Certain Competition Graphs Based on Intuitionistic Neutrosophic Environment

Abstract: The concept of intuitionistic neutrosophic sets provides an additional possibility to represent imprecise, uncertain, inconsistent and incomplete information, which exists in real situations. This research article first presents the notion of intuitionistic neutrosophic competition graphs. Then, p-competition intuitionistic neutrosophic graphs and m-step intuitionistic neutrosophic competition graphs are discussed. Further, applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition are described.


Introduction
Euler [1] introduced the concept of graph theory in 1736, which has applications in various fields, including image capturing, data mining, clustering and computer science [2][3][4][5].A graph is also used to develop an interconnection between objects in a known set of objects.Every object can be illustrated by a vertex, and interconnection between them can be illustrated by an edge.The notion of competition graphs was developed by Cohen [6] in 1968, depending on 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 and intuitionistic fuzzy sets theory are useful models for dealing with uncertainty and incomplete information.However, they may not be sufficient in modeling of indeterminate and inconsistent information encountered in the real world.In order to cope with this issue, neutrosophic set theory was proposed by Smarandache [7] as a generalization of fuzzy sets and intuitionistic fuzzy sets.However, since neutrosophic sets are identified by three functions called truth-membership (t), indeterminacy-membership (i) and falsity-membership ( f ), whose values are the 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, Smarandache in 1998 [8] and Wang et al. [9] in 2010 defined the concept of single-valued neutrosophic sets and their operations as a generalization of intuitionistic fuzzy sets.Yang et al. [10] introduced the concept of the single-valued neutrosophic relation based on the single-valued neutrosophic set.They also developed kernels and closures of a single-valued neutrosophic set.The concept of the single-valued intuitionistic neutrosophic set was proposed by Bhowmik and Pal [11,12].
The valuable contribution of fuzzy graph and generalized structures has been studied by several researchers [13][14][15][16][17][18][19][20][21][22].Smarandache [23] proposed the notion of the neutrosophic graph and separated them into four main categories.Wu [24] discussed fuzzy digraphs.Fuzzy m-competition and p-competition graphs were introduced by Samanta and Pal [25].Samanta et al. [26] introduced m-step fuzzy competition graphs.Dhavaseelan et al. [27] defined strong neutrosophic graphs.Akram and Shahzadi [28] introduced the notion of a single-valued neutrosophic graph and studied some of its operations.They also discussed the properties of single-valued neutrosophic graphs by level graphs.Akram and Shahzadi [29] introduced the concept of neutrosophic soft graphs with applications.Broumi et al. [30] proposed single-valued neutrosophic graphs and discussed some properties.Ye [31][32][33] has presented several novel concepts of neutrosophic sets with applications.In this paper, we first introduce the concept of intuitionistic neutrosophic competition graphs.We then discuss m-step intuitionistic neutrosophic competition graphs.Further, we describe applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition.Finally, we present our developed methods by algorithms.
Our paper is divided into the following sections: In Section 2, we introduce certain competition graphs using the intuitionistic neutrosophic environment.In Section 3, we present applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition.Finally, Section 4 provides conclusions and future research directions.

Intuitionistic Neutrosophic Competition Graphs
We have used standard definitions and terminologies in this paper.For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [34][35][36][37][38][39][40][41][42][43][44].Definition 1. [38] Let X be a fixed set.A generalized intuitionistic fuzzy set I of X is an object having the form I={(u, µ I (u), ν I (u))|u ∈ U}, where the functions µ I (u) :→ [0, 1] and ν I (u) :→ [0, 1] define the degree of membership and degree of non-membership of an element u ∈ X, respectively, such that: This condition is called the generalized intuitionistic condition.Definition 2. [11] An intuitionistic neutrosophic set (IN-set) is defined as Ȃ = (w, t Ȃ(w), i Ȃ(w), f Ȃ(w)), where: for all, w ∈ X, such that: Definition 3. [12] An intuitionistic neutrosophic relation (IN-relation) is defined as an intuitionistic neutrosophic subset of X × Y, which has of the form: where t R , i R and f R are intuitionistic neutrosophic subsets of X × Y satisfying the conditions: 1. one of these t R (w, z), i R (w, z) and f R (w, z) is greater than or equal to 0.5, denote the truth-membership, indeterminacy-membership and falsity-membership of an element w ∈ X and: denote the truth-membership, indeterminacy-membership and falsity-membership of an element (w, z) ∈ E (edge set).
By direct calculations, we have Tables 1 and 2 representing IN-out and in-neighborhoods, respectively.Therefore, there is an edge between two vertices in INC-graph C( − → G ), whose truth-membership, indeterminacy-membership and falsity-membership values are given by the above formula.

Definition 9. For an IN
Otherwise, it is called weak.
, where p, q and r are the truth-membership, indeterminacy-membership and falsity-membership values of either the edge (w, a) or the edge (z, a), respectively.Here, Then, Therefore, the edge (w, z) in C( − → G ) is independent strong if and only if p > 0.5, q < 0.5 and r < 0.5.Hence, the edge (w, z) of C( − → G ) is independent strong if and only if: We illustrate the theorem with an example as shown in Figure 5.

Theorem 2. If all the edges of an IN-digraph
− → G are independent strong, then: Proof.Suppose all the edges of IN-digraph − → G are independent strong.Then: for all the edges (w, z) in − → G .Let the corresponding INC-graph be C( − → G ). Case (1): When N + (w) ∩ N + (z) = ∅ for all w, z ∈ X, then there does not exist any edge in C( − → G ) between w and z.Thus, we have nothing to prove in this case.
Proof.Using similar arguments as in Theorem 2.1.[39], it can be proven. 3 We now construct the INC-graph G , where w = (t w , i w , f w ) and k = (t k , i k , f k ), from C( − → G 1 ) * and C( − → G 2 ) * using Theorem 2.14.We obtained two sets of edges by using Condition (1).

All the truth-membership, indeterminacy-membership and falsity-membership degrees of adjacent edges of G
* and G are given in Table 5.
and t w : For every vertex w ∈ X, the intuitionistic neutrosophic singleton set, A w = (w, h 1 , h 2 , h 3 ), such that: ), which has the same intuitionistic neutrosophic set of vertices in G and has an intuitionistic neutrosophic edge between two vertices w, z ∈ X in N(G) if and only if N(w) ∩ N(z) is a non-empty IN-set in G.The truth-membership, indeterminacy-membership and falsity-membership values of the edge (w, z) are given by: ), which has the same intuitionistic neutrosophic set of vertices in G and has an intuitionistic neutrosophic edge between two vertices w, z The truth-membership, indeterminacy-membership and falsity-membership values of the edge (w, z) are given by: Thus, for each edge (w, z) in IN-graph G, there exists an edge (w, z) in N[G].
Definition 13.The support of an IN-set Ȃ = (w, t Ȃ, i Ȃ, f Ȃ) in X is the subset Â of X defined by: and |supp( Â)| is the number of elements in the set.
We now discuss p-competition intuitionistic neutrosophic graphs.Suppose p is a positive integer.Then, p-competition The three-competition IN-graph is illustrated by the following example.
For p = 3, t(w 1 , w 2 ) = 0.003, i(w 1 , w 2 ) = 0.003 and f (w 1 , w 2 ) = 0.02.As shown in Figure 14.Now, there is an edge between human and bald eagle; snake and bald eagle; salamander and trout; salamander and frog; trout and frog; trout and dragonfly; trout and mayfly; dragonfly and mayfly in the INC-graph, which highlights the competition between them; and for the other pair of species, there is no edge, which indicates that there is no competition in the INC-graph Figure 18.For example, there is an edge between human and bald eagle indicating a 12% degree of likeness to prey on the same species, a 3% degree of indeterminacy and a 4% degree of non-likeness between them.We present our method, which is used in our ecosystem application in Algorithm 1.
Step 1. Input the truth-membership, indeterminacy-membership and falsity-membership values for set of n species.
Step 2. If for any two distinct vertices w i and w j , t(w i w j ) > 0, i(w i w j ) > 0, f (w i w j ) > 0, then (w j , t(w i w j ), i(w i w j ), f (w i w j )) ∈ N + (w i ).
Step 3. Repeat Step 2 for all vertices w i and w j to calculate IN-out-neighborhoods N + (w i ).
Step 4. Calculate N + (w i ) ∩ N + (w j ) for each pair of distinct vertices w i and w j .
Step 6.If N + (w i ) ∩ N + (w j ) = ∅, then draw an edge w i w j .
Step 7. Repeat Step 6 for all pairs of distinct vertices.
Step 8. Assign membership values to each edge w i w j using the conditions: t(w i w j ) = (w i ∧ w j )H 1 [N + (w i ) ∩ N + (w j )] i(w i w j ) = (w i ∧ w j )H 2 [N + (w i ) ∩ N + (w j )] f (w i w j ) = (w i ∨ w j )H 3 [N + (w i ) ∩ N + (w j )].
The INC-graph is shown in Figure 20.The solids lines indicate the strength of competition between two applicants, and dashed lines indicate the applicant competing for the particular career.For example, Nazneen and Rosaleen both are competing for the career, surgery, and the strength of competition between them is (0.06, 0.1, 0.08).In Table 10, W(z, c) represents the competition of applicant z for career c with respect to loyalty quality, indeterminacy and disloyalty to compete with the others.The strength to compete with the other applicants with respect to a particular career is calculated in Table 10.
From Table 10, Nazneen and Rosaleen have equal strength to compete with the other for the career, surgery.Abner and Casper have equal strength of competition for the career, anatomy.Amara competes with the others for the career, pharmacy and medicine.

Definition 6 .
Let − → G be an IN-digraph, then the intuitionistic neutrosophic in-neighborhoods (IN-in-neighborhoods) of a vertex w are an IN-set:

Table 3 .
IN

Table 4 .
IN which has the same intuitionistic neutrosophic set of vertices as in − → G and has an intuitionistic neutrosophic edge between two vertices w, z ∈ X in C p ( − → G ) if and only if |supp(N + (w) ∩ N + (z))| ≥ p.The truth-membership value of edge (w, z) in

Table 7 .
Likeness, indeterminacy and dislikeness of preys and predators.