An Introduction to Bipolar Single Valued Neutrosophic Graph Theory

In this paper, we first define the concept of bipolar single neutrosophic graphs as the generalization of bipolar fuzzy graphs, N-graphs, intuitionistic fuzzy graph, single valued neutrosophic graphs and bipolar intuitionistic fuzzy graphs.


Introduction
Zadeh [9] coined the term 'degree of membership' and defined the concept of fuzzy set in order to deal with uncertainty. Atanassov [8] incorporated the degree of non-membership in the concept of fuzzy set as an independent component and defined the concept of intuitionistic fuzzy set. Smarandache [2] grounded the term 'degree of indeterminacy' as an independent component and defined the concept of neutrosophic set from the philosophical point of view to deal with incomplete, indeterminate and inconsistent information in real world. The concept of neutrosophic set is a generalization of the theory of fuzzy set, intuitionistic fuzzy set. Each element of a neutrosophic set has three membership degrees including a truth membership degree, an indeterminacy membership degree, and a falsity membership degree which are within the real standard or nonstandard unit interval ] − 0, 1 + [. Therefore, if their range is restrained within the real standard unit interval [0, 1], the neutrosophic set is easily applied to engineering problems. For this purpose, Wang et al. [6] introduced the concept of the single-valued neutrosophic set (SVNS) as a subclass of the neutrosophic set. Recently, Deli et al. [7] defined the concept of bipolar neutrosophic, as a generalization of single valued neutrosophic set, and bipolar fuzzy graph, also studying some of their related properties. The neutrosophic set theory of and their extensions have been applied in various domains [22] (refer to the site http://fs.gallup.unm.edu/NSS/).
When the relations between nodes (or vertices) in problems are indeterminate, the concept of fuzzy graphs [15] and its extensions, such as intuitionistic fuzzy graphs [11,16], N-graphs [13], bipolar fuzzy graphs [11,12,14], bipolar intuitionistic fuzzy graphs [1] are not suitable. For this purpose, Smarandache [3] defined four main categories of neutrosophic graphs, two based on literal indeterminacy (I), calling them I-edge neutrosophic graph and I-vertex neutrosophic graph; these concepts are deeply studied and gained popularity among some researchers [4,5,19,20,21] due to their applications in the real world problems. The two others graphs are based on (t, i, f) components, and are called: (t, i, f)-edge neutrosophic graph and (t, i, f)-vertex neutrosophic graph; but these new concepts are not developed at all yet. Later on, Broumi et al. [18] introduced a third neutrosophic graph model. The single valued neutrosophic graph is the generalization of fuzzy graph and intuitionstic fuzzy graph. Also, the same authors [17] introduced neighborhood degree of a vertex and closed neighborhood degree of a vertex in the single valued neutrosophic graph, as a generalization of neighborhood degree of a vertex and closed neighborhood degree of vertex in fuzzy graph and intuitionistic fuzzy graph.
In this paper, motivated by the works of Deli et al. [7] and Broumi et al. [18], we introduced the concept of bipolar single valued neutrosophic graph and proved some propositions.

Preliminaries
In this section, we mainly recall some notions, which we are also going to use in the rest of the paper. The readers are referred to [6,7,10,11,13,15,18] for further details and background.
Definition 2.1 [6] Let U be an universe of a discourse; then, the neutrosophic set A is an object having the form A = {< x: , , >, x ∈ U}, where the functions T, I, F: U→] − 0,1 + [ define respectively the degree of membership, the degree of indeterminacy, and the degree of non-membership of the element x ∈ U to the set A with the condition: − 0 ≤ + + ≤ 3 + . Definition 2.2 [7] A bipolar neutrosophic set A in X is defined as an object of the form A={<x, Definition 2.4 [15] A fuzzy graph with V as the underlying set is a pair G = (σ, μ), where σ: V → [0, 1] is a fuzzy subset and μ: V × V → [0, 1] is a fuzzy relation on σ such that μ(x, y) σ(x) σ(y) for all x, y ∈ V where stands for minimum. Definition 2.5 [13] By a N-graph G of a graph , we mean a pair G= ( , ) where is an N-function in V and is an N-relation on E such that (u,v) max ( (u), (v)) all u, v V.

Bipolar Single Valued Neutrosophic Graphs
In this section, we firstly define the concept of a bipolar single valued neutrosophic relation. ii. If one of the inequalities is not satisfied, then G is not a BSVNG.

Conclusion
In this paper, we have introduced the concept of bipolar single valued neutrosophic graphs and also proved that the most widely used extensions of fuzzy graphs are particular cases of bipolar single valued neutrosophic graphs. So our future work will focus on: (1) The study of certains types of bipolar single valued neutrosophic graphs such as, complete bipolar single valued neutrosophic graphs, strong bipolar single valued neutrosophic graphs, regular bipolar single valued neutrosophic graphs. (2) The concept of energy of bipolar single valued neutrosophic graphs. (3) The study about applications, especially in traffic light problem.