Strong Degrees in Single Valued Neutrosophic Graphs

—The concept of single valued neutrosophic graphs (SVNGs) generalizes the concept of fuzzy graphs and intuitionistic fuzzy graphs. The purpose of this research paper is to define different types of strong degrees in SVNGs and introduce novel concepts, such as the vertex truth-membership, vertex indeterminacy-membership and falsity-membership sequence in SVNG with proof and numerical illustrations


I. INTRODUCTION
In [1], [3] Smarandache explored the notion of neutrosophic sets (NS in short) as a powerful tool which extends the concepts of crisp set, fuzzy sets and intuitionistic fuzzy sets [2]- [6].This concept deals with uncertain, incomplete and indeterminate information that exist in real world.The concept of NS sets associate to each element of the set a degree of membership ( ) A T x ,a degree of indeterminacy and a degree of falsity , in which each membership degree is a real standard or non-standard subset of the nonstandard unit ] -0, 1 + [.Smaranadache [1], [2] and Wang [7] defined the concept of single valued neutrosophic sets (SVNS), an instance of NS, to deal with real application.In [8], the readers can found a rich literature on SVNS.
In more recent times, combining the concepts of NSs, interval valued neutrosophic sets (IVNSs) and bipolar neutrosophic sets with graph theory, Broumi et al. introduced various types of neutrosophic graphs including single valued neutrosophic graphs (SVNGs for short) [9], [11], [14], interval valued neutrosophic graphs [13], [18], [20], bipolar neutrosophic graphs [10], [12], all these graphs are studied deeply.Later on, the same authors presented some papers for solving the shortest path problem on a network having single valued neutrosophic edges length [17], interval valued neutrosophic edge length [32], bipolar neutrosophic edge length [21], trapezoidal neutrosophic numbers [15], SVtrapezoidal neutrosophic numbers [16], triangular fuzzy neutrosophic [19].Our approach of neutrosophic graphs are different from that of Akram et al. [26]- [28] since while Akram considers, for the neutrosophic environment (<=, <=, >=) we do (<=, >=, >=) which is better, since while T is a positive quality, I, F are considered negative qualities.Akram et al. include "I" as a positive quality together with "T".So our papers improve Akram et al.'s papers.After that, several authors are focused on the study of SVNGs and many extensions of SVNGs have been developed.Hamidi and Borumand Saeid [25] defined the notion of accessible-SVNGs and apply it social networks.In [24], Mehra and Manjeet defined the notion of single valued neutrosophic signed graphs.Hassan et al. [30] proposed some kinds of bipolar neutrosophic graphs.Naz et al. [23] studied some basic operations on SVNGs and introduced vertex degree of these operations for SVNGs and provided an application of single valued neutrosophic digraph (SVNDG) in travel time.Ashraf et al. [22] defined new classes of SVNGs and studied some of its important properties.They solved a multi-attribute decision making problem using a SVNDG.Mullai [31] solved the spanning tree problem in bipolar neutrosophic environment and gave a numerical example.
Motivated by the Karunambigai work's [29].The concept of strong degree of intuitionistic fuzzy graphs is extended to strong degree of SVNGs This paper has been organized in five sections.In Section 2, we firstly review some basic concepts related to neutrosophic set, single valued neutrosophic sets and SVNGs.In Section 3, different strong degree of SVNGs are proposed and studied with proof and example.In Section 4, the concepts of vertex truth-membership, vertex indeterminacy-

II. PRELIMINAREIS AND DEFINTIONS
In the following, we briefly describe some basic concepts related to neutrosophic sets, single valued neutrosophic sets and SVNGs. .For all x , x=(x, , A is a single valued neutrosophic element of A.
The single valued neutrosophic set can be written in the following form: Where , , denotes the truth-membership function, indeterminacy membership function and falsity membership function of the edge ( ) B respectively where , i, j {1,2,…,n} A is called the vertex set of G and B is the edge set of G.
The following Fig. 1 represented a graphical representation of single valued neutrosophic graph.

III. STRONG DEGREE IN SINGLE VALUEDNEUTROSOPHIC GRAPH
The following section introduces new concepts and proves their properties.

Definition 3.1 Given the SVN-graph G= (V, E). The Tstrong degree of a vertex
Definition 3.6 Given the SVN-graph G=(V, E).The maximum strong degree of G is defined as , where 10 Let G be a SVNG, the total strong degree of a vertex i vV  in G is defined as Definition 3.12 Given the SVN-graph G = (V, E).The maximum total strong degree of G is defined as: , where Definition 3. 13 Given the SVN-graph G=(V,E).The Tstrong size of a SVNG is defined as Definition 3.14 Given the SVN-graph G=( V, E).The Istrong size of a SVNG is defined as Definition 3.15 Given the SVN-graph G=(V, E).The Fstrong size of a SVNG is defined as 16 Given the SVN-graph G=(V, E).The strong size of a SVNG is defined as 17 Given the SVN-graph G=(V,E).The Tstrong order of a SVNG is defined as Definition 3.18 Given the SVN-graph G=(V, E).The Istrong order of a SVNG is defined as Definition 3. 19 Given the SVN-graph G=(V, E).The Fstrong order of a SVNG is defined as Definition 3.20 Given the SVN-graph G=(V, E).The strong order of a SVNG is defined as () () () ()   is minimum edge truth membership, is the maximum edge indeterminacy membership and is the maximum edge falsity membership of emits from for all j = 2,3,4,…, n.
2) is maximum edge truth membership, is the minimum edge indeterminacy membership and is the minimum edge falsity membership of among all edges from emits from to for all i = 1, 2,3,4,…, n-1. 3 , and t ( ) = (G) = ∑ .

1)
To prove that is minimum edge truth membership, is the maximum edge indeterminacy membership and is the maximum edge falsity membership of emits from v 1 j=2,3,….,n.Assume the contrary i.e.
is not an edge of minimum true membership, maximum indeterminate membership and maximum false membership emits from .Also let , 2 ≤ k ≤ n,k l be an edge with minimum true membership,, maximum indeterminate membership and maximum false membership emits from .Also since max { , } max { , }, so either or .
Since l, k 1, this is contradiction to our vertex assumption that is the unique minimum vertex true membership, is the maximum vertex indeterminate membership and is the maximum vertex false membership.

Hence
is minimum edge true membership, is the maximum edge indeterminate membership and is the maximum edge false membership of emits from to for all j = 2, 3, 4,…, n.
2) On the contrary, assume let is not an edge with maximum true membership, minimum indeterminate membership and minimum false membership emits from for 1 k n-1.On the other hand, let be an edge with maximum true membership, minimum indeterminate membership and minimum false membership emits from from 1 r n-1, k r.

Therefore, t ( ) = (G).
Also, Suppose that t (v 1 ) (G) and let , k 1 be a vertex in G with maximum F-total degree.Hence, t ( ) = ∑ Suppose that t ( ) (G).Let , 1 ≤ l ≤ n-1 be a vertex in G such that t ( ) (G) and t ( ) t ) .In addition, Also, suppose that t ( ) (G).Let , 1 ≤ l ≤ n-1 be a vertex in G such that t ( ) (G) and t ( ) t ) .In addition, Hence the lemma is proved.2) If the vertex truth membership sequence of G is of the form { , }, vertex indeterminacy membership of G is of the form { , } and vertex falsity membership sequence of G is of the form { , } with 0 n-2, then there exists exactly vertices with minimum T-total degree (G), maximum I-total degree and maximum F-total degree and exactly (n-) vertices with maximum T-total degree (G) , minimum I-total degree (G) and minimum F-total degree (G).(G), maximum I-total degree and maximum F-total degree .Also, there exists exactly vertices with maximum T-total degree (G), minimum I-total degree (G) and minimum F-total degree (G).
Proof: The proof of ( 1) and ( 2 So, there exist exactly vertices with degree (G).

V. CONCLUSION
In this paper, the idea of strong degree is imposed on the existing concepts of degrees in SVNGs.After that, we defined the vertex truth-membership, vertex indeterminacymembership and vertex falsity membership sequence in SVNG with proofs and suitable examples.In the next research, the proposed concepts can be extended to labeling neutrosophic graph and also characterize the corresponding properties.

Definition 2 . 1 [ 1 ]Definition 2 . 2 [ 7 ]
Given the universal set . 1 + [.For all x , x=(x, A is neutrosophic element of A. The neutrosophic set can be written in the following form: Given the universal set .A single valued neutrosophic set A on is characterized by a truth membership function , an indeterminacy membership function and falsity membership function , where , , :

Future
. Hence is an edge with maximum true membership, minimum indeterminate membership and minimum false membership among all edges emits from to .i = 1, 2, 3, …, n and for all other indices j, G) and t ( ) t ( ) .In addition, t ( ) = [ ∑ + ∑

Remark 3 . 32 1 )Definition 4 . 1 Note 4 . 2
In a complete SVNG G, There exists at least one pair of vertices and such that G) + (G).IV.VERTEX TRUTH MEMBERSHIP , VERTEX INDTERMINACY MEMBERSHIP AND VERTEX FALSITY MEMEBERSHIP SEQUENCE IN SVNG In this section, vertex truth membership, vertex indeterminacy membership and vertex falsity membership sequences are defined in SVNGs.Given a SVN-graph G with | | = n.The vertex truth membership sequence of G is defined to be with … where , 0 1, is the truth membership value of the vertex when vertices are arranged so that their truth membership values are nondecreasing.Particular, is smallest vertex truth membership value and is largest vertex truth membership value in G.If vertex truth membership sequence is repeated more than once in G, say r 1 times, then it is denoted by in the sequence.

Fig. 2 .
Fig. 2. Vertex truth membership sequence.Definition 4.4 Let G be a SVNG with | | = n.The vertex indeterminacy membership sequence of G is defined to be with … where , 0 1, is the indeterminacy membership value of the vertex when vertices are arranged so that their indeterminacy membership values are non-increasing.Particular, is largest vertex indeterminacy membership value and is smallest vertex indeterminacy membership value in G.

Note 4 . 5 Example 4 . 6
If vertex indeterminacy membership sequence is repeated more than once in G, say r 1 times, then it is denoted by in the sequence.In Fig.3the vertex indeterminacy membership sequence of G is {0.7, 0.6, 0.6, 0.5, 0.4, 0.4 } or { 0

Fig. 3 .Definition 4 . 7 Example 4 . 9
Fig. 3. Vertex indeterminacy membership sequence.Definition 4.7 Let G be a SVNG with | | = n.The vertex falsity membership sequence of G is defined to be with … where , 0 1, is the falsity membership value of the vertex when vertices are arranged so that their falsity membership values are non-increasing.Particular, is largest vertex falsi Y membership value and is smallest vertex falsity membership value in G.

Fig. 4 .
Fig. 4. Vertex falsity membership sequence.Definition 4.10 If a SVNG with | | = n has vertex truth membership sequence , vertex indeterminacy membership sequence and vertex falsity membership sequence in same order, then it said to have vertex single valued neutrosophic sequence and denoted by .

Fig. 5 .
Fig. 5. Vertex single valued neutrosophic sequence.The properties of vertex truth membership, vertex indeterminacy membership and vertex falsity sequences of complete SVNGs are discussed below: Theorem 4.12 Let G=(V,E) be a complete SVNG with| | = n.Then 1) If the vertex truth membership sequence of G is of the form { , }, vertex indeterminacy membership sequence of G is of the form { , } and vertex falsity membership sequence of G is of the form { , }, then a. (G) = n.and (G)= ∑ (G) = n.and (G)=∑