Representation of competitions by complex neutrosophic information

Abstract

The concept of generalized complex neutrosophic graph of type 1 is an extended approach of generalized neutrosophic graph of type 1. It is an effective model to handle inconsistent information of periodic nature. In this research article, we discuss certain notions, including isomorphism, competition graph, minimal graph and competition number corresponding to generalized complex neutrosophic graphs. Further, we describe these concepts by several examples and present some of their properties. Moreover, we analyze that a competition graph corresponding to a generalized complex neutrosophic graph can be represented by an adjacency matrix with suitable real life examples. Also, we enumerate the utility of generalized complex neutrosophic competition graphs for computing the strength of competition between the objects. Finally, we highlight the significance of our proposed model by comparative analysis with the already existing models.

Keywords

Isomorphism competition graphs of type 1 minimal graphs competition matrix comparative analysis

1 Introduction

Graph theory is a conceptual framework for solving different problems in algebra, computer science, biological science and social networks. Cohen [10] introduced the notion of competition graphs to represent the competition between species in the food web. Competition graphs have several applications, including modeling of communication over a noisy channel, economic systems and channel assignment, etc.

Zadeh [37] originated the definition of a fuzzy set as a strong mathematical tool in handling and solving uncertain systems which exists in industry, nature and humanity. Atanassov [8] generalized the notion of fuzzy sets and proposed the concept of an intuitionistic fuzzy set whose characteristic function has two components, i.e., truth membership function and falsity membership function, which are more or less independent from each other such that their sum should not be greater than one. Intuitionistic fuzzy sets discuss higher order uncertainty as compared to fuzzy sets and they are comparatively intensive and meaningful. Smarandache [31, 32] introduced the notion of neutrosophic set in which the membership value is determined by truth, indeterminacy and falsity components. Moreover, all these membership values are real standard or non-standard subsets of the non-standard unit interval ] 0^-, 1⁺ [ and also their sum is not restricted. Later on, Wang et al. [33] proposed the idea of single-valued neutrosophic sets to apply neutrosophic sets in real life situations more easily and conveniently. Recently, a new concept in neutrosophic sets called neutrosophic structured element was introduced by Edalatpanah [13]. This concept has proven to be a fruitful tool to deal with neutrosophic decision making problems. Moreover, Edalatpanah contributed a lot of research work in the field of neutrosophic set theory (Reader can see [14 –16]). Sheng et al. [29] discussed a new fractal approach for describing induced-fracture porosity/permeability/compressibility in stimulated unconventional reservoirs. Further, Zhao et al. [39] developed a new workflow for fast optimization of waterflooding performance. Kumar et al. [21] presented an efficient method to solve Gaussian valued neutrosophic shortest path problems. The uncertainty in many real life problems is often change with respect to time period. Ramot et al. [23, 24] generalized the membership range to “unit circle in the complex plane”, conflict the others who limited the range to [0, 1] and introduced the concept of complex fuzzy set. Ramot et al. [24] discussed the union, intersection and compliment of complex fuzzy sets with the help of illustrative examples. A systematic review of complex fuzzy sets was proposed by Yazdanbakhsh and Dick [35]. Complex intuitionistic fuzzy sets were introduced by Alkouri and Salleh [6]. Further, complex neutrosophic sets were proposed by Ali and Smarandache [7].

Kaufmann [20] presented the first definition of a fuzzy graph by considering Zadeh’s fuzzy relations. Rosenfeld [25] was the first person who studied fuzzy relations on fuzzy sets and established the theory of fuzzy graphs. He studied several basic graph-theoretic concepts, including bridges, trees and established some of their properties. Mordeson and Peng [22] discussed some operations on fuzzy graphs. Dey et al. [12] proposed fuzzy minimum spanning tree problem. Fuzzy minimum spanning tree has emerged from various real life applications in different areas by considering uncertainty that exists in arc lengths of a fuzzy graph. Based on [33], Karasslan and Davvaz [17] investigated the properties of single-valued neutrosophic graphs which are the extensions of fuzzy graphs and suitable to handle inconsistent information. Hamidi and Saeid [18] discussed single-valued neutrosophic graphs in wireless sensor networks. Further, several new concepts related to single-valued neutrosophic graphs with their applications were discussed in [1–5 , 38]. Zuo et al. [40] introduced picture fuzzy graph. It can work very efficiently in uncertain scenarios which involve more answers to these type: yes, no, abstain and refusal. Kartick et al. [19] presented the concept of m-polar neutrosophic graph. An m-polar neutrosophic graph is much efficient for such real word problems which can construct more precise, flexible and comparable system as compared to the fuzzy and neutrosophic graph model. Fuzzy graphs have a property according to which edge membership value should be less than or equal to the minimum of the membership value of its end vertices. Due to this restriction on edge membership values, fuzzy graphs may fail to communicate many real life problems. Therefore, to remove this restriction from the edge membership values, the concept of generalized fuzzy graphs was introduced by Samanta and Sarkar [27]. These graphs are appropriate to represent many real life uncertain networks, where fuzzy graphs may fail to communicate suitable results. Further, Broumi et al. [9] introduced the notion of complex generalized neutrosophic graphs of type 1, which are the extensions of generalized single-valued neutrosophic graphs of type 1. Recently, Saba et al. [30] discussed certain properties of generalized complex neutrosophic graphs of type 1. Cohen’s competition graphs [10] were not sufficient to represent the strength of competition among the species. Samanta and Sarkar [28] introduced generalized fuzzy competition graphs. Sahoo and Pal [26] investigated competitions under intuitionistic fuzzy graphs. Further, Das et al. [11] presented competition in generalized neutrosophic environment. There exist many competitions in this World which are periodic in nature. The existing competition graphs are insufficient to represent periodic nature competitions or competitions depend upon 2-dimensional information. For example, in an ecosystem, some species (predators) may be strong or weak within specific time interval. Similarly, some species (preys) may be digestive or harmful under some specific time interval. Such type of periodic information or two-dimensional information about predators and preys can not be handled by using already existing competition graphs. Nowadays, complex form of memberships is a very active fruitful research topic. Many researchers have addressed this topic. This motivate us to discuss competition in the framework of generalized complex neutrosophic graph of type 1. It is an effective model to handle competitions under inconsistent information of periodic nature. The contents of this research paper are as follows: In Section 2, we first recall some basic definitions for fully benefit of this paper. Then, we discuss the notions, including isomorphism, competition graphs, minimal graphs and competition number corresponding to generalized complex neutrosophic graphs. We elaborate these notions by several examples and present some of their interesting proved results. Further, we figure out that a generalized complex neutrosophic competition graph can be represented by an adjacency matrix, called a competition matrix with a suitable real life example. In Section 3, we specify the utility of generalized complex neutrosophic competition graph to represent the competition in job seeking environment and business marketing environment and find out the most competing object by computing strength of the competition. Moreover, we present the procedure of our developed method as an algorithm. Section 4 highlights the significance of our proposed model by comparative analysis with the already existing models. Finally, section 5 deals with conclusion of our proposed model and directions in future.

2 Representation of competitions by complex neutrosophic graphs

Definition 2.1. [5] An ordered pair G = (μ, σ) on a non-empty set X is called a single-valued neutrosophic graph, if μ is a single-valued neutrosophic set on X and σ is a single-valued neutrosophic relation on X such that σ_T (x, y) ≤ min {μ_T (x) , μ_T (y)}, σ_I (x, y) ≤ min {μ_I (x) , μ_I (y)}, σ_F (x, y) ≤ max {μ_F (x) , μ_F (y)}, for all x, y ∈ X.

Definition 2.2. [11] Let X be a non-empty set and E ⊆ X × X. Two functions are considered as follows: μ = (μ_T, μ_I, μ_F) : X → [0, 1] ³ and σ = (σ_T, σ_I, σ_F) : E → [0, 1] ³. We suppose A_T = {(μ_T (x) , μ_T (y)) |σ_T (x, y) ≥0}, A_I = {(μ_I (x) , μ_I (y)) |σ_I (x, y) ≥0}, A_F = {(μ_F (x) , μ_F (y)) |σ_F (x, y) ≥0}. We have considered σ_T, σ_I, σ_F ≥ 0, for all sets A_T, A_I, A_F, since it is possible to have edge membership = 0 (for T or I or F). The pair (μ, σ) is said to be a generalized single-valued neutrosophic graph of type 1 if there exist functions ϕ : A_T → [0, 1], ψ : A_I → [0, 1], δ : A_F → [0, 1] such that $\begin{matrix} σ_{T} (x, y) = ϕ (μ_{T} (x), μ_{T} (y)), \\ σ_{I} (x, y) = ψ (μ_{I} (x), μ_{I} (y)), \\ σ_{F} (x, y) = δ (μ_{F} (x), μ_{F} (y)), \end{matrix}$ where x, y ∈ X. Here, μ (x), x ∈ X is the membership value of the vertex x and σ (x, y) = (σ_T (x, y) , σ_I (x, y) , σ_F (x, y)), x, y ∈ X is the membership value of the edge (x, y).

Definition 2.3. [7] A complex neutrosophic set N on a non-empty set X is an object having the form N = {(x, m_T (x) e^iw_T(x), m_I (x) e^iw_I(x), m_F (x) e^iw_F(x)) : x ∈ X}, where $i = \sqrt{- 1}$ , m_T (x) , m_I (x) , m_F (x) ∈ [0, 1] are known as amplitude terms, w_T (x) , w_I (x) , w_F (x) ∈ [0, 2π] are called phase terms.

Definition 2.4. [7] Let N₁ = {(x, m_T (x) e^iw_T(x), m_I (x) e^iw_I(x), m_F (x) e^iw_F(x)) : x ∈ X} and $N_{2} = {(x, m_{T}^{'} (x) e^{i w_{T}^{'} (x)}, m_{I}^{'} (x) e^{i w_{I}^{'} (x)}, m_{F}^{'} (x) e^{i w_{F}^{'} (x)}) : x \in X}$ be two complex neutrosophic sets on a non-empty set X, then $N_{1} ⋂ N_{2} = {(x, min {m_{T} (x), m_{T}^{'} (x)} e^{i min {w_{T} (x), w_{T}^{'} (x)}}, min {m_{I} (x), m_{I}^{'} (x)} e^{i min {w_{I} (x), w_{I}^{'} (x)}}, max {m_{F} (x), m_{F}^{'} (x)} e^{i max {w_{F} (x), w_{F}^{'} (x)}} : x \in N_{1} \cap N_{2}}$ .

Definition 2.5. [30] Let X be a non-empty set and E ⊆ X × X. Two functions are considered as follows: μ = (μ_T, μ_I, μ_F) : X → ({a_T|a_T ∈ C ∧ |a_T|≤1} × {a_I|a_I ∈ C ∧ |a_I|≤1} × {a_F|a_F ∈ C ∧ |a_F|≤1}) and σ = (σ_T, σ_I, σ_F) : E → ({b_T|b_T ∈ C ∧ |b_T|≤1} × {b_I|b_I ∈ C ∧ |b_I|≤1} × {b_F|b_F ∈ C ∧ |b_F|≤1}). We suppose $\begin{matrix} A_{T} = {(μ_{T} (x), μ_{T} (y)) | r_{T} (x, y) \geq 0 \land w_{T} (x, y) \geq 0}, \\ A_{I} = {(μ_{I} (x), μ_{I} (y)) | r_{I} (x, y) \geq 0 \land w_{I} (x, y) \geq 0}, \\ A_{F} = {(μ_{F} (x), μ_{F} (y)) | r_{F} (x, y) \geq 0 \land w_{F} (x, y) \geq 0} . \end{matrix}$ We have considered r_T, r_I, r_F, w_T, w_I, w_F ≥ 0, for all sets A_T, A_I, A_F, since it is possible to have edge degree = 0 (for T or I or F). The pair (μ, σ) is said to be a generalized complex neutrosophic graph of type 1(G_ICNG) if there exist functions ϕ : A_T → {c_T|c_T ∈ C ∧ |c_T|≤1}, ψ : A_I → {c_I|c_I ∈ C ∧ |c_I|≤1}, δ : A_F → {c_F|c_F ∈ C ∧ |c_F|≤1} such that $\begin{matrix} σ_{T} (x, y) = ϕ (μ_{T} (x), μ_{T} (y)), \\ σ_{I} (x, y) = ψ (μ_{I} (x), μ_{I} (y)), \\ σ_{F} (x, y) = δ (μ_{F} (x), μ_{F} (y)), \end{matrix}$ where x, y ∈ X. Here, μ (x) = (μ_T (x) , μ_I (x) , μ_F (x)), x ∈ X is the complex-valued membership of the vertex x and σ (x, y) = (σ_T (x, y) , σ_I (x, y) , σ_F (x, y)), x, y ∈ X is the complex-valued membership of the edge (x, y).

Definition 2.6. [30] Let G = (μ, σ) be a G_ICNG and (x, y) be an edge of G. Then, the score of an edge (x, y) of G is given by $S (x, y) = \frac{1 + n_{T} (x, y) - 2 n_{I} (x, y) - n_{F} (x, y)}{2} + \frac{2 π + w_{T} (x, y) - 2 w_{I} (x, y) - w_{F} (x, y)}{4 π}$ , where n_T (x, y), n_I (x, y), n_F (x, y) are the generalized truth, indeterminacy and falsity amplitude membership values of edge (x, y), respectively and w_T (x, y), w_I (x, y), w_F (x, y) are the generalized truth, indeterminacy and falsity phase membership values of edge (x, y), respectively.

Definition 2.7. [30] Let G = (μ, σ) be a G_ICNG. Then, the degree of vertex x in G is denoted by D (x) = (D_T (x) , D_I (x) , D_F (x)) and is defined to be D (x) = (∑_y∈Xσ_T (x, y) , ∑_y∈Xσ_I (x, y) , ∑_y∈Xσ_F (x, y)).

In simple graph-theoretic concepts, two graphs are isomorphic if they have same number of vertices and edges. We now discuss the situation when two G_ICNGs are isomorphic. For this purpose, we first present the definition of homomorphism between two G_ICNGs.

Definition 2.8. Let G₁ = (μ₁, σ₁) and G₂ = (μ₂, σ₂) be two G_ICNGs. Let H : X₁ → X₂, ξ₁ : A₁ → ({a_T|a_T ∈ C ∧ |a_T|≤1} × {a_I|a_I ∈ C ∧ |a_I|≤1} × {a_F|a_F ∈ C ∧ |a_F|≤1}) and ξ₂ : A₂ → ({b_T|b_T ∈ C ∧ |b_T|≤1} × {b_I|b_I ∈ C ∧ |b_I|≤1} × {b_F|b_F ∈ C ∧ |b_F|≤1}) be the functions, where A₁ = {(μ_T (x) , μ_I (x) , μ_F (x)) |x ∈ X₂} and A₂ = {((μ_T (x) , μ_T (y)) , (μ_I (x) , μ_I (y)) , (μ_F (x) , μ_F (y))) |x, y ∈ X₂, (x, y) ∈ E₂} such that μ_{1_T} (x) = ξ_{1_T} (μ_{2_T} (H (x))), μ_{1_I} (x) = ξ_{1_I} (μ_{2_I} (H (x))), μ_{1_F} (x) = ξ_{1_F} (μ_{2_F} (H (x))) and σ_{1_T} (x, y) = ξ_{2_T} (σ_{2_T} (H (x) , H (y))), σ_{1_I} (x, y) = ξ_{2_I} (σ_{2_I} (H (x) , H (y))), σ_{1_F} (x, y) = ξ_{2_F} (σ_{2_F} (H (x) , H (y))). Then, the function H is said to be a homomorphism between two G_ICNGs.

Definition 2.9. Let G₁ = (μ₁, σ₁) and G₂ = (μ₂, σ₂) be two G_ICNGs. Also, let H : X₁ → X₂ be a bijective homomorphism and ξ₁, ξ₂ be two identity mappings. Then, the function H is said to be an isomorphism between G_ICNGs G₁ and G₂.

Before introducing the concept of competition graphs in the framework of G_ICNG, some related definitions are discussed below which will be useful to understand generalized complex neutrosophic competition graphs of type 1 easily.

Definition 2.10. Let X be a non-empty set and $\vec{E} \subseteq X \times X$ . Two functions are considered as follows: μ = (μ_T, μ_I, μ_F) : X → ({a_T|a_T ∈ C ∧ |a_T|≤1} × {a_I|a_I ∈ C ∧ |a_I|≤1} × {a_F|a_F ∈ C ∧ |a_F|≤1}) and $σ = (σ_{T}, σ_{I}, σ_{F}) : \vec{E} \to ({b_{T} | b_{T} \in C \land | b_{T} | \leq 1} \times {b_{I} | b_{I} \in C \land | b_{I} | \leq 1} \times {b_{F} | b_{F} \in C \land | b_{F} | \leq 1})$ . We suppose $\begin{matrix} A_{T} = {(μ_{T} (x), μ_{T} (y)) | r_{T} (\vec{x, y}) \geq 0 \land w_{T} (\vec{x, y}) \geq 0}, \\ A_{I} = {(μ_{I} (x), μ_{I} (y)) | r_{I} (\vec{x, y}) \geq 0 \land w_{I} (\vec{x, y}) \geq 0}, \\ A_{F} = {(μ_{F} (x), μ_{F} (y)) | r_{F} (\vec{x, y}) \geq 0 \land w_{F} (\vec{x, y}) \geq 0} . \end{matrix}$ We have considered r_T, r_I, r_F, w_T, w_I, w_F ≥ 0, for all sets A_T, A_I, A_F, since it is possible to have edge degree = 0 (for T or I or F). The pair (μ, σ) is said to be a generalized complex neutrosophic digraph of type 1(G_ICND) if there exist functions ϕ : A_T → {c_T|c_T ∈ C ∧ |c_T|≤1}, ψ : A_I → {c_I|c_I ∈ C ∧ |c_I|≤1}, δ : A_F → {c_F|c_F ∈ C ∧ |c_F|≤1} such that $\begin{matrix} σ_{T} (\vec{x, y}) = ϕ (μ_{T} (x), μ_{T} (y)), \\ σ_{I} (\vec{x, y}) = ψ (μ_{I} (x), μ_{I} (y)), \\ σ_{F} (\vec{x, y}) = δ (μ_{F} (x), μ_{F} (y)), \end{matrix}$ where x, y ∈ X. Here, μ (x) = (μ_T (x) , μ_I (x) , μ_F (x)), x ∈ X is the complex-valued membership of the vertex x and $σ (\vec{x, y}) = (σ_{T} (\vec{x, y}), σ_{I} (\vec{x, y}), σ_{F} (\vec{x, y}))$ , x, y ∈ X is the complex-valued membership of the edge $(\vec{x, y})$ .

Example 2.1. Let X = {x₁, x₂, x₃} be the vertex set and $\vec{E} = {(\vec{x_{2}, x_{1}}), (\vec{x_{3}, x_{1}}), (\vec{x_{3}, x_{2}})}$ be the edge set. The vertex membership values are given in Table 1.

Table 1
Membership values of vertices

μ x ₁ x ₂ x ₃

μ _T 0.6 $e^{i \frac{π}{2}}$ 0.5e^iπ 0.7 $e^{i \frac{π}{6}}$

μ _I 0.3 $e^{i \frac{π}{6}}$ 0.7 $e^{i \frac{π}{2}}$ 0.2 $e^{i \frac{π}{3}}$

μ _F 0.5 $e^{i \frac{π}{3}}$ 0.3 $e^{i \frac{π}{4}}$ 0.6 $e^{i \frac{π}{2}}$

μ	x ₁	x ₂	x ₃
μ _T	0.6 $e^{i \frac{π}{2}}$	0.5e^iπ	0.7 $e^{i \frac{π}{6}}$
μ _I	0.3 $e^{i \frac{π}{6}}$	0.7 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{3}}$
μ _F	0.5 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{2}}$

Consider the functions

$\begin{matrix} ϕ (m_{T}, n_{T}) = m_{T} \lor n_{T}, ϕ (w_{m_{T}}, w_{n_{T}}) = w_{m_{T}} \lor w_{n_{T}}, \\ ψ (m_{I}, n_{I}) \leq m_{I} \lor n_{I}, ψ (w_{m_{I}}, w_{n_{I}}) \leq w_{m_{I}} \lor w_{n_{I}}, \\ δ (m_{F}, n_{F}) = m_{F} \land n_{F}, δ (w_{m_{F}}, w_{n_{F}}) = w_{m_{F}} \land w_{n_{F}} . \end{matrix}$

Here,

$\begin{matrix} A_{T} = {(0.5 e^{i π}, 0.6 e^{i \frac{π}{2}}), (0.7 e^{i \frac{π}{6}}, 0.6 e^{i \frac{π}{2}}), (0.7 e^{i \frac{π}{6}}, 0.5 e^{i π})}, \\ A_{I} = {(0.7 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{6}}), (0.2 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{6}}), (0.2 e^{i \frac{π}{3}}, 0.7 e^{i \frac{π}{2}})}, \\ A_{F} = {(0.3 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{3}}), (0.6 e^{i \frac{π}{2}}, 0.5 e^{i \frac{π}{3}}), (0.6 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{4}})} . \end{matrix}$

Then, the generalized membership values of edges are given in Table 2.

Table 2

Generalized membership values of edges

σ	$(\vec{x_{2}, x_{1}})$	$(\vec{x_{3}, x_{1}})$	$(\vec{x_{3}, x_{2}})$
σ _T	0.6e^iπ	0.7 $e^{i \frac{π}{2}}$	0.7e^iπ
σ _I	0.4 $e^{i \frac{π}{3}}$	0.2 $e^{i \frac{5 π}{12}}$	0.5 $e^{i \frac{7 π}{12}}$
σ _F	0.3 $e^{i \frac{π}{4}}$	0.5 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$

The corresponding G_ICND is shown in Fig. 1.

Fig. 1

G_ICND

Definition 2.11. Let $\vec{G} = (μ, σ)$ be a G_ICND. Then, generalized out-neighbourhood of vertex x ∈ X is a complex neutrosophic set(CNS) $N^{+} (x) = (X_{x}^{+}, σ_{x}^{+})$ , where $X_{x}^{+} = {y | n_{T} (\vec{x, y}) \geq 0, n_{I} (\vec{x, y}) \geq 0, n_{F} (\vec{x, y}) \geq 0, w_{T} (\vec{x, y}) \geq 0, w_{I} (\vec{x, y}) \geq 0, w_{F} (\vec{x, y}) \geq 0}$ and $σ_{x}^{+} = (σ_{T . x}^{+}, σ_{I . x}^{+}, σ_{F . x}^{+}) : X_{x}^{+} \to ({z_{T} | z_{T} \in C \land | z_{T} | \leq 1} \times {z_{I} | z_{I} \in C \land | z_{I} | \leq 1} \times {z_{F} | z_{F} \in C \land | z_{F} | \leq 1})$ defined by

$\begin{matrix} σ_{T . x}^{+} (y) = n_{T} (\vec{x, y}) e^{i w_{T} (\vec{x, y})}, σ_{I . x}^{+} (y) = n_{I} (\vec{x, y}) e^{i w_{I} (\vec{x, y})}, \\ σ_{F . x}^{+} (y) = n_{F} (\vec{x, y}) e^{i w_{F} (\vec{x, y})}, (\vec{x, y}) \in \vec{E} . \end{matrix}$

Example 2.2. Consider G_ICND(Fig. 1), where X = {x₁, x₂, x₃} and $\vec{E} = {(\vec{x_{2}, x_{1}}), (\vec{x_{3}, x_{1}}), (\vec{x_{3}, x_{2}})}$ . The generalized out-neighbourhoods of vertices x₁, x₂ and x₃ are N⁺ (x₁) =∅, $N^{+} (x_{2}) = {x_{1} (0.6 e^{i π}, 0.4 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{3}})}$ and $N^{+} (x_{3}) = {x_{1} (0.7 e^{i \frac{π}{2}}, 0.2 e^{i \frac{π}{2}}, 0.5 e^{i \frac{π}{3}}), x_{2} (0.7 e^{i π}, 0.5 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{4}})}$ , respectively.

Definition 2.12. Let $\vec{G} = (μ, σ)$ be a G_ICND. Then, generalized complex neutrosophic competition graph of type 1(G_ICNCG) $C (\vec{G})$ of $\vec{G}$ is a G_ICNG which has same vertex set X with same membership values as in $\vec{G}$ and has a generalized complex neutrosophic edge between two vertices x, y ∈ X in $C (\vec{G})$ if and only if N⁺ (x)⋂ N⁺ (y) ≠ ∅ and the generalized membership value of edge (x, y) in G_ICNCG is (σ_T (x, y) , σ_I (x, y) , σ_F (x, y)) if there exists functions ϕ : C_T → {c_T|c_T ∈ C ∧ |c_T|≤1}, ψ : C_I → {c_I|c_I ∈ C ∧ |c_I|≤1} and δ : C_F → {c_F|c_F ∈ C ∧ |c_F|≤1}, where $C_{T} = {(μ_{T} (x), μ_{T} (y) | N_{T}^{+} (x) ⋂ N_{T}^{+} (y) \neq \emptyset}$ , $C_{I} = {(μ_{I} (x), μ_{I} (y) | N_{I}^{+} (x) ⋂ N_{I}^{+} (y) \neq \emptyset}$ and $C_{F} = {(μ_{F} (x), μ_{F} (y) | N_{F}^{+} (x) ⋂ N_{F}^{+} (y) \neq \emptyset}$ such that

$\begin{matrix} σ_{T} (x, y) = ϕ (μ_{T} (x), μ_{T} (y)) = max {N_{T}^{+} (x) ⋂ N_{T}^{+} (y)}, \\ σ_{I} (x, y) = ψ (μ_{I} (x), μ_{I} (y)) = max {N_{I}^{+} (x) ⋂ N_{I}^{+} (y)}, \\ σ_{F} (x, y) = δ (μ_{F} (x), μ_{F} (y)) = min {N_{F}^{+} (x) ⋂ N_{F}^{+} (y)}, \end{matrix}$

where x, y ∈ X.

Example 2.3. Consider G_ICND (Fig. 2(a)). Since, $N^{+} (x_{2}) ⋂ N^{+} (x_{3}) = {x_{1} (0.6 e^{i \frac{π}{2}}, 0.2 e^{i \frac{π}{2}}, 0.5 e^{i \frac{π}{3}})}$ . Therefore, there is an edge between vertices x₂ and x₃ in the corresponding G_ICNCG (Fig. 2(b)).

Fig. 2

Generalized complex neutrosophic competition graph of type 1

A G_ICNCG can represent any competition with much effective way. For every G_ICNG, there always exist a G_ICND such that $C (\vec{D}) = G$ . Following is the theorem, which clarifies the statement.

Theorem 2.1. Let G be a G_ICNG. Then, there exists a G_ICND $\vec{D}$ such that $C (\vec{D}) = G$ .

Proof. Let G = (μ, σ) be a G_ICNG and x, y be two arbitrary vertices of G. Now, a G_ICND $\vec{D}$ is to be constructed such that $C (\vec{D}) = G$ . Let x₁ and y₁ be the corresponding vertices in $\vec{D}$ of x and y in G with same membership values. Also, we can draw two directed edges from vertices x₁, y₁ to at least one common vertex z₁ in $\vec{D}$ such that z₁ ∈ N⁺ (x₁) ⋂ N⁺ (y₁). Similarly, we can do the same process for all other vertices of graph G. Hence, $C (\vec{D}) = G$ .□ According to Theorem 2.1, any G_ICNG can be considered as G_ICNCG of some G_ICND. However, this G_ICND may not be unique, i.e., there may be infinitely many G_ICNDs whose competition graph is the given G_ICNG. Therefore, among these infinitely many G_ICNDs whose competition graph is the given G_ICNG, some types of graphs have minimum number of edges. These graphs are called minimal graphs.

Definition 2.13. Let G be a G_ICNG. Minimal graph of G is a G_ICND $\vec{D}$ with minimum number of edges such that $C (\vec{D}) = G$ , i.e., if there exist another G_ICND $\vec{D^{'}}$ satisfying $C (\vec{D^{'}}) = G$ , Then, the number of edges of $\vec{D^{'}}$ is greater than or equal to the number of edges of $\vec{D}$ .

Since, infinitely many G_ICNDs can be constructed for any G_ICNG with different edge membership values. Therefore, minimal graphs are not unique. However, the number of edges in each minimal graph is same. Hence, crisp underlying graphs of the minimal graphs corresponding to a G_ICNG are isomorphic.

Theorem 2.2. Let $\vec{D} = (μ_{1}, σ_{1})$ be a minimal graph of a G_ICNG G = (μ, σ) and (x, y) be an edge of G. Then, there exist vertices x₁, y₁, z₁ in $\vec{D}$ such that $σ_{T} (x, y) = σ_{1_{T}} (\vec{x_{1}, z_{1}})$ , $σ_{I} (x, y) = σ_{1_{I}} (\vec{x_{1}, z_{1}})$ , $σ_{F} (x, y) = σ_{1_{F}} (\vec{x_{1}, z_{1}})$ or $σ_{T} (x, y) = σ_{1_{T}} (\vec{y_{1}, z_{1}})$ , $σ_{I} (x, y) = σ_{1_{I}} (\vec{y_{1}, z_{1}})$ , $σ_{F} (x, y) = σ_{1_{F}} (\vec{y_{1}, z_{1}})$ .

Proof. Let G = (μ, σ) be a G_ICNG and $\vec{D} = (μ_{1}, σ_{1})$ be a minimal graph of G. Also, let (x, y) be an edge of G. Minimal graphs are drawn in such a way that there exists at least one common vertex, say z₁, of x₁ and y₁, where x₁, y₁ are the corresponding vertices of x, y, respectively. Therefore, there exists at least two directed edges $(\vec{x_{1}, z_{1}})$ and $(\vec{y_{1}, z_{1}})$ . Since, $σ_{T} (x, y) = max {N_{T}^{+} (x_{1}) ⋂ N_{T}^{+} (y_{1})}$ , $σ_{I} (x, y) = max {N_{I}^{+} (x_{1}) ⋂ N_{I}^{+} (y_{1})}$ and $σ_{F} (x, y) = min {N_{F}^{+} (x_{1}) ⋂ N_{F}^{+} (y_{1})}$ , where $N^{+} (x_{1}) = {z_{1} (σ_{1_{T}} (\vec{x_{1}, z_{1}}), σ_{1_{I}} (\vec{x_{1}, z_{1}}), σ_{1_{F}} (\vec{x_{1}, z_{1}}))}$ and $N^{+} (y_{1}) = {z_{1} (σ_{1_{T}} (\vec{y_{1}, z_{1}}), σ_{1_{I}} (\vec{y_{1}, z_{1}}), σ_{1_{F}} (\vec{y_{1}, z_{1}}))}$ . Hence, $σ_{T} (x, y) = σ_{1_{T}} (\vec{x_{1}, z_{1}})$ , $σ_{I} (x, y) = σ_{1_{I}} (\vec{x_{1}, z_{1}})$ , $σ_{F} (x, y) = σ_{1_{F}} (\vec{x_{1}, z_{1}})$ or $σ_{T} (x, y) = σ_{1_{T}} (\vec{y_{1}, z_{1}})$ , $σ_{I} (x, y) = σ_{1_{I}} (\vec{y_{1}, z_{1}})$ , $σ_{F} (x, y) = σ_{1_{F}} (\vec{y_{1}, z_{1}})$ .□ Theorem 2.3. Isomorphic G_ICNDs are minimal graphs of some G_ICNGs. But, converse may not be true, i.e., minimal graphs are not necessarily isomorphic G_ICNDs.

Proof. We have to show that isomorphic G_ICNDs are minimal graphs of some G_ICNGs. Since, the isomorphic G_ICNDs have same number of edges as well as same membership values of corresponding vertices and edges. Therefore, if one G_ICND is a minimal graph of some G_ICNG, Then, its isomorphic G_ICNDs are also minimal. Now, we have to show that the converse of the theorem may not be true. For this purpose, let (x, y) be an edge in a G_ICNG G = (μ, σ). Also, let $\vec{D} = (μ_{1}, σ_{1})$ be a minimal graph of G. Now, according to Theorem 2.2, there exists vertices x₁, y₁, z₁ in $\vec{D}$ such that $σ_{T} (x, y) = σ_{1_{T}} (\vec{x_{1}, z_{1}})$ , $σ_{I} (x, y) = σ_{1_{I}} (\vec{x_{1}, z_{1}})$ , $σ_{F} (x, y) = σ_{1_{F}} (\vec{x_{1}, z_{1}})$ or $σ_{T} (x, y) = σ_{1_{T}} (\vec{y_{1}, z_{1}})$ , $σ_{I} (x, y) = σ_{1_{I}} (\vec{y_{1}, z_{1}})$ , $σ_{F} (x, y) = σ_{1_{F}} (\vec{y_{1}, z_{1}})$ . Without the loss of generality, let us assume that (x, y) has the same membership as (x₁, z₁). Similarly, consider another minimal graph ${\vec{D}}^{'}$ of G. Since, all minimal graphs have same number of edges. Therefore, there exists edges $(\vec{x_{1}^{'}, z_{1}^{'}}), (\vec{y_{1}^{'}, z_{1}^{'}}) \in {\vec{D}}^{'}$ corresponding to edges $(\vec{x_{1}, z_{1}}), (\vec{y_{1}, z_{1}}) \in \vec{D}$ . But, this does not shows that edges $(\vec{x_{1}, z_{1}})$ and $(\vec{x_{1}^{'}, z_{1}^{'}})$ have the same membership values. Thus, minimal graphs of some G_ICNG are not necessarily isomorphic.□ The characterization of minimal graphs such as their number of edges for different G_ICNGs are discussed bellow:

Theorem 2.4. Let G be a connected G_ICNG whose underlying crisp is a complete graph with n vertices. Then, the number of edges in a minimal graph of G is equal to 2n, n ≥ 3.

Proof. Let G = (μ, σ) be a connected G_ICNG whose underlying crisp is a complete graph(see Fig. 3(a)) with n vertices, i.e., each vertex of G is connected with all remaining vertices of G. Let x, y be two adjacent vertices of G and x′, y′ be the corresponding vertices of the minimal graph $\vec{D}$ . Consider a G_ICND $\vec{D_{1}}$ in such a way that every vertex of $\vec{D_{1}}$ other than x′ has the only out-neighbourhood, i.e., x′(see Fig. 3(b)). Thus, $\vec{D_{1}}$ has (n - 1) edges. Similarly, consider a G_ICND $\vec{D_{2}}$ in such a way that every vertex of $\vec{D_{2}}$ other than y′ has the only generalized out-neighbourhood, i.e., y′(see Fig. 3(c)). Thus, $\vec{D_{2}}$ also has (n - 1) edges. Now, consider a G_ICND $\vec{D_{3}}$ with only two directed edges $(\vec{x^{'}, z^{'}}), (\vec{y^{'}, z^{'}})$ (see Fig. 3(d)). Finally, consider the union of $\vec{D_{1}}, \vec{D_{2}}$ and ${\vec{D}}_{3}$ . This union will give us $\vec{D}$ , i.e., $\vec{D} = \vec{D_{1}} ⋃ \vec{D_{2}} ⋃ \vec{D_{3}}$ . This shows that the number of edges in $\vec{D}$ is 2n = (n - 1) + (n - 1) +2. Hence, the theorem is proved.

□

Fig. 3

G_ICNG and its minimal graph

Fig. 4

G_ICNG and its minimal graph

Theorem 2.5. Let G be a connected G_ICNG whose underlying crisp is a cycle with n vertices. Then, the number of edges in a minimal graph of G is equal to 2n, n ≥ 3.

Proof. Let G = (μ, σ) be a connected G_ICNG whose underlying crisp is a cycle with n vertices shown in Fig. 4(a). Let x₁, x₂, . . . , x_n be the vertices of G. Also, let $\vec{D}$ be the minimal graph of G and $x_{1}^{'}, x_{2}^{'}, . . ., x_{n}^{'}$ be the corresponding vertices in $\vec{D}$ of vertices x₁, x₂, . . . , x_n, respectively. Now, for every edge (x₁, x₂) ∈ G, there will only one common out-neighbourhood of $x_{1}^{'}, x_{2}^{'} \in \vec{D}$ , say $x_{3}^{'}$ . Also, $x_{3}^{'}$ is not common out-neighbourhood of any other vertex in $\vec{D}$ . Similarly, there will only one common neighbourhood out-neighbourhood of $x_{2}^{'}, x_{3}^{'} \in \vec{D}$ , say $x_{4}^{'}$ and so on, shown in Fig. 4(b). Hence, for n edges in G, there exists 2n edges in the corresponding minimal graph $\vec{D}$ .□

Definition 2.14. Let G = (μ, σ) be a G_ICNG and x be the vertex of G. Then, x is said to be an isolated vertex if S (x, x_i) =0, i = 1, 2, …, n, i.e., if x with all of its adjacent vertices x_i, i = 1, 2, …, n has a zero score.

Note 2.1. If n_T (x, y) =1 = n_I (x, y), n_F (x, y) =0 and w_T (x, y) =2π = w_I (x, y), w_F (x, y) =0, Then, edge (x, y) has a zero score.

Definition 2.15. Let G = (μ, σ) be a G_ICNG. A cycle of length greater than and equal to 4 in G is said to be a hole if score of all edges of this cycle is non-zero.

Example 2.4. Consider a G_ICNG G₁ = (μ₁, σ₁), where X₁ = {x₁, x₂, x₃, x₄} and E₁ = {(x₁, x₂) , (x₁, x₄), (x₂, x₃) , (x₃, x₄)}. Consider the functions $\begin{matrix} ϕ (m_{T}, n_{T}) = m_{T} \lor n_{T}, ψ (m_{I}, n_{I}) = m_{I} \lor n_{I}, δ (m_{F}, n_{F}) = m_{F} \land n_{F}, \\ ϕ (w_{m_{T}}, w_{n_{T}}) = w_{m_{T}} \lor w_{n_{T}}, ψ (w_{m_{I}}, w_{n_{I}}) = w_{m_{I}} \lor w_{n_{I}}, δ (w_{m_{F}}, w_{n_{F}}) = w_{m_{F}} \land w_{n_{F}} . \end{matrix}$ Then, the generalized membership values of edges are calculated in Table 3 and the corresponding G_ICNG is shown in Fig. 5. The scores of edges are S (x₁, x₂) =0.4333, S (x₁, x₄) =0.5250, S (x₂, x₃) =0.5458 and S (x₃, x₄) =0.4500. Here, x₁ - x₂ - x₃ - x₄ - x₁ is a hole, since length of this cycle is 4 and all of its edges has a non-zero score.

Table 3

Generalized memberships of edges

σ ₁	(x₁, x₂)	(x₁, x₄)	(x₂, x₃)	(x₃, x₄)
σ _{1_T}	0.5 $e^{i \frac{π}{2}}$	0.9 $e^{i \frac{π}{2}}$	0.7 $e^{i \frac{π}{4}}$	0.9 $e^{i \frac{π}{5}}$
σ _{1_I}	0.5 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{2}}$
σ _{1_F}	0.3 $e^{i \frac{π}{2}}$	0.1 $e^{i \frac{π}{5}}$	0.3 $e^{i \frac{π}{5}}$	0.1 $e^{i \frac{π}{5}}$

Fig. 5

G_ICNG

Definition 2.16. Let G = (μ, σ) be a G_ICNG. Then, competition number of G is defined to be the smallest number of isolated vertices in G and it is denoted by K (G).

Lemma 2.6. Let G = (μ, σ) be a G_ICNG with exactly one hole. Then, competition number of G_ICNG may be greater than 2 (competition number of a crisp graph with one hole is at most 2).

Example 2.5. Let G = (μ, σ) be a G_ICNG and it has vertex set X = {x₁, x₂, x₃, x₄, x₅, x₆, x₇} and edge set E = {(x₁, x₂) , (x₁, x₆) , (x₁, x₇) , (x₂, x₄) , (x₂, x₆) , (x₆, x₇)}. The vertex membership values are given in Table 4 and Consider the functions

$\begin{matrix} m_{T} \lor n_{T} < ϕ (m_{T}, n_{T}) \leq 1, w_{m_{T}} \lor w_{n_{T}} < ϕ (w_{m_{T}}, w_{n_{T}}) \leq 2 π, \\ m_{I} \lor n_{I} < ψ (m_{I}, n_{I}) \leq 1, w_{m_{I}} \lor w_{n_{I}} < ψ (w_{m_{I}}, w_{n_{I}}) \leq 2 π, \\ 0 \leq δ (m_{F}, n_{F}) < m_{F} \land n_{F}, 0 \leq δ (w_{m_{F}}, w_{n_{F}}) < w_{m_{F}} \land w_{n_{F}} . \end{matrix}$

Table 4

Membership values of vertices

μ	x ₁	x ₂	x ₃	x ₄	x ₅	x ₆	x ₇
μ _T	0.6 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{5}}$	0.3 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{4}}$	0.8 $e^{i \frac{π}{7}}$
μ _I	0.3 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{4}}$	0.5 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{2}}$	0.7 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$
μ _F	0.4 $e^{i \frac{π}{2}}$	0.1 $e^{i \frac{π}{2}}$	0.1 $e^{i \frac{π}{7}}$	0.4 $e^{i \frac{π}{3}}$	0.2 $e^{i \frac{π}{5}}$	0.2 $e^{i \frac{π}{4}}$	0.1 $e^{i \frac{π}{2}}$

Then, generalized membership values of edges are given in Table 5.

Table 5

Generalized membership values of edges

σ	(x₁, x₂)	(x₁, x₆)	(x₁, x₇)	(x₂, x₄)	(x₂, x₆)	(x₆, x₇)
σ _T	0.7e^iπ	0.8e^i2π	0.85e^iπ	1e^i2π	0.7 $e^{i \frac{π}{2}}$	0.9 $e^{i \frac{π}{3}}$
σ _I	0.8 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{2}}$	1e^i2π	0.9e^iπ	0.5 $e^{i \frac{π}{2}}$
σ _F	0.05 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{5}}$	0.02 $e^{i \frac{π}{4}}$	0eⁱ⁰	0.07 $e^{i \frac{π}{5}}$	0.05 $e^{i \frac{π}{6}}$

The corresponding G_ICNG is shown in Fig. 6.

Fig. 6

G_ICNG with K (G) =3

It can be easily seen that the crisp underlying graph G^* of this G_ICNG has exactly one hole with competition number 2. It may be noted that the scores of edges of hole x₁ - x₂ - x₆ - x₇ - x₁ are non-zero. Here, S (x₂, x₄) =0. Therefore, vertex x₄ together with vertices x₃ and x₅ is an isolated vertex of G_ICNG. Hence, K (G) =3.

Definition 2.17. Let G = (μ, σ) be a G_ICNG. Then, G is said to be a chordal G_ICNG if the chord of all of its holes has non-zero score. For example, consider G_ICNG of Exp. 2. Here, G has only one hole having a chord (x₁, x₆) with S (x₁, x₆) ≠0. Hence, G is a chordal graph.

If a G_ICND $\vec{G}$ is given then its corresponding G_ICNCG $C (\vec{G})$ can be determined with the help of an adjacency matrix. We call this adjacency matrix of G_ICNCG, a competition matrix. The entries of competition matrix are calculated in the following manner.

Step. 1: Draw the crisp underlying digraph ${\vec{G}}^{*}$ to construct the G_ICND $\vec{G}$ .

Step. 2: Input the truth, indeterminacy and falsity membership values of all n vertices of $\vec{G}$ .

Step. 3: Calculate the generalized truth, indeterminacy and falsity membership values of all directed edges of $\vec{G}$ by applying ϕ, ψ, δ according the situation, by keeping just in view that the amplitude terms and the phase terms of the membership values should remain in the closed intervals [0, 1] and [0, 2π], respectively. That is, $\begin{matrix} 0 \leq n_{T} (x, y), n_{I} (x, y), n_{F} (x, y) \leq 1, \\ 0 \leq w_{n_{T}} (x, y), w_{n_{I}} (x, y), w_{n_{F}} (x, y) \leq 2 π, \end{matrix}$ where x, y ∈ X.

Step. 4: Calculate generalized out-neighbourhoods of all n vertices, i.e., N⁺ (x_i), i = 1, 2, …, n.

Step. 5: Find out the set N⁺ (x_j) ⋂ N⁺ (x_k), j, k = 1, 2, …, m, j ≠ k.

Step. 6: If N⁺ (x_j)⋂ N⁺ (x_k) ≠ ∅ then there is an edge between vertices x_j and x_k, j, k = 1, 2, …, m, j ≠ k of G_ICNCG.

Step. 7: Calculate truth, indeterminacy and falsity membership values of edge (x_j, x_k), if N⁺ (x_j)⋂ N⁺ (x_k) ≠ ∅, by the following formula:

$\begin{matrix} σ_{T} (x_{j}, x_{k}) = ϕ (μ_{T} (x_{j}), μ_{T} (x_{k})) = max {N_{T}^{+} (x_{j}) ⋂ N_{T}^{+} (x_{k})}, \\ σ_{I} (x_{j}, x_{k}) = ψ (μ_{I} (x_{j}), μ_{I} (x_{k})) = max {N_{I}^{+} (x_{j}) ⋂ N_{I}^{+} (x_{k})}, \\ σ_{F} (x_{j}, x_{k}) = δ (μ_{F} (x_{j}), μ_{F} (x_{k})) = min {N_{F}^{+} (x_{j}) ⋂ N_{F}^{+} (k)}, \end{matrix}$ s

where x_j, x_k ∈ X, j, k = 1, 2, …, m, j ≠ k.

Step. 8: The entries of competition matrix are given as follows: $e_{jk} = {\begin{matrix} (σ_{T} (x_{j}, x_{k}), σ_{I} (x_{j}, x_{k}), σ_{F} (x_{j}, x_{k})) & if N^{+} (x_{j}) ⋂ N^{+} (x_{k}) \neq \emptyset \\ (0, 0, 0) & if N^{+} (x_{j}) ⋂ N^{+} (x_{k}) = \emptyset, \end{matrix}$

where x_j, x_k ∈ X, j, k = 1, 2, …, m, j ≠ k.

Example 2.6. An example of constructing competition matrix is given below:

Step. 1: The crisp underlying digraph ${\vec{G}}^{*} = (X, E)$ is shown in Fig. 7(a).

Fig. 7

G_ICND

Step. 2: The membership values of all vertices are shown in Table 6.

Table 6

Membership values of vertices

μ	x ₁	x ₂	x ₃	x ₄
μ _T	0.6 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{5}}$	0.7 $e^{i \frac{π}{4}}$	0.7 $e^{i \frac{π}{3}}$
μ _I	0.3 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{2}}$
μ _F	0.2 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{7}}$	0.3 $e^{i \frac{π}{4}}$

Step. 3: Consider the functions

$\begin{matrix} ϕ (m_{T}, n_{T}) = m_{T} \land n_{T}, ϕ (w_{m_{T}}, w_{n_{T}}) = w_{m_{T}} \land w_{n_{T}}, \\ ψ (m_{I}, n_{I}) = m_{I} \land n_{I}, ψ (w_{m_{I}}, w_{n_{I}}) = w_{m_{I}} \land w_{n_{I}}, \\ δ (m_{F}, n_{F}) = m_{F} \lor n_{F}, δ (w_{m_{F}}, w_{n_{F}}) = w_{m_{F}} \lor w_{n_{F}} . \end{matrix}$

Then, generalized membership values of directed edges are given in Table 7.

Table 7

Generalized membership values of edges

σ	$(\vec{x_{2}, x_{1}})$	$(\vec{x_{2}, x_{4}})$	$(\vec{x_{3}, x_{1}})$	$(\vec{x_{3}, x_{2}})$	$(\vec{x_{3}, x_{4}})$	$(\vec{x_{4}, x_{1}})$
σ _T	0.5 $e^{i \frac{π}{5}}$	0.5 $e^{i \frac{π}{5}}$	0.6 $e^{i \frac{π}{4}}$	0.5 $e^{i \frac{π}{5}}$	0.7 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{3}}$
σ _I	0.2 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$	0.4 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$
σ _F	0.4 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{3}}$	0.5 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{4}}$	0.3 $e^{i \frac{π}{3}}$

Step. 4: The out-neighbourhoods of vertices x₁, x₂, x₃, x₄ are N⁺ (x₁) =∅, $N^{+} (x_{2}) = {x_{1} (0.5 e^{i \frac{π}{5}}$ , $0.2 e^{i \frac{π}{4}}$ , $0.4 e^{i \frac{π}{2}}), x_{4} (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{3}}, 0.4 e^{i \frac{π}{2}})}$ , $N^{+} (x_{3}) = {x_{1} (0.6 e^{i \frac{π}{4}}, 0.3 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{3}})$ , $x_{2} (0.5 e^{i \frac{π}{5}}, 0.4 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{2}}), x_{4} (0.7 e^{i \frac{π}{4}}$ , $0.3 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{4}})}$ , $N^{+} (x_{4}) = {x_{1} (0.6 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{4}}, 0.3 e^{i \frac{π}{3}})}$ , respectively.

Step. 5: The intersection of out-neighbourhoods are: $N^{+} (x_{2}) ⋂ N^{+} (x_{3}) = {x_{1} (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{2}}), x_{4} (0.5 e^{i \frac{π}{5}}$ , $0.2 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{2}})}$ , $N^{+} (x_{2}) ⋂ N^{+} (x_{4}) = {x_{1} (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{4}}, 0.4 e^{i \frac{π}{2}})}$ , $N^{+} (x_{3}) ⋂ N^{+} (x_{4}) = {x_{1} (0.6 e^{i \frac{π}{4}}, 0.3 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{3}})}$ .

Step. 6: Since, N⁺ (x₂) ⋂ N⁺ (x₃), N⁺ (x₂) ⋂ N⁺ (x₄), N⁺ (x₃) ⋂ N⁺ (x₄) are non-empty sets. Therefore, there exit edges (x₂, x₃), (x₂, x₄), (x₃, x₄) showing competition between the vertices of corresponding G_ICNCG shown in Fig. 7(b).

Step. 7: The membership values of competition edges are calculated in Table. 8.

Table 8

Membership values of competition edges

σ	(x₂, x₃)	(x₂, x₄)	(x₃, x₄)
σ _T	0.5 $e^{i \frac{π}{5}}$	0.5 $e^{i \frac{π}{5}}$	0.6 $e^{i \frac{π}{4}}$
σ _I	0.2 $e^{i \frac{π}{3}}$	0.2 $e^{i \frac{π}{4}}$	0.3 $e^{i \frac{π}{4}}$
σ _F	0.5 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{4}}$	0.5 $e^{i \frac{π}{3}}$

Step. 8: The corresponding competition matrix is given below: $(\begin{matrix} (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) \\ (0, 0, 0) & (0, 0, 0) & (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{2}}) & (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{4}}, 0.4 e^{i \frac{π}{4}}) \\ (0, 0, 0) & (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{2}}) & (0, 0, 0) & (0.6 e^{i \frac{π}{4}}, 0.3 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{3}}) \\ (0, 0, 0) & (0.5 e^{i \frac{π}{5}}, 0.2 e^{i \frac{π}{4}}, 0.4 e^{i \frac{π}{4}}) & (0.6 e^{i \frac{π}{4}}, 0.3 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{3}}) & (0, 0, 0) \end{matrix})$ .

Example 2.7. Competition in ecosystem: Let us take a real life competition in an ecosystem. Let X = {Grass(G), Grass Hopper(GH), Hawk(H), Mouse(M), Snake(S), Coyote(C), Vulture(V)} be the set of vertices(taken as species) and $E = {(\vec{GH, G})$ , $(\vec{M, G})$ , $(\vec{H, GH})$ , $(\vec{H, M})$ , $(\vec{C, M})$ , $(\vec{V, M})$ , $(\vec{S, M})$ , $(\vec{C, H})$ , $(\vec{V, S})$ , $(\vec{V, C})}$ be the set of directed edges (showing that one specie attacks another specie in the food web). The crisp underlying diagraph ${\vec{G}}^{*}$ of this small food web between Herbivore, Omnivore and Carnivore is shown in Fig. 8.

Fig. 8

Crisp diagraph showing food web in ecosystem

Since, the existence of species in an ecosystem and the number of attacks among species in the food web are different in different time period. Therefore, we can assign 2-dimensional truth, indeterminacy and falsity membership values to the vertices and edges in our considered food web. First, we talk about vertices(species). Consider the specie Mouse(M), M has 80% ability to survive in 17% of specific time period, 20% chances that M may or may not be able to survive in 25% of specific time period, while 10% chances that Mouse is not be able to survive in 33% of specific time period in the environment. We model this information about specie M as: M $(0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}})$ . The 2-dimensional membership values of vertices are given in Table 9.

Table 9

Membership values of vertices

μ	G	GH	H	M	S	C	V
μ _T	0.8e^iπ	0.6 $e^{i \frac{π}{4}}$	0.4 $e^{i \frac{π}{3}}$	0.8 $e^{i \frac{π}{6}}$	0.2 $e^{i \frac{π}{8}}$	0.3 $e^{i \frac{π}{5}}$	0.7 $e^{i \frac{π}{3}}$
μ _I	0.1 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{4}}$	0.4 $e^{i \frac{π}{6}}$	0.2 $e^{i \frac{π}{4}}$
μ _F	0.1 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{5}}$	0.1 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{8}}$	0.5 $e^{i \frac{π}{2}}$

Consider the functions

$\begin{matrix} ϕ (m_{T}, n_{T}) = m_{T} \lor n_{T}, ϕ (w_{m_{T}}, w_{n_{T}}) = w_{m_{T}} \lor w_{n_{T}}, \\ ψ (m_{I}, n_{I}) = m_{I} \land n_{I}, ψ (w_{m_{I}}, w_{n_{I}}) = w_{m_{I}} \land w_{n_{I}}, \\ δ (m_{F}, n_{F}) = m_{F} \land n_{F}, δ (w_{m_{F}}, w_{n_{F}}) = w_{m_{F}} \land w_{n_{F}} . \end{matrix}$

Then, generalized membership values of directed edges are given in Table 10. Consider the directed edge $(\vec{S, M})$ , its 2-dimensional membership value shows that the Snake(S) 80% likes to eat Mouse(M) in 17% of specific time period, 20% indeterminate to eat M in 25% of specific time period and 10% dislikes to eat M in 33% of specific time period. We model this information as: $(\vec{S, M}) (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}})$ . The generalized 2-dimensional membership values of each directed edge is given in Table 10.

Table 10

Generalized membership values of edges

σ	$(\vec{GH, G})$	$(\vec{M, G})$	$(\vec{H, GH})$	$(\vec{H, M})$	$(\vec{C, M})$	$(\vec{V, M})$	$(\vec{S, M})$	$(\vec{C, H})$	$(\vec{V, S})$	$(\vec{V, C})$
σ _T	0.8e^iπ	0.8e^iπ	0.6 $e^{i \frac{π}{3}}$	0.8 $e^{i \frac{π}{3}}$	0.8 $e^{i \frac{π}{6}}$	0.8 $e^{i \frac{π}{3}}$	0.8 $e^{i \frac{π}{6}}$	0.4 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{3}}$
σ _I	0.1 $e^{i \frac{π}{2}}$	0.1 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{6}}$	0.2 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{4}}$	0.3 $e^{i \frac{π}{6}}$	0.2 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{6}}$
σ _F	0.1 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{5}}$	0.1 $e^{i \frac{π}{5}}$	0.1 $e^{i \frac{π}{8}}$	0.1 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{8}}$	0.5 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{8}}$

The generalized out-neighbourhoods of each specie are: N⁺ (G) =∅, $N^{+} (GH) = {G (0.8 e^{i π}, 0.1 e^{i \frac{π}{2}}, 0.1 e^{i \frac{π}{3}})}$ , $N^{+} (H) = {GH (0.6 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{5}}), M (0.8 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{5}})}$ , $N^{+} (M) = {G (0.8 e^{i π}, 0.1 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}})}$ , $N^{+} (S) = {M (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}})}$ , $N^{+} (C) = {M (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{8}}), H (0.4 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{6}}, 0.3 e^{i \frac{π}{8}})}$ , $N^{+} (V) = {M (0.8 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}), S (0.7 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{4}}, 0.5 e^{i \frac{π}{2}})$ , $C (0.7 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{6}}, 0.3 e^{i \frac{π}{8}})}$ . The non-empty intersections of generalized out-neighbourhoods of species are: N⁺ (GH) ⋂ N⁺ (M), N⁺ (H) ⋂ N⁺ (S), N⁺ (H) ⋂ N⁺ (C), N⁺ (H) ⋂ N⁺ (V), N⁺ (S) ⋂ N⁺ (C), N⁺ (S) ⋂ N⁺ (V), N⁺ (C) ⋂ N⁺ (V). So, (GH, M), (H, S), (H, C), (H, V), (S, C), (S, V), (C, V) will be the edges in the corresponding competition graph shown in Fig. 9. The dark color lines represent competition among the species, while dim color lines indicate that species are competing for their food in the food web. The corresponding competition matrix(showing that different species are competing with each other for their food) is given on the next page.

$(\begin{matrix} (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) \\ (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0.8 e^{i π}, 0.1 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) \\ [0.3 cm] (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{5}}) & (0.8 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) \\ (0, 0, 0) & (0.8 e^{i π}, 0.1 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) & (0, 0, 0) \\ (0, 0, 0) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) & (0, 0, 0) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{3}}) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) \\ (0, 0, 0) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{5}}) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{3}}) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{3}}) \\ (0, 0, 0) & (0, 0, 0) & (0.8 e^{i \frac{π}{3}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) & (0, 0, 0) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{4}}, 0.1 e^{i \frac{π}{3}}) & (0.8 e^{i \frac{π}{6}}, 0.2 e^{i \frac{π}{6}}, 0.1 e^{i \frac{π}{3}}) & (0, 0, 0) \end{matrix})$

Fig. 9

G_ICNCG

3 Application

Fuzzy competition graph is a fruitful source of graph theory to represent strength of competition between objects. But, fuzzy competition graphs may fail to depict the strength of competition in many real life problems due to natural presence of periodic nature inconsistent information. The existence of periodic nature uncertain competitions between objects increases the significance of our proposed model. In this research article, we apply G_ICNCG model in the environment of competition among different suppliants for particular jobs and find out which suppliant is the most competitive one among all suppliants. Let {S₁, S₂, S₃, S₄, S₅} be the set of 5 suppliants competing with each other to get a job in the set {J₁, J₂, J₃, J₄}. A G_ICND $\vec{G}$ is given in Fig. 10 indicating that the suppliants are competing for particular jobs.

Fig. 10

G_ICND

Here, we take 5 suppliants S₁, S₂, S₃, S₄, S₅ and 4 jobs J₁, J₂, J₃, J₄ as a vertex set. The membership values of each vertex is given in Table 11. The amplitude terms of truth, indeterminacy and falsity membership values of each suppliant indicate that the specific suppliant is eligible, may or may not be eligible and ineligible for a particular job, respectively. While, phase terms of truth, indeterminacy and falsity membership values of each suppliant show that how much the specific suppliant is experienced, may or may not be experienced and inexperienced towards a particular job. The amplitude terms of truth, indeterminacy and falsity membership values of each job indicate the availability, inconsistent availability and unavailability of that particular job, respectively. While, phase terms of truth, indeterminacy and falsity membership values of each job show that how much the specific job is beneficial, may or may not be beneficial and disadvantageous for suppliants.

Table 11

Membership values of vertices

μ	S ₁	S ₂	S ₃	S ₄	S ₅	J ₁	J ₂	J ₃	J ₄
μ _T	0.7 $e^{i \frac{π}{3}}$	0.6 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{5}}$	0.3 $e^{i \frac{π}{5}}$	0.7 $e^{i \frac{π}{5}}$	0.8 $e^{i \frac{π}{2}}$	0.7 $e^{i \frac{π}{3}}$	0.9 $e^{i \frac{π}{2}}$
μ _I	0.3 $e^{i \frac{π}{2}}$	0.1 $e^{i \frac{π}{3}}$	0.5 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{5}}$	0.3 $e^{i \frac{π}{4}}$	0.2 $e^{i \frac{π}{5}}$	0.4 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{2}}$
μ _F	0.7 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{5}}$	0.2 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{7}}$	0.6 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{4}}$	0.3 $e^{i \frac{π}{5}}$

Consider the functions

$\begin{matrix} ϕ (m_{T}, n_{T}) = m_{T} \lor n_{T}, ϕ (w_{m_{T}}, w_{n_{T}}) = w_{m_{T}} \lor w_{n_{T}}, \\ ψ (m_{I}, n_{I}) = m_{I} \lor n_{I}, ψ (w_{m_{I}}, w_{n_{I}}) = w_{m_{I}} \lor w_{n_{I}}, \\ δ (m_{F}, n_{F}) = m_{F} \land n_{F}, δ (w_{m_{F}}, w_{n_{F}}) = w_{m_{F}} \land w_{n_{F}} . \end{matrix}$

Then, generalized membership values of directed edges are given in Table 13. The amplitude terms of truth, indeterminacy and falsity membership values of each directed edge between a suppliant and a job indicate that the specific suppliant is able to get that particular job, may or may not be able to get that particular job and not able to get that particular job, respectively. While, phase terms of truth, indeterminacy and falsity membership values of each directed edge between a suppliant and a job indicate the chances to get particular job, there may or may not be the chances to get particular job and not chances to get a particular job.

Table 12

Generalized membership values of edges

σ	σ _T	σ _I	σ _F
$(\vec{S_{1}, J_{1}})$	0.7 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{7}}$
$(\vec{S_{1}, J_{4}})$	0.9 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{5}}$
$(\vec{S_{2}, J_{1}})$	0.7 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{7}}$
$(\vec{S_{2}, J_{2}})$	0.8 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{5}}$
$(\vec{S_{3}, J_{2}})$	0.8 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{2}}$	0.2 $e^{i \frac{π}{3}}$
$(\vec{S_{3}, J_{3}})$	0.7 $e^{i \frac{π}{3}}$	0.5 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{4}}$
$(\vec{S_{4}, J_{3}})$	0.7 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{4}}$
$(\vec{S_{4}, J_{4}})$	0.9 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{5}}$
$(\vec{S_{5}, J_{1}})$	0.7 $e^{i \frac{π}{5}}$	0.3 $e^{i \frac{π}{4}}$	0.4 $e^{i \frac{π}{7}}$
$(\vec{S_{5}, J_{2}})$	0.8 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{2}}$	0.6 $e^{i \frac{π}{3}}$
$(\vec{S_{5}, J_{3}})$	0.7 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{4}}$
$(\vec{S_{5}, J_{4}})$	0.9 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{5}}$

Table 13

Strength of competition among all suppliants

Suppliant	Suppliant’s Competitors	Degree of Suppliant	Competition Strenght
S ₁	S₂, S₄, S₅	(2.5e^i1.33π, 0.8e^i1.25π, 1e^i0.49π)	0.54
S ₂	S₁, S₃, S₅	(2.3e^i1.33π, 1e^i1.33π, 1.1e^i0.62π)	0.11
S ₃	S₂, S₄, S₅	(2.3e^i1.33π, 0.16e^i1.33π, 0.5e^i0.83π)	1.2
S ₄	S₁, S₃, S₅	(2.5e^i1.33π, 1e^i1.33π, 0.5e^i1.33π)	-0.62
S ₅	S₁, S₂, S₃, S₄	(3.4e^i2π, 1.4e^i1.75π, 0.9e^i0.74π)	0.11

The generalized out-neighbourhoods of vertices are: $N^{+} (S_{1}) = {J_{1} (0.7 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{7}}), J_{4} (0.9 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{2}}$ , $0.3 e^{i \frac{π}{5}})}$ , $N^{+} (S_{2}) = {J_{1} (0.7 e^{i \frac{π}{2}}, 0.2 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{7}})$ , $J_{2} (0.8 e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{5}})}$ , $N^{+} (S_{3}) = {J_{2} (0.8 e^{i \frac{π}{2}}, 0.5 e^{i \frac{π}{2}}, 0.2$ $e^{i \frac{π}{3}})$ , $J_{3} (0.7 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{3}}, 0.1 e^{i \frac{π}{4}})}$ , $N^{+} (S_{4}) = {J_{3} (0.7 e^{i \frac{π}{3}}, 0.4 e^{i \frac{π}{3}}, 0.1 e^{i \frac{π}{4}}), J_{4} (0.9 e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{5}})}$ , $N^{+} (S_{5}) = {J_{1} (0.7 e^{i \frac{π}{5}}, 0.3 e^{i \frac{π}{4}}, 0.4 e^{i \frac{π}{7}}), J_{2} (0.8$ $e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{2}}, 0.6 e^{i \frac{π}{3}}), J_{3} (0.7 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{3}}, 0.1 e^{i \frac{π}{4}}), J_{4} (0.9 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{5}})}$ , N⁺ (J₁) = ∅ = N⁺ (J₂) = N⁺ (J₃) = N⁺ (J₄). The non-empty intersections of generalized out-neighbourhoods of vertices are: N⁺ (S₁) ⋂ N⁺ (S₂), N⁺ (S₁) ⋂ N⁺ (S₄), N⁺ (S₁) ⋂ N⁺ (S₅), N⁺ (S₂) ⋂ N⁺ (S₃), N⁺ (S₂) ⋂ N⁺ (S₅), N⁺ (S₃) ⋂ N⁺ (S₄) N⁺ (S₃) ⋂ N⁺ (S₅), N⁺ (S₄) ⋂ N⁺ (S₅). So, (S₁, S₂), (S₁, S₄), (S₁, S₅), (S₂, S₃), (S₂, S₅), (S₃, S₄), (S₃, S₅), (S₄, S₅) will be the edges in the corresponding competition graph shown in Fig. 11. The dark color lines represent competition between suppliants, while dim color lines indicate that suppliants are competing for which particular job.

Fig. 11

G_ICNCG

Now, to highlight that which suppliant is the most competitive suppliant among all suppliants, the strength of competition of each suppliant is calculated in Table 12. Thus, Table 12 indicates that suppliant S₃ is the most competitive suppliant.

We now present the general procedure of our method which is used in our application from the algorithm given in Table 14.

Table 14

Finding the most competing suppliant or object

Algorithm
Step 1.	Input the membership values of all n vertices(jobs and suppliants).
Step 2.	Calculate the generalized membership values of all directed edges(from suppliants to particular jobs).
Step 3.	Calculate generalized out-neighbourhoods of all competing vertices(suppliants).
Step 4.	Draw an edge between two suppliants in the corresponding G_ICNCG if N⁺ (x_i)⋂ N⁺ (x_j) ≠ ∅,
	i ≠ j, i, j = 1, 2, …, k.
Step 5.	Assign membership values to each edge in G_ICNCG according to formula given in definition 2.12.
Step 6.	Calculate the degree of each suppliant in G_ICNCG according to the formula given in definition 2.7.
Step 7.	Find out the most competing suppliant by calculating the score value of degree of each suppliant by using the formula given in definition 2.6.
Step 8.	By ignoring negative sign, that suppliant is considered to be the most competing suppliant whose degree has greater score value than the other suppliants.

3.1 Business environment competitions

In business marketing environment, there always exists competition among several companies due to selling identical products to the customers, other companies and retailers. Each company tries to achieve consumer’s attention to their products by adopting different strategies. Although, competition graphs are amazing tool to represent competition between objects. But, these graph theoretic competitions are not flexible to handle uncertain 2-dimensional competitions. For example, companies are different according to their annual profit and loss due to adopting different strategies within a certain time period. Consider an example of business marketing competition among 5 electronic companies Haier, LG, Mitshubishi, Panasonic, Gree, 2 brands, 2 retailers and 1 outlet retailer. The corresponding G_ICND is shown in Fig. 12.

Fig. 12

G_ICND

Let X = {Haier, LG, Mitsubishi, Panasonic, Gree, Brand-1, Brand-2, Retailor-1, Retailor-2, Outlet} be the vertex set. The membership values of vertices are given in Table 15. The amplitude terms of truth, indeterminacy and falsity membership values of each vertex represent its annual profit, annual undecided profit and annual loss. While, phase terms of truth, indeterminacy and falsity membership values of each vertex indicate that the strategy implemented by a company/brand/retailer/outlet is a strong, may or may not be strong and weak planning within a specific time period for that particular company/brand/retailer/outlet.

Table 15

Membership values of vertices

μ	Haier	LG	Mitsubishi	Panasonic	Gree	Brand-1	Brand-2	Retailer-1	Retailer-2	Outlet
μ _T	0.6 $e^{i \frac{π}{2}}$	0.7 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{4}}$	0.5 $e^{i \frac{π}{3}}$	0.6 $e^{i \frac{π}{2}}$	0.7 $e^{i \frac{π}{3}}$	0.9 $e^{i \frac{π}{4}}$	0.8 $e^{i \frac{π}{4}}$	0.5 $e^{i \frac{π}{7}}$
μ _I	0.4 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{3}}$	0.6 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{2}}$	0.6 $e^{i \frac{π}{2}}$	0.1 $e^{i \frac{π}{4}}$	0.4 $e^{i \frac{π}{6}}$	0.7 $e^{i \frac{π}{3}}$	0.8 $e^{i \frac{π}{2}}$
μ _F	0.9 $e^{i \frac{π}{3}}$	0.7 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{5}}$	0.7 $e^{i \frac{π}{4}}$	0.3 $e^{i \frac{π}{5}}$	0.9 $e^{i \frac{π}{3}}$	0.2 $e^{i \frac{π}{2}}$	0.5 $e^{i \frac{π}{3}}$	0.8 $e^{i \frac{π}{3}}$

Consider the functions

Then, generalized membership values of directed edges are given in Table 16. The amplitude terms of truth, indeterminacy and falsity membership values of each directed edge $(\vec{x, y})$ shows that the x company wants to purchase products of y company, x company is undecided to purchase products of y company and x company do not purchase products of y company. While, phase terms of truth, indeterminacy and falsity membership values of each directed edge $(\vec{x, y})$ shows that the x company’s strategy attracts y company, x company’s strategy may not attracts y company and x company’s strategy do not repels y company within a specific time period.

Table 16

Generalized membership values of directed edges

$(\vec{x, y})$	σ _T	σ _I	σ _F
(Haier, Brand-1)	0.6 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{3}}$	0.3 $e^{i \frac{π}{5}}$
(Haier, Retailer-2)	0.8 $e^{i \frac{π}{2}}$	0.4 $e^{i \frac{π}{3}}$	0.5 $e^{i \frac{π}{3}}$
(LG, Brand-2)	0.7 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{4}}$	0.7 $e^{i \frac{π}{4}}$
(LG, Retailer-2)	0.8 $e^{i \frac{π}{4}}$	0.3 $e^{i \frac{π}{3}}$	0.5 $e^{i \frac{π}{5}}$
(Mitsubishi, Brand-2)	0.7 $e^{i \frac{π}{3}}$	0.1 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{3}}$
(Mitsubishi, Outlet)	0.6 $e^{i \frac{π}{3}}$	0.4 $e^{i \frac{π}{3}}$	0.6 $e^{i \frac{π}{3}}$
(Gree, Brand-1)	0.6 $e^{i \frac{π}{2}}$	0.6 $e^{i \frac{π}{2}}$	0.3 $e^{i \frac{π}{5}}$
(Gree, Retailer-2)	0.4 $e^{i \frac{π}{7}}$	0.3 $e^{i \frac{π}{5}}$	0.3 $e^{i \frac{π}{7}}$
(Panasonic, Outlet)	0.5 $e^{i \frac{π}{4}}$	0.6 $e^{i \frac{π}{3}}$	0.5 $e^{i \frac{π}{5}}$

The generalized out-neighbourhoods of companies are given in Table 17.

Table 17

Generalized out-neighbourhoods of companies

Company	Generalized out-neighbourhoods
Haier	{Brand-1( $0.6 e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{3}}, 0.3 e^{i \frac{π}{5}}$ ), Retailer-2( $0.8 e^{i \frac{π}{2}}, 0.4 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{3}}$ )}
LG	{Brand-2( $0.7 e^{i \frac{π}{3}} 0.1 e^{i \frac{π}{4}} 0.7 e^{i \frac{π}{4}}$ ), Retailer-2( $0.8 e^{i \frac{π}{4}}, 0.3 e^{i \frac{π}{3}} 0.5 e^{i \frac{π}{5}}$ )}
Mitsubishi	{Brand-2( $0.7 e^{i \frac{π}{3}}, 0.1 e^{i \frac{π}{4}}, 0.6 e^{i \frac{π}{3}}$ ), Outlet( $0.6 e^{i \frac{π}{3}}, 0.4 e^{i \frac{π}{3}}, 0.6 e^{i \frac{π}{3}}$ )}
Panasonic	{Outlet( $0.5 e^{i \frac{π}{4}}, 0.6 e^{i \frac{π}{3}}, 0.5 e^{i \frac{π}{5}}$ )}
Gree	{Brand-1( $0.6 e^{i \frac{π}{2}}, 0.6 e^{i \frac{π}{2}}, 0.3 e^{i \frac{π}{5}}$ ), Retailer-2( $0.4 e^{i \frac{π}{7}}, 0.3 e^{i \frac{π}{5}}, 0.3 e^{i \frac{π}{7}}$ )}

From Table 17, it is easy to see that Haier and LG, Haier and Gree, LG and Mitsubishi, LG and Gree, Mitsubishi and Panasonic have common generalized out-neighbourhoods, i.e., Retailer-2, Brand-1 together with Retailer-2, Brand-2, Retailer-2 and Outlet, respectively. Therefore, there exist edges between these pair of companies in the corresponding G_ICNCG given in Fig. 13. The dark color lines represent competition between companies, while dim color lines indicate that companies are competing for which particular brand/retailer/outlet.

Fig. 13

G_ICNCG

Now, to highlight that which company is the most competitive company among all, the strength of competition of each company is calculated in Table 18. Thus, Table 18 indicates that Panasonic is the most competitive company in the business marketing competition.

Table 18

Strength of competition among all companies

Company	Company’s Competitors	Degree of Company	Competition Strenght
Haier	LG, Gree	(1.4e^i0.75π, 0.7e^i0.6667π, 0.8e^i0.5333π)	0.3208
LG	Haier, Mitsubishi, Gree	(1.9e^i0.7262π, 0.7e^i0.7833π, 1.8e^iπ)	-0.1101
Mitsubishi	LG, Panasonic	(1.2e^i0.5833π, 0.5e^i0.5833π, 1.3e^i0.6667π)	0.1875
Panasonic	Mitsubishi	(0.5e^i0.25π, 0.4e^i0.75π, 0.6e^i0.75π)	0.3625
Gree	Haier, LG	(1e^i0.6429π, 0.7e^i1.5333π, 0.9e^i0.5333π)	0.1107

4 Comparative analysis and discussion

Competitions are everywhere in this World. Cohen’s competition graphs are useful to represent competition among competing objects in a discrete manner. Therefore, Cohen’s competition graphs are not flexible. Fuzzy competition graphs deal with real life competitions in a flexible way. Fuzzy competition graphs not only represent the competition between objects but also the strength of competition between objects. In fuzzy competition graphs each vertex and edge has truth membership value in the real unit interval. Intuitionistic fuzzy competition graphs and neutrosophic competition graphs are the extensions of fuzzy competition graphs as these graphs assign falsity and indeterminacy membership values to each of the vertex and edge. However, all these model have restriction on edge membership values. Generalized fuzzy graphs remove this restriction and able to handle more real life competitions. Further extended form is generalized neutrosophic competition graphs. All these previous models may fail to represent competitions in many real life problems due to the natural existence of inconsistent periodic nature information. For example, the existence of species in an ecosystem and the number of attacks among species in the food web are different in different time period. Similarly, profit and loss of companies in business marketing environment are different in different time period. These are inconsistent competitions of periodic nature. Such type of uncertain periodic nature competitions among objects increase the significance of our proposed model. Our proposed model, i.e., generalized complex neutrosophic competition graph is the most flexible model to deal any real life competition with inconsistent and incomplete competitions of periodic nature in real life. The potential of our proposed model for representing the periodic uncertain information having no restriction on edge membership values makes it superior than the other existing models. The comparison of our proposed model with other models can be easily seen from Table 19.

Table 19
Comparative Analysis

Model Positive Amplitude Phase Undecided Amplitude Phase Negative Amplitude Phase Dimension Edge membership/Restriction

Cohen’s C ✓ × × × × × 1 Binary set/No

FCG ✓ × × × × × 1 Unit interval/Yes

IFCG ✓ × × × ✓ × 1 Unit interval/Yes

NCG ✓ × ✓ × ✓ × 1 Unit interval/Yes

GFCG ✓ × × × × × 1 Unit interval/No

GNCG ✓ × ✓ × ✓ × 1 Unit interval/No

G₁CNG ✓ ✓ ✓ ✓ ✓ ✓ 2 Unit disc/No

Model	Positive Amplitude Phase	Undecided Amplitude Phase	Negative Amplitude Phase	Dimension	Edge membership/Restriction
Cohen’s C	✓ ×	× ×	× ×	1	Binary set/No
FCG	✓ ×	× ×	× ×	1	Unit interval/Yes
IFCG	✓ ×	× ×	✓ ×	1	Unit interval/Yes
NCG	✓ ×	✓ ×	✓ ×	1	Unit interval/Yes
GFCG	✓ ×	× ×	× ×	1	Unit interval/No
GNCG	✓ ×	✓ ×	✓ ×	1	Unit interval/No
G₁CNG	✓ ✓	✓ ✓	✓ ✓	2	Unit disc/No

5 Conclusion

Complex form of membership plays a vital role in modeling and handling uncertain research domain of periodic nature. It provides more accuracy and flexibility as compared to fuzzy form of membership. The G_ICNG model is able to discuss various real life problems, where the truth, indeterminacy and falsity membership values of the given information are periodic in nature. In this research paper, we have discussed some interesting notions, including isomorphism, competition graphs, minimal graphs and competition number related to G_ICNGs. We also described these notions by several examples and presented some of their interesting proved results. Further, we represented a G_ICNCG with the help of an adjacency matrix. Moreover, we specify the utility of G_ICNGs to represent the strength of competition between different objects. Then, we presented the procedure of our developed decision making method as an algorithm. We aim to extend our research work to (1) Generalized complex bipolar neutrosophic competition graphs, (2) Generalized complex neutrosophic competition hypergraphs, (3) Generalized complex bipolar neutrosophic competition hypergraphs, (4) Generalized n-step complex neutrosophic graphs, (5) Generalized n-step complex bipolar neutrosophic graphs.

Conflict of interest

The authors declare no conflict of interest.

References

Akram

, Single-valued neutrosophic graphs, Infosys Science Foundation Series in Mathematical Sciences, Springer (2018), DOI: 10.1007/978-981-13-3522-8.

Akram

, Bashir

and Samanta

, Complex Pythagorean Fuzzy Planar Graphs, Int J Appl Comput Math 6(58) (2020), DOI: 10.1007/s40819-020-00817-2.

Akram

and Sattar

, Competition graphs under complex Pythagorean fuzzy information, Journal of Applied Mathematics and Computing (2020), DOI: 10.1007/s12190-020-01329-4.

Akram

, Siddique

and Davvaz

, New Concepts in neutrosophic graphs with application, Journal of Applied Mathematics and Computing 57(12) (2018), 279–302.

Akram

and Siddique

, Neutrosophic competition graphs with applications, Journal of Intelligent & Fuzzy Systems 33(2) (2017), 921–935.

Alkouri

A.M.

and Salleh

A.R.

, Complex intuitionistic fuzzy sets, AIP Conf Proc 1482(1) (2012), 464–470.

Ali

and Samarandache

, Complex neutrosophic set, Neural Comput & Appl 28(7) (2017), 1817–1834.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1) (1986), 87–96.

Broumi

, Bakali

, Talea

and Smarandache

, Complex neutrosophic graphs of type 1, Proceedings of 2017 IEEE International Conference on Innovations in Intelligent Systems and Applications (INISTA) Gdynia Maritime University, Gdynia, Poland (2017), 432–437.

10.

Cohen

J.E.

, Novel Interval graphs and food webs: a finding and a problem, Document 17696 (1968), PR, RAND Corporation, Santa Monica, CA.

11.

Das

, Samanta

and De

, Generalized neutrosophic competition graphs, Neutrosophic Sets & Systems 31 (2020), 156–171.

12.

Dey

, Son

L.H.

, Pal

and Long

H.V.

, Fuzzy Minimum Spanning Tree with Interval Type 2 Fuzzy Arc Length: Formulation and A New Genetic Algorithm, Soft Computing 24 (2020), 3963–3974.

13.

Edalatpanah

S.A.

, Neutrosophic structured element, Expert Systems (2020), DOI: 10.1111/exsy.12542.

14.

Edalatpanah

S.A.

, Neutrosophic perspective on DEA, Journal of Applied Research on Industrial Engineering 5(4) (2018), 339–345.

15.

Edalatpanah

S.A.

, Systems of neutrosophic linear equations, Neutrosophic Sets and Systems 33(1) (2020), 92–104.

16.

Edalatpanah

S.A.

, Data envelopment analysis based on triangular neutrosophic numbers, CAAI Transactions on Intelligence Technology (2020), DOI: 10.1049/trit.2020.0016.

17.

Faruk

and Bijan

, Properties of single-valued neutrosophic graphs, Journal of Intelligent and Fuzzy Systems 34(1) (2018), 57–79.

18.

Hamidi

and Saeid

A.B.

, Achievable single-valued neutrosophic graphs in wireless sensor networks, New Mathematics and Natural Computation 14(2) (2018), 157–185.

19.

Kartick

, Dey

, Pal

, Long

H.V.

and Son

L.H.

, A Study of m-Polar Neutrosophic Graph with Applications, Journal of Intelligent & Fuzzy Systems 38(4) (2020), 4809–4828.

20.

Kaufmann

, Introduction a la Thiorie des Sous-Ensemble Flous, Masson et Cie 1 (1973).

21.

Kumar

, Edalatpanah

S.A.

, Jha

and Singh

, A novel approach to solve gaussian valued neutrosophic shortest path problems, International Journal of Engineering and Advanced Technology 8(3) (2019), 347–353.

22.

Mordeson

J.N.

and Nair

P.S.

, Fuzzy graphs and fuzzy hypergraphs, Physica Verlag, Heidelberg (2001).

23.

Ramot

, Milo

, Friedman

and Kandel

, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems 10(2) (2002), 171–186.

24.

Ramot

, Friedman

, Langholz

and Kandel

, Complex fuzzy logic, IEEE Transactions on Fuzzy Systems 11(4) (2003), 450–461.

25.

Rosenfeld

, Fuzzy graphs, Fuzzy Sets and Their Applications, Academic Press, New York (1975), 77–95.

26.

Sahoo

and Pal

, Intuitionistic fuzzy competition graphs, Journal of Applied Mathematics and Computing 52 (2016), 37–57.

27.

Samanta

and Sarkar

, A study on generalized fuzzy graphs, Journal of Intelligent & Fuzzy Systems 35(3) (2018), 3405–3412.

28.

Samanta

and Sarkar

, Representation of competitions by generalized fuzzy graphs, Int. J. of Computational & Intelligence Systems 11 (2018), 1005–1015.

29.

Sheng

G.L.

, Su

Y.L.

and Wang

W.D.

, A new fractal approach for describing induced-fracture porosity/permeability/compressibility in stimulated unconventional reservoirs, Journal of Petroleum Science and Engineering 179 (2019), 855–866.

30.

Siddique

, Ahmad

and Akram

, A study on generalized graphs representations of complex neutrosophic information, Journal of Applied Mathematics & Computing (2020), DOI : 10.1007/s12190-020-01400-0.

31.

Smarandache

, A Unifying Field in Logics: Neutrosophic Logic, American Research Press Rehoboth (1995).

32.

Smarandache

, Neutrosophic set - a generalization of the intuitionistic fuzzy set, Granular Computing 2006 IEEE International Conference (2006), 38–42, DOI: 10.1109/GRC.2006.1635754.

33.

Wang

, Smarandache

, Zhang

Y.Q.

and Sunderraman

, Single-valued neutrosophic sets, Multisspace and Multistruct 4 (2010), 410–413.

34.

Yaqoob

and Akram

, Complex neutrosophic graphs, Bull Comput Appl Math 6 (2018), 85–109.

35.

Yazdanbakhsh

and Dick

, A systematic review of complex fuzzy sets and logic, Fuzzy Sets Syst 338 (2018), 1–22.

36.

, Single-valued neutrosophic minimum spanning tree and its clustering method, Journal of Intelligent Systems 23(3) (2014), 311–324.

37.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

38.

Zhan

, Akram

and Sitara

, Novel decision-making method based on bipolar neutrosophic information, Soft Computing 23(20) (2019), 9955–9977.

39.

Zhao

, Xu

, Guo

, Liu

, Zhang

, Ning

, Li

and Shi

, A new and fast waterflooding optimization workflow based on INSIM-derived injection efficiency with a field application, Journal of Petroleum Science and Engineering 179 (2019), 1186–1200.

40.

Zuo

, Pal

and Dey

, New Concepts of Picture Fuzzy Graphs with Application, Mathematics 7(5) (2019), 470.

Representation of competitions by complex neutrosophic information

Abstract

Keywords

1 Introduction

2 Representation of competitions by complex neutrosophic graphs

Table 1 Membership values of vertices μ x 1 x 2 x 3 μ T 0.6 e i π 2 0.5e iπ 0.7 e i π 6 μ I 0.3 e i π 6 0.7 e i π 2 0.2 e i π 3 μ F 0.5 e i π 3 0.3 e i π 4 0.6 e i π 2

Conflict of interest

References

Table 1
Membership values of vertices

μ x ₁ x ₂ x ₃

μ _T 0.6 $e^{i \frac{π}{2}}$ 0.5e^iπ 0.7 $e^{i \frac{π}{6}}$

μ _I 0.3 $e^{i \frac{π}{6}}$ 0.7 $e^{i \frac{π}{2}}$ 0.2 $e^{i \frac{π}{3}}$

μ _F 0.5 $e^{i \frac{π}{3}}$ 0.3 $e^{i \frac{π}{4}}$ 0.6 $e^{i \frac{π}{2}}$