Single valued neutrosophic graphs: Degree, order and size

The single valued neutrosophic graph is a new version of graph theory presented recently as a generalization of fuzzy graph and intuitionistic fuzzy graph. The single valued neutrosophic graph (SVN-graph) is used when the relation between nodes (or vertices) in problems are indeterminate. In this paper, we examine the properties of various types of degrees, order and size of single valued neutrosophic graphs and a new definition for regular single valued neutrosophic graph is given.


I. INTRODUCTION
Neutrosophic set (NS for short) proposed by Smarandache [11,12] is a powerful tool to deal with incomplete, indeterminate and inconsistent information in real world. It is a generalization of the theory of fuzzy set [16], intuitionistic fuzzy sets [22,24], interval-valued fuzzy sets [18] and interval-valued intuitionistic fuzzy sets [23], then the neutrosophic set is characterized by a truth-membership degree (t), an indeterminacy-membership degree (i) and a falsity-membership degree (f) independently, 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], Nevertheless, NSs are hard to be apply in practical problems since the values of the functions of truth, indeterminacy and falsity lie in] • 0, 1 + [. Therefore, Wang et al. [14] presented single-valued neutrosophic sets (SVNSs) whose functions of truth, indeterminacy and falsity lie in [0, 1]. The same authors introduced the notion of interval valued neutrosophic sets [15] as subclass of neutrosophic sets in which the value of truth-membership, indeterminacymembership and falsity-membership degrees are intervals of numbers instead of the real numbers. neutrosophic sets and its extensions such as single valued neutrosophic sets, interval neutrosophic sets, simplified neutrosophic sets and so on have been applied in a wide variety of fields including computer science, engineering, mathematics, medicine and economic [1,2,3,7,8,10,11,12,13,17,19,20,21,27,33,34,35].
Many works on fuzzy graphs and intuitionistic fuzzy graphs [4,5,6,27,28,41] have been carried out and all of them have considered the vertex sets and edge sets as fuzzy and /or intuitionistic fuzzy sets. But, when the relations between nodes (or vertices) in problems are indeterminate, the fuzzy graphs and intuitionistic fuzzy graphs are failed. For this purpose, Smarandache [9] have defined four main categories of neutrosophic graphs, two based on literal indeterminacy (I), which called them; I-edge neutrosophic graph and I-vertex neutrosophic graph, these concepts are studied deeply and has gained popularity among the researchers due to its applications via real world problems [38,39,40]. The two others graphs are based on (t, i, f) components and called them; The (t, i, f)-edge neutrosophic graph and the (t, i, f)vertex neutrosophic graph, these concepts are not developed at all. Later on, Broumi et al. [30] introduced a third neutrosophic graph model combined the (t, i, f)-edge and and the (t, i, f)-vertex neutrosophic graph and investigated some of their properties. The third neutrosophic graph model is called 'single valued neutrosophic graph' (SVNG for short). The single valued neutrosophic graph is the generalization of fuzzy graph and intuitionistic fuzzy graph. Also, Broumi et al. [31] introduced the concept of bipolar single valued neutrosophic graph as a generalization of fuzzy graphs, intuitionistic fuzzy graph, N-graph, bipolar fuzzy graph and single valued neutrosophic graph and studied some of their related properties. The same authors [32,33,34], introduced the concept of interval valued neutrosophic graph as a generalization of single valued neutrosophic graph and have discussed some of their properties with proof and examples. The remainder of this paper is organized as follows. In Section 2, we review some basic concepts about neutrosophic sets, single valued neutrosophic sets, single valued neutrosophic graph and complete single valued neutrosophic graph. The type of degrees in single valued neutrosophic graphs such as degree of vertex, total degree, effective degree, neighborhood degree, closed neighborhood degree are defined in Section 3. Furthermore, some properties of the proposed degrees are discussed with numerical examples. In Section 4, we present the concept of regular single valued neutrosophic graph and proved some propositions. In addition, Section 5 also present the concept of order and size of single valued neutrosophic graph. Finally, Section 6 outlines the conclusion of this paper and suggests several directions for future research.

II. PRELIMINARIES
In this section, we mainly recall some notions related to neutrosophic sets, single valued neutrosophic sets, fuzzy graph, intuitionistic fuzzy graph, single valued neutrosophic graphs, relevant to the present work. See especially [12,14,26,28] for further details and background. Definition 2.1 [12]. Let X be a space of points (objects) with generic elements in X denoted by x; then the neutrosophic set A (NS A) is an object having the form A = {< x: Since it is difficult to apply NSs to practical problems, Wang et al. [14] introduced the concept of a SVNS, which is an instance of a NS and can be used in real scientific and engineering applications.   [28]. A single valued neutrosophic graph (SVN-graph) with underlying set V is defined to be a pair G= of indeterminacy-membership and falsity-membership of the element i v ∈ V, respectively, and Denotes the degree of truth-membership, indeterminacymembership and falsity-membership of the edge ( , ) We call A the single valued neutrosophic vertex set of V, B the single valued neutrosophic edge set of E, respectively.

Definition 3.2:
Let G= (A, B) be a single valued neutrosophic graph. Then the total degree of a vertex i v ∈ G is defined by ( )

Definition 3.3:
The maximum degree of G is ( ) (the empty set).

Definition 3.9:
The effective degree of a vertex 'v' in G is defined by [ ] [ ] d v |v ∈ V} denotes the maximum effective F-degree. Example 3.12: Consider a SVN-graph as in Fig.3. By usual computation, we have the effective degrees for all vertices Here 2 3 v v is only effective degree.

Definition 3.15:
The minimum neighbourhood degree is defined as It is clear from calculation that G is regular single valued neutrosophic graph (RSVN-graph).
By definition, the closed neighborhood T-degree of each vertex is the sum of the truth-membership values of the vertices and itself, the closed neighborhood I-degree of each vertex is the sum of the indeterminacy-membership values of the vertices and itself and the closed neighborhood F-degree of each vertex is the sum of the falsitymembership values of the vertices and itself, Therefore all the vertices will have the same closed neighborhood Tdegree, closed neighborhood -degree and closed neighborhood F -degree. This implies minimum closed neighborhood degree is equal to maximum closed neighborhood degree (i. VI. CONCLUSION In this paper we have described degree of a vertex, total degree, effective degree, neighborhood degree, closed neighborhood, order and size of single valued neutrosophic graphs. The necessary and sufficient conditions for a single valued neutrosophic graph to be the regular single valued neutrosophic graphs have been presented. Further, we are going to study some types of single valued neutrosophic graphs such irregular and totally irregular single valued neutrosophic graphs and single valued neutrosophic hypergraphs.