝑖)<0 for 𝑖=1,2,…,𝑛}, for any 𝑥∈𝑆. Let 𝑥,𝑦∈𝑆. If 𝜇^𝑃_𝐴(𝑥)=0 or 𝜇^𝑃_𝐴(𝑦)=0, then 𝜇^𝑃_𝐴(𝑥)∧𝜇^𝑃_𝐴(𝑦)=0 and 𝜇^𝑁_𝐴(𝑥)=0 or 𝜇^𝑁_𝐴(𝑦)=0, and then 𝜇^𝑁_𝐴(𝑥)∨𝜇^𝑁_𝐴(𝑦)=0. Therefore, 𝜇́^𝑃_𝐵(𝑥𝑦)≥𝜇^𝑃_𝐴(𝑥)∧𝜇^𝑃_𝐴(𝑦) and 𝜇́^𝑁_𝐵(𝑥𝑦)≤𝜇^𝑁_𝐴(𝑥)∨𝜇^𝑁_𝐴(𝑦). Again, if 𝜇^𝑃_𝐴(𝑥)≠0,𝜇^𝑁_𝐴(𝑥)≠0, then by definition of 𝜇́^𝑃_𝐴(𝑥) and 𝜇́^𝑁_𝐴(𝑥), we have 𝜇́^𝑃_𝐵(𝑥𝑦)≥𝜇^𝑃_𝐴(𝑥)∧𝜇^𝑃_𝐴(𝑦) and 𝜇́^𝑁_𝐵(𝑥𝑦)≤𝜇^𝑁_𝐴(𝑥)∨𝜇^𝑁_𝐴(𝑦). Hence (𝜇́^𝑃_𝐴,𝜇́^𝑁_𝐴) is a bipolar fuzzy subsemigroup of 𝑆 containing (𝜇^𝑃_𝐴,𝜇^𝑁_𝐴). Now let 𝐿 be any bipolar fuzzy subsemigroup of 𝑆 containing (𝜇^𝑃_𝐴,𝜇^𝑁_𝐴). Then for any 𝑥∈𝑆 with 𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with 𝜇^𝑃_𝐴(𝑥_𝑖)>0,𝜇^𝑁_𝐴(𝑥_𝑖)<0, for 𝑖=1,2,…,𝑛, we have 𝜇^𝑃_𝐿(𝑥_𝑖)≥𝜇^𝑃_𝐿(𝑥₁)∧𝜇^𝑃_𝐿(𝑥₂)∧⋯∧𝜇^𝑃_𝐿(𝑥_𝑛)≥𝜇^𝑃_𝐴(𝑥₁)∧𝜇^𝑃_𝐴(𝑥₂)∧⋯∧𝜇^𝑃_𝐴(𝑥_𝑛) and 𝜇^𝑁_𝐿(𝑥_𝑖)≥𝜇^𝑁_𝐿(𝑥₁)∧𝜇^𝑁_𝐿(𝑥₂)∧⋯∧𝜇^𝑁_𝐿(𝑥_𝑛)≥𝜇^𝑁_𝐴(𝑥₁)∧𝜇^𝑁_𝐴(𝑥₂)∧⋯∧𝜇^𝑁_𝐴(𝑥_𝑛). Thus 𝜇^𝑃_𝐿(𝑥)≥∨{𝜇^𝑃_𝐴(𝑥₁)∧𝜇^𝑃_𝐴(𝑥₂)∧⋯∧𝜇^𝑃_𝐴(𝑥_𝑛)∶𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with 𝜇^𝑃_𝐴(𝑥_𝑖)>0 for 𝑖=1,2,…,𝑛}, 𝜇^𝑁_𝐿(𝑥)≤∧{𝜇^𝑁_𝐴(𝑥₁)∨𝜇^𝑁_𝐴(𝑥₂)∨⋯∨𝜇^𝑁_𝐴(𝑥_𝑛)∶𝑥=𝑥₁𝑥₂⋯𝑥_𝑛 with 𝜇^𝑁_𝐴(𝑥_𝑖)<0 for 𝑖=1,2,…,𝑛}, for any 𝑥∈𝑆. Hence 𝜇^𝑃_𝐿(𝑥)≥𝜇́^𝑃_𝐴(𝑥),𝜇^𝑁_𝐿(𝑥)≤𝜇́^𝑁_𝐴(𝑥), for all 𝑥∈𝑆. Thus 𝜇́^𝑃_𝐴(𝑥)≤𝜇^𝑃_𝐿(𝑥), 𝜇́^𝑁_𝐴(𝑥)≥𝜇^𝑃_𝐴(𝑥). Thus 𝐴^′=(𝜇́^𝑃_𝐴,𝜇́^𝑁_𝐴) is the meeting of all bipolar fuzzy subsemigroups containing (𝜇^𝑃_𝐴,𝜇^𝑁_𝐴).

Theorem 30. Let (𝑆,∗) be a semigroup and 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be a bipolar fuzzy subset of 𝑆. Then for any 𝛼∈[0,1], (⟨𝜇^𝑃_𝛼⟩,⟨𝜇^𝑁_𝛼⟩)=(⟨𝜇^𝑃⟩_𝛼,⟨𝜇^𝑁⟩_𝛼) and (⟨(𝜇⁺)^𝑃_𝛼⟩,⟨(𝜇⁺)^𝑁_𝛼⟩)=(⟨𝜇^𝑃⟩⁺_𝛼,⟨𝜇^𝑁⟩⁺_𝛼), where (⟨𝜇^𝑃_𝛼⟩,⟨𝜇^𝑁_𝛼⟩) denotes the subsemigroup generated by (𝜇^𝑃_𝛼,𝜇^𝑁_𝛼) and ⟨(𝜇^𝑃,𝜇^𝑁)⟩ denotes bipolar fuzzy subsemigroup generated by (𝜇^𝑃,𝜇^𝑁).

Proof. 𝑥∈𝜇^𝑃_𝛼,𝜇^𝑁_𝛼⟺thereexists𝑥₁,𝑥₂,…,𝑥_𝑛in𝜇^𝑃_𝛼,𝜇^𝑁_𝛼suchthat𝑥=𝑥₁𝑥₂⋯𝑥_𝑛⟺thereexists𝑥₁,𝑥₂,…,𝑥_𝑛in𝑆suchthat𝜇^𝑃𝑥_𝑖≥𝛼,𝜇^𝑁𝑥_𝑖≤𝛼,∀𝑖=1,2,…,𝑛,𝑥=𝑥₁𝑥₂⋯𝑥_𝑛⟺𝜇^𝑃(𝑥)≥𝛼,𝜇^𝑁𝑥)𝜇(≤𝛼⟺𝑥∈^𝑃_𝛼,𝑥∈𝜇^𝑁_𝛼. (15) Therefore (⟨𝜇^𝑃_𝛼⟩,⟨𝜇^𝑁_𝛼⟩)=(⟨𝜇^𝑃⟩_𝛼,⟨𝜇^𝑁⟩_𝛼). Similarly, we have (⟨(𝜇⁺)^𝑃_𝛼⟩,⟨(𝜇⁺)^𝑁_𝛼⟩)=(⟨𝜇^𝑃⟩⁺_𝛼,⟨𝜇^𝑁⟩⁺_𝛼).

Remark 31. Let (𝑆,∗) be a semigroup and 𝐴=(𝜇^𝑃_𝐴,𝜇^𝑁_𝐴) be a bipolar fuzzy subset of 𝑆. Then by Theorem 30, we have ⟨supp(𝐴)=𝐴⁺⟩=supp⟨𝐴⟩.

Let 𝐺 denote the Cayley bipolar fuzzy graphs 𝐺=(𝑉,𝑅) induced by (𝑉,∗,𝜇^𝑃,𝜇^𝑁). Then we have the following results.

Theorem 32. Let 𝐴 be any subset of 𝑉^′ and 𝐺^′=(𝑉^′,𝑅^′) be the Cayley graph induced by (𝑉^′,∗,𝐴). Then 𝐺^′ is connected if and only if ⟨𝐴⟩⊇𝑉−𝑣₁.

Theorem 33. 𝐺 is connected if and only ifsupp⟨𝐴⟩⊇𝑉−𝑣₁.

Theorem 34. Let 𝐴 be any subset of a set 𝑉^′ and let 𝐺^′=(𝑉^′,𝑅^′) be the Cayley graph induced by the triplet (𝑉^′,∗,𝐴). Then 𝐺^′ is weakly connected if and only if ⟨𝐴∪𝐴⁻¹⟩⊇𝑉−𝑣₁, where 𝐴⁻¹={𝑥⁻¹∶𝑥∈𝐴}.

Definition 35. Let (𝑆,∗) be a group and let 𝐴 be a bipolar fuzzy subset of 𝑆. Then we define 𝐴⁻¹ as bipolar fuzzy subset of 𝑆 given by 𝐴⁻¹(𝑥)=𝐴(𝑥⁻¹) for all 𝑥∈𝑆.

Theorem 36. 𝐺 is weakly connected if and only if supp(⟨𝐴∪𝐴⁻¹⟩)⊇𝑉−𝑣₁.

Proof. ⟺𝐺isweaklyconnected𝑉,𝑅⁺₀⟺𝐴isweaklyconnected₊₀∪𝐴⁺₀⁻¹⊇𝑉−𝑣₁⟺supp(𝐴)∪supp(𝐴)⁻¹⊇𝑉−𝑣₁⟺supp𝐴∪(𝐴)⁻¹⊇𝑉−𝑣₁⟺supp𝐴∪𝐴⁻¹⊇𝑉−𝑣₁. (16)

Theorem 37. Let 𝐴 be any subset of a set 𝑉^′ and let 𝐺^′=(𝑉^′,𝑅^′) be the Cayley graph induced by the triplet (𝑉^′,∗,𝐴). Then 𝐺^′ is semiconnected if and only if ⟨𝐴⟩∪⟨𝐴⁻¹⟩⊇𝑉−𝑣₁, where 𝐴⁻¹={𝑥⁻¹∶𝑥∈𝐴}.

Theorem 38. 𝐺 is semi-connected if and only if supp(⟨𝐴⟩∪⟨𝐴⁻¹⟩)⊇𝑉−𝑣₁.

Proof. ⟺𝐺issemiconnected𝑉,𝑅⁺₀⟺𝐴issemiconnected₊₀∪𝐴⁺₀⁻¹⊇𝑉−𝑣₁⟺supp(𝐴)∪supp(𝐴)⁻¹⊇𝑉−𝑣₁⟺supp⟨𝐴⟩∪(𝐴)⁻¹⊇𝑉−𝑣₁⟺supp⟨𝐴⟩∪𝐴⁻¹⊇𝑉−𝑣₁. (17)

Theorem 39. Let 𝐺^′=(𝑉^′,𝑅^′) be the Cayley graph induced by the triplet (𝑉^′,∗,𝐴). Then 𝐺^′ is locally connected if and only if ⟨𝐴⟩=⟨𝐴⁻¹⟩, where 𝐴⁻¹=(𝑥⁻¹∶𝑥∈𝐴).

Theorem 40. Let 𝐺 is locally connected if and only if supp(⟨𝐴⟩)=supp(⟨𝐴⁻¹⟩).

Proof. 𝐺islocallyconnected⟺𝑉,𝑅⁺₀⟺𝐴islocallyconnected₊₀=𝐴⁺₀⁻¹⟺supp(𝐴)=supp(𝐴)⁻¹⟺supp⟨𝐴⟩=supp𝐴⁻¹. (18)

Theorem 41. Let 𝐺^′=(𝑉^′,𝑅^′) be the Cayley graph induced by the triplet (𝑉^′,∗,𝐴), where 𝑉^′ is finite. Then 𝐺^′ is quasi-connected if and only if it is connected.

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

Proof. 𝐺isquasi-connected⟺𝑉,𝑅⁺₀⟺isquasi-connected𝑉,𝑅⁺₀isconnected⟺𝐺isconnected. (19)

Definition 43. The 𝜇^𝑃 strength of a path 𝑃=𝑣₁,𝑣₂,…,𝑣_𝑛 is defined as min(𝜇^𝑃₂(𝑣_𝑖,𝑣_𝑗)) for all 𝑖 and 𝑗 and is denoted by 𝑆^𝑃_𝜇. The 𝜇^𝑁 strength of a path 𝑃=𝑣₁,𝑣₂,…,𝑣_𝑛 is defined as max(𝜇^𝑁₂(𝑣_𝑖,𝑣_𝑗)) for all 𝑖 and 𝑗 and is denoted by 𝑆^𝑁_𝜇.

Definition 44. Let 𝐺=(𝑉,𝜇^𝑃,𝜇^𝑁) be a bipolar fuzzy graph. Then 𝐺 is said to be(1)𝛼-connected if for every pair of vertices 𝑥,𝑦∈𝐺, there is a path 𝑃 from 𝑥 to 𝑦 such that strength (𝑃)≥𝛼,(2)weakly 𝛼-connected if a bipolar fuzzy graph (𝑉,𝑅∨𝑅⁻¹) is 𝛼-connected,(3)semi-𝛼-connected if for every 𝑥,𝑦∈𝑉, there is a path of strength greater than or equal to 𝛼 from 𝑥 to 𝑦 or from 𝑦 to 𝑥 in 𝐺,(4)locally 𝛼-connected if for every pair of vertices 𝑥 and 𝑦, there is a path 𝑃 of strength greater than or equal to 𝛼 from 𝑥 to 𝑦 whenever there is a path 𝑃^′ of strength greater than or equal to 𝛼 from 𝑦 to 𝑥,(5)quasi-𝛼-connected if for every pair 𝑥,𝑦∈𝑉, there is some 𝑧∈𝑉 such that there is directed path from 𝑧 to 𝑥 of strength greater than or equal to 𝛼 and there is a directed path from 𝑧 to 𝑦 of strength greater than or equal to 𝛼.

Remark 45. Let 𝐺=(𝑉,𝑅) be any bipolar fuzzy graph; then 𝐺 is 𝛼-connected (weakly 𝛼-connected, semi 𝛼-connected, locally 𝛼-connected or quasi 𝛼-connected) if and only if the induce fuzzy graph (𝑉,𝑅⁺₀) is connected (weakly connected, semiconnected, locally connected, or quasi-connected).

Let 𝐺 denote the Cayley bipolar fuzzy graphs 𝐺=(𝑉,𝑅) induced by (𝑉,∗,𝜇^𝑃,𝜇^𝑁). Also for any 𝛼∈[−1,1], we have the following results.

Theorem 46. 𝐺 is 𝛼-connected if and only if ⟨𝐴⟩_𝛼⊇𝑉−𝑣₁.

Proof. 𝐺isconnected⟺𝑉,𝑅_𝛼⟺𝐴isconnected_𝛼⊇𝑉−𝑣₁⟺⟨𝐴⟩_𝛼⊇𝑉−𝑣₁. (20)

Theorem 47. 𝐺 is weakly 𝛼-connected if and only if ⟨𝐴∪𝐴⁻¹⟩_𝛼⊇𝑉−𝑣₁.

Proof. 𝐺isweaklyconnected⟺𝑉,𝑅_𝛼⟺𝐴isweaklyconnected_𝛼∪𝐴_𝛼⁻¹⊇𝑉−𝑣₁⟺𝐴∪𝐴⁻¹_𝛼⊇𝑉−𝑣₁⟺𝐴∪(𝐴)⁻¹_𝛼⊇𝑉−𝑣₁. (21)

Theorem 48. 𝐺 is semi-𝛼-connected if and only if (⟨𝐴⟩_𝛼∪⟨𝐴⁻¹⟩_𝛼)⊇𝑉−𝑣₁.

Theorem 49. Let 𝐺 be locally 𝛼-connected if and only if ⟨𝐴⟩_𝛼=⟨𝐴_𝛼⁻¹⟩.

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

4. Conclusions

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

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.