CERTAIN BIPOLAR NEUTROSOPHIC COMPETITION GRAPHS

Bipolarity plays an important role in many research domains. A bipolar fuzzy model is a very important model in which positive information represents what is possible or preferred, while negative information represents what is forbidden or surely false. In this research paper, we first introduce the concept of p-competition bipolar neutrosophic graphs. We then define generalization of bipolar neutrosophic competition graphs called m-step bipolar neutrosophic competition graphs. Moreover, we present some related concepts of bipolar neutrosophic graphs. Finally, we describe an application of m-step bipolar neutrosophic competition graphs.


Introduction
The notion of competition graphs was introduced by Cohen [10] in 1968, depending upon a problem in ecology.The competition graphs have many utilizations in solving daily life problems, including channel assignment, modeling of complex economic, phytogenetic tree reconstruction, coding and energy systems.Fuzzy set theory [26] and intuitionistic fuzzy set theory [6] are useful models for dealing with uncertainty and incomplete information.But they may not be sufficient in modeling of indeterminate and inconsistent information encountered in real world.In order to cope with this issue, neutrosophic set theory was proposed by Smarandache [18] as a generalization of fuzzy sets and intuitionistic fuzzy sets.However, since neutrosophic sets are identified by three functions called truthmembership (t), indeterminacy-membership (i) and falsity-membership (f ) whose values are real standard or non-standard subset of unit interval ]0 − , 1 + [.There are some difficulties in modeling of some problems in engineering and sciences.To overcome these difficulties, in 2010, concept of single-valued neutrosophic sets and its operations defined by Wang et al. [22] as a generalization of intuitionistic fuzzy sets.Ye [24,25] has presented several novel applications of neutrosophic sets.Deli et al. [11] extended the ideas of bipolar fuzzy sets [28] and neutrosophic sets to bipolar neutrosophic sets and studied its operations and applications in decision making problems.Smarandache [20] proposed notion of neutrosophic graph and they separated them to four main categories.Wu [23] discussed fuzzy digraphs.The concept of fuzzy k-competition graphs and p-competition fuzzy graphs was first introduced by Samanta and Pal in [15], it was further studied in [5,13,16,17].Cho et al. [9] proposed the generalization of a digraphs known as m-step competition graphs.Samanta et al. [16] introduced the generalization of fuzzy competition graphs, called m-step fuzzy competition graphs.On the other hand, the concepts of bipolar fuzzy competition graphs and intuitionistic fuzzy competition graphs are discussed in [17,13].Samanta et al. [16] also introduced the concepts of fuzzy mstep neighbouthood graphs.The notion of bipolar fuzzy graphs was first introduced by Akram [1] in 2011 as a generalization of fuzzy graphs.On the other hand, Akram and Shahzadi [4] first introduced the notion of neutrosophic soft graphs and gave its applications.Akram [2] introduced the notion of single-valued neutrosophic planar graphs.Akram and Sarwar have shown that there are some flaws in Broumi et al. [8] 's definition, which cannot be applied in network models.All the predator-prey relations cannot only be represented by bipolar neutrosophic competition graphs.For example, in a food web, species may have a chain consisting of same number of preys by which they can reach to their common preys.This idea motivates the necessity of m-step bipolar neutrosophic competition graphs.In this research paper, we first introduce the concept of p-competition bipolar neutrosophic graphs.We then define generalization of bipolar neutrosophic competition graphs called m-step bipolar neutrosophic competition graphs.Moreover, we present some related concepts of bipolar neutrosophic graphs.Finally, we describe an application of m-step bipolar neutrosophic competition graphs.

Certain Bipolar Neutrosophic Competition Graphs
Definition 2.1.[26,27]A fuzzy set µ in a universe X is a mapping µ : X → [0, 1].A fuzzy relation on X is a fuzzy set ν in X × X. Definition 2.2.[28]A bipolar fuzzy set on a non-empty set X has the form The positive membership value µ P A (x) represents the strength of truth or satisfaction of an element x to a certain property corresponding to bipolar fuzzy set A and µ N A (x) denotes the strength of satisfaction of an element x to some counter property of bipolar fuzzy set A. If µ P A (x) = 0 and µ N A (x) = 0 it is the situation when x has only truth satisfaction degree for property A. If µ N A (x) = 0 and µ P A (x) = 0, it is the case that x is not satisfying the property of A but satisfying the counter property to A. It is possible for x that µ P A (x) = 0 and µ N A (x) = 0 when x satisfies the property of A as well as its counter property in some part of X.
Definition 2.5.[21]A neutrosophic set A on a non-empty set X is characterized by a truth-membership fuction t A : X → [0, 1], indeterminacy-membership function i A : X → [0, 1] and a falsity-membership function f A : X → [0, 1].There is no restriction on the sum of t A (x), i A (x) and f A (x) for all x ∈ X.
Definition 2.6.[11]A bipolar neutrosophic set A on a non-empty set X is an object of the form The positive values t P A (x), i P A (x), f P A (x) denote respectively the truth, indeterminacy and falsememberships degrees of an element x ∈ X, whereas, t N A (x), i N A (x), f N A (x) denote the implicit counter property of the truth,indeterminacy and false-memberships degrees of the element x ∈ X corresponding to the bipolar neutrosophic set A.
Definition 2.7.The height of bipolar neutrosophic set A = (t P A (x), i P A (x), G be a bipolar neutrosophic digraph then bipolar neutrosophic out-neighbourhoods of a vertex x is a bipolar neutrosophic set where, x → [0, 1], defined by t x → [0, 1], defined by f Definition 2.9.Let − → G be a bipolar neutrosophic digraph then bipolar neutrosophic in-neighbourhoods of a vertex x is a bipolar neutrosophic set where, G and there is an edge between two vertices x and y if and only if N + (x) ∩ N + (y) is non-empty.The positive truthmembership, indeterminacy-membership, falsity-membership and negative truthmembership, indeterminacy-membership, falsity-membership values of the edge (x, y) are defined as, (1)  By direct calculations we have Table 1 representing bipolar single-valued neutrosophic out-neighbourhoods.
Table 1.Bipolar single-valued neutrosophic out-neighbourhoods Then bipolar single-valued neutrosophic competition graph of Fig. 1 is shown in Fig. 2.
This gives the result, Hence, the edge (x, y) is strong.This proves the result.
We now define another extension of bipolar neutrosophic competition graph known as m-step bipolar neutrosophic competition graph.
In this paper, we will use the following notations: ), where X + x = {y| there exists a directed bipolar neutrosophic path of length m from x to y, − → P m x,y }, t ), where X − x = {y| there exists a directed bipolar neutrosophic path of length m from y to x, − → P m y,x }, t The 2−step bipolar neutrosophic competition graph is illustrated by the following example.
G as all preys are strong.So, the edge (x, y), x, y ∈ X in C m ( − → G ) have the memberships values ), and hence, all the edges are strong.
A relation is established between m-step bipolar neutrosophic competition graph of a bipolar neutrosophic digraph and bipolar neutrosophic competition graph of m-step bipolar neutrosophic digraph. where, Let Hence, N m (x) = (X x , t P x , i P x , f P x , t N x , i N x , f N x ), where X x = {y| there exists a directed bipolar neutrosophic path of length m from x to y, P m x,y }, t P x : X x → [0, 1], i P x : X x → [0, 1], f P x : X x → [0, 1], t N x : X x → [−1, 0], i N x : X x → [−1, 0], f N x : X x → [−1, 0], are defined by t P x = min{t P (x 1 , x 2 ), (x 1 , x 2 ) is an edge of P m x,y }, i P x = min{i P (x 1 , x 2 ), (x 1 , x 2 ) is an edge of P m x,y }, f P x = max{f P (x 1 , x 2 ), (x 1 , x 2 ) is an edge of P m x,y }, t N x = max{t N (x 1 , x 2 ), (x 1 , x 2 ) is an edge of P m x,y }, i N x = max{i N (x 1 , x 2 ), (x 1 , x 2 ) is an edge of P m x,y }, f N x = min{f N (x 1 , x 2 ), (x 1 , x 2 ) is an edge of P m x,y }, respectively.3. Strength of competition of applicants for international games (x, y) T (x, y) S(x, y) (Abigail, Alex) (0.12, 0.06, 0.30, −0.04, −0.08, −0.24) 1.04 (Abigail, Amelia) (0.16, 0.14, 0.30, −0.12, −0.10, −0.30) 1 (Alex, Amelia) (0.24, 0.06, 0.24, −0.08, −0.08, −0.30) 1.24 The strength to compete the others players with respect hardwork in order to achieve success is calculated in Table 3.In Table 3, T (x, y) represents the value of strength of competition between players x and y with respect to hardwork to  3, it is clear that the strength of competition between Alex and Amelia to achieve the success in particular game in international level is 1.24, while strength of competition between between Abigail and Amelia is 1, and strength of competition between between Abigail and Alex is 1.04.It is also clear from the Table 3, that Alex and Amelia are strongest contestants, as the strength of competition between them has the largest value than the other contestants.
We now elaborate this method with the help of an algorithm.

1
implies there exists a bipolar neutrosophic directed path from x to a i of length m, (x, a i ) = min{− → B P 1 (u, v)|(u, v) is an edge in