Bipolar Picture Fuzzy Graphs with Application

: In this manuscript, we introduce and discuss the term bipolar picture fuzzy graphs along with some of its fundamental characteristics and applications. We also initiate the concepts of complete bipolar picture fuzzy graphs and strong bipolar picture fuzzy graphs. Firstly, we apply different types of operations to bipolar picture fuzzy graphs and then we introduce various products of bipolar picture fuzzy graphs. Several other terms such as order and size, path, neighbourhood degrees, busy values of vertices and edges of bipolar picture fuzzy graphs are also discussed. These terminologies also lay the foundations for the discussion about the regular bipolar picture fuzzy graphs. Moreover, we also discuss isomorphisms, weak and co-weak isomorphisms and automorphisms of bipolar picture fuzzy graphs. Finally, at the base of bipolar picture fuzzy graph we present the construction of a bipolar picture fuzzy acquaintanceship graph, which would be an important tool to measure the symmetry or asymmetry of acquaintanceship levels of social networks, computer networks etc.


Introduction
In 1965, Zadeh [1] introduced the term fuzzy sets (FSs), which is extensively used in different fields such as life sciences, social sciences, engineering, theory of decision making, computer sciences etc. Subsequently, many generalizations of the fuzzy sets have been explored in the literature like interval-valued fuzzy sets (IVFSs), bipolar fuzzy sets (BPFs), intuitionistic fuzzy sets (IFSs), picture fuzzy sets (PFSs) and so on (see e.g., [2,3]). The term interval-valued fuzzy set (IVFS) was also introduced by Zadeh [4]. Another generalization of fuzzy sets termed bipolar fuzzy sets (BPFSs) was introduced in [5]. In bipolar fuzzy sets (BPFSs) the membership value was considered in the interval [-1, 1]. In continuation, recently, the term bipolar Pythagorean fuzzy sets along with its applications towards decision making theory is explored in [6]. Various types of relations on BFSs were introduced in [7]. Basically, the term bipolar fuzzy relations (BPFRs) is the direct extension of fuzzy relations. BPFRs were also given a name "bifuzzy relations". Some new types of bipolar fuzzy relations and bipolar fuzzy equivalence relation were discussed in [7]. Atanassov [8] introduced the notion of intuitionistic fuzzy sets which was another generalized form of the fuzzy sets. Similarly, the generalization of both the fuzzy sets and intuitionistics fuzzy sets termed picture fuzzy sets (PFSs) was initiated by Cuong [9]. He also studied several operations and characteristics of PFSs. PFS is described by assigning three memberships values to the object which are neutral, positive and negative. After this, Bo et al. [10] introduced few new operations and relations on PFSs. Cuong et al. [11] introduced various types of fuzzy logical operators in the setting of PFSs.
On the other hand, Rosenfeld [12] extended the scope of fuzzy sets towards graph theory by initiating the notion of fuzzy graphs(FGs). Later on, Bhattacharya [13] added Definition 13. [35] A bipolar intuitionistic fuzzy graph (BPIFG) on V is the pair G = (A, B), where A = (α P A (u), α N A (u), β P A (u), β N A (u)) is a BPIFS on V and B = (α P B (u), α N B (u), β P B (u), β N B (u)) is a BPIFS on E ⊆ V × V satisfying for all u, v ∈ E.
Following [36], for each x in X, α P (x) stands for the positive membership degree, β P (x) for the positive non-membership degree and γ P (x) for the positive neutral degree. Alternatively, α N (x) represents the negative membership degree, β N (x) is the negative non-membership degree and γ N (x) is a negative neutral degree. On the other hand, if α P (x) = 0 while all other mappings are mapped to zero then it means that x has only a positive membership property of the bipolar picture fuzzy set. Similarly, if α N (x) = 0 while all other mappings matched to zero (or equal to zero) then it reflects that x has only the negative membership property of a BPPFS. Additionally, if γ P (x) = 0 and remaining mappings are mapped to zero then it reflects that x has only the positive neutral property of a BPPFS. By γ N (x) = 0 and the other mapping goes to zero then we mean that x has only the negative neutral property of a BPPFS. However, if β P (x) = 0 while all other mapping matched to zero then it implies that x has only the positive nonmembership property of a BPPFS. Finally, if β N (x) = 0 while remaining are zero then it implies that x has only the negative nonmembership property in a BPPFS. Definition 17. Let G * = (V, E) be a graph. A pair G = (C, D) is said to be a bipolar picture fuzzy graph (BPPFG) on G * , where } is a bipolar picture fuzzy set on E ⊆ V × V such that for every edge uv ∈ E, Example 1. One can easily verify that the graphs shown in Figure 1a,b are BPPFGs.
Definition 20. Let J * = (V 1 , E 1 ) and K * = (V 2 , E 2 ) be two graphs. Let J = (C 1 , } is a BPPFS on E 2 be the two BPPFGs. Then the operations union and intersection between J and K can be defined as For any vertex u: Case (i): For any vertex u: Case (i): Definition 21. Let G 1 = (C 1 , D 1 ) and G 2 = (C 2 , D 2 ) be the two BPPFGs on G * = (V 1 , E 1 ) and G * * = (V 2 , E 2 ), respectively. Then the ring sum where uv represents an edge between the two vertices u, v while E 1 , E 2 represent edges sets in G 1 and G 2 , respectively. Theorem 1. Ring sum of two BPPFGs is a BPPFG.

Proof. Let us consider two BPPFGs
). Then we have the following cases. Case 1: Finally, to prove H ⊕ H = ∅. Let u ∈ V be any vertex, then by Definition (21) we have Similarly, for any edge (u, v) ∈ E. Following Definition 21, we have Otherwise, it is a free vertex.

Definition 24. The busy value of a vertex u of a BPPFG H
Definition 27. Two vertices u and v are connected by a path p i.e., p : If H contains a "x − y" walk of length k, then H contains a "x − y" path of length k.

Different Types of Products of Bipolar Picture Fuzzy Graphs
Definition 28. The strong product of two BPPFGs The strong product of two BPPFGs is always a BPPFG.

Example 2.
Let us consider two BPPFGs graphs given in Figure 1a,b. Then their semi-strong product is as follows.

Homomorphism of Bipolar Picture Fuzzy Graphs
Definition 32. Let H 1 and H 2 be the two BPPFGs. A homomorphism f : H 1 → H 2 is the map f :

Definition 33.
Let H 1 and H 2 be the two BPPFGs. An isomorphism f : H 1 → H 2 is a bijective mapping f :

Proposition 4. Let H = (C, D) be a BPPFG and Aut(H) be the set of all automorphisms of H.
is a bipolar picture fuzzy group on Aut(H).

Complete and Strong Bipolar Picture Fuzzy Graphs
Example 3. One can easily verify that the graph shown in Figure 1a is a complete BPPFG.
Proof. As we know that the strong product of BPPFGs is a BPPFG and each pair of vertices are adjacent, . Similarly, one can easily verify that Hence, H 1 ⊗ H 2 is a complete BPPFG.
} is said to be a strong bipolar picture fuzzy graph (in short, BPPFG) if for all u, v ∈ E. Figure 3 is a strong BPPFG.

Example 4. The graph shown in
Example 5. Graph in Figure 4 is the complement of a strong BPPFG shown in Figure 3. Theorem 6. Let G 1 = (C 1 , D 1 ) and G 2 = (C 2 , D 2 ) be the two strong BPPFGs. Then G 1 G 2 is strong BPPFG.

Application
Modelling by using graphs has vast applications in various fields of computer science, mathematics, chemistry, physics, social sciences etc. Usually such types of models require more arrangements than merely the adjacencies among the vertices. In the study of social circuits, it is found that two people know each other i.e., if they are familiar (acquainted), or whether they are friends of each others (in the real world or in the virtual world such as Instagram) and so on. We can label each person in a particular group of people by a vertex u. There is an undirected edge between a vertex u and v if two people has a relationship with each other. In such type of graphs no multiple edges and usually no loops are needed. There is an edge between the vertices u and v when there is any acquaintanceship exists between them. In such graphs there does not exist any loop or multiple edges. In acquaintanceship graphs, the vertex (node) represents the level of acquaintanceship (how much a person is socialized or familiar/friendly) of a person while the the edge is the acquaintanceship between two persons in the social network. Since each vertex has equal importance in the classical graphs, it is not possible to graph the social networks model properly through them. In addition, all social units (individual or organization) present in social groups must be considered with equal importance in the classical graph theory. However, in the real life, the situation is different. Similarly, every edge (relationship) has an equal strength in the classical graphs. Moreover, in classical graphs it is assumed that the relationship between two social units are of equal strength, however, in real life it is not possible. Thus the acquaintance of the person has fuzzy boundary and hence can be better represented through the fuzzy graphs. In fuzzy acquaintanceship graph, each vertex represents the person and its membership value which reflects the strength of acquaintance of the person within the social group. Hence we present a fuzzy acquaintanceship graph, a bipolar fuzzy acquaintanceship and consequently a bipolar picture fuzzy acquaintanceship graph models to find out that how much the person is acquainted (social) within a group. Bipolar picture fuzzy acquaintanceship graph models would be more efficient to detect the symmetry or asymmetry existing between entities through the levels of acquaintanceships in social networks, computer networks etc.

Fuzzy Acquaintanceship Graph
We take a fuzzy acquaintanceship graph of a social network which is shown in Figure 5. In which the nodes represent the degree of the level of acquaintance of a person within the social group. The degree of the level of acquaintance is expressed in its membership value. Degree of membership states that how much a person is acquainted e.g., X is 60% acquainted within the group. The edges of a graph describe the acquaintanceship level of one person with the other person. The membership degree of edges can be considered in terms of positive percentage e.g., Y has 40% acquaintanceship level with X and so.

Bipolar Fuzzy Acquaintanceship Graph
The acquaintanceship of a person may be positive or negative. Suppose if a person A and B belong to a social network but having not a good relationships between them then the acquaintanceship between them is negative. We can depict such circumstances through the bipolar fuzzy acquaintanceship graph. Consider a bipolar fuzzy acquaintanceship graph of a social group shown in Figure 6. In which the nodes are reflecting the degree of the level of acquaintanceship of a person belongs to a social group and the edges represent the degree of acquaintanceship levels among the persons. Degree of positive membership can be interpreted as how much a person acquainted and negative membership tells us that how much a person losses the the level of acquaintance, X has 50% level of acquaintance within the group but it loses 20% level in the same group. Edges of the graph reflect the acquaintance of one person with the other persons in the group. The positive and negative memberships degrees of edges describes the percentage of positive and negative acquaintance,for instance e.g., X is acquainted 10% with W and W is not acquainted 10% with X.

Bipolar Picture Fuzzy Acquaintanceship Graph
The degree of the acquaintanceship of a person is defined in terms of its membership (positive, negative), non-membership (positive, negative) and neutral membership (positive, negative) values. The degree of the membership (positive, negative) can be interpreted as a good acquaintanceship (gaining, losing). By a good acquaintanceship, we mean the acquaintance with intimacy. The degree of non-membership (positive, negative) can be interpreted as a bad acquaintanceship (gaining, losing). Bad acquaintanceship means acquaintance with ill-famed. The degree of neutral membership (positive, negative) represents that the person having a loose acquaintanceship (gaining, losing). By a loose acquaintanceship, we mean someone we do not know well enough but we probably see them around occasionally. In Figure 7, X gains (resp., loses) 30% (resp., 50%) good acquaintanceship, he gains 20% (resp., loses 10%) bad acquaintanceship but he gains (resp., loses) 30% (resp., loses 20%) loose acquaintanceship within the social group. On the other hands, the edges of a graph (Figure 7) reflect the acquaintanceship of one person with another person. The degree of a membership (positive and negative), non-membership (positive and negative) and neutral membership(positive and negative) of the edges can be interpreted as the percentage of good acquaintanceship (gaining, losing), bad acquaintanceship (gaining, losing) and non-acquaintanceships (gaining, losing). Furthermore, it is easy to verify that the values of the edges of a graph in Figure 7 are satisfying the below conditions. α P D (uv) ≤ min(α P C (u), α P C (v)), α N D (uv) ≥ max(α N C (u), α N C (v)) γ P D (uv) ≤ min(γ P C (u), γ P C (u)), γ N D (uv) ≥ max(γ N C (u), γ N C (v)) β P D (uv) ≥ max(β P C (u), β P C (v)), β N D (uv) ≤ min(β N C (u), β N C (v)).
Refer to the graph shown in Figure 7, we have α P D (UV) ≤ min (α P C (U), α P C (V)) ⇒ α P D (UV) ≤ min (0.4, 0.2) ⇒ 0.1 ≤ 0. Hence by doing same calculations for the other vertices and edges of the graph shown in Figure 7, it is easy to verify that the graph given in Figure 7 is a bipolar picture fuzzy acquaintanceship graph. Similarly, by the values of vertices and edges, one can easily deduce that the graph in Figure 7 is asymmetric.

Conclusions
Fuzzy graphs theory plays a significant role in modeling many real world problems containing uncertainties in different fields such as decision making theory, computer science, optimization problems, data analysis, networking etc. In this perspective, a number of generalizations of fuzzy graph have been introduced to deal with the difficult and complex real life problems. The picture fuzzy set is a direct extension of both the fuzzy sets and intuitonistic fuzzy sets. Bipolar fuzzy set is another generalized form of fuzzy set which is also an effective tool for the multiagent decision analysis. The main goal of this manuscript is to initiate the concepts of bipolar picture fuzzy graph and its different characterizations. In this article, first we propose the definition of bipolar picture fuzzy graphs based on the bipolar picture fuzzy relation. In this article, we have introduced the terms bipolar picture fuzzy graphs, complete bipolar picture fuzzy graphs and strong bipolar picture fuzzy graphs along with their several fundamental properties. For the sake of investigations, we have introduced and applied numerous operations like union, intersection, complement, ring sum etc. on bipolar picture fuzzy graphs. We also introduce different types of products of bipolar picture fuzzy graphs like semi-strong product, direct product, normal products etc. Several other terms such as order and size, path neighborhood degrees, busy values of vertices and edges of bipolar picture fuzzy graphs are also studied. These terminologies also laid the foundation for the discussion of regular bipolar picture fuzzy graphs. Furthermore, we also discuss isomorphisms, weak and co-weak isomorphisms and automorphisms of bipolar picture fuzzy graphs. During this, we have proved that the set of all automorphisms of a bipolar picture fuzzy graph forms a group. Finally, we construct a bipolar picture fuzzy acquaintanceship graph which reflects the importance of our theoretical results produced in this article. Evidently, the network modelled through a bipolar picture fuzzy acquaintanceship graph shown in Figure 7 has no any symmetry. However, we can also model a symmetric relation through the bipolar picture fuzzy acquaintanceship graph. On the same patterns, one could express collaboration graph, computer networking, social networking, web graphs in the frame of bipolar picture fuzzy graphs. In general, numbers of applications of bipolar fuzzy graphs and picture fuzzy graphs have been explored in different fields of social, natural and computer sciences. Evidently, bipolar picture fuzzy graphs would be an important tool to deal with real world problems containing uncertainties. Finally, one can extend this work by introducing bipolar interval-valued picture fuzzy graphs.