Edge regular intuitionistic fuzzy soft graphs

Abstract

In this research article, we present a novel frame work for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We present the concepts of intuitionistic fuzzy soft graphs, regular intuitionistic fuzzy soft graphs and irregular intuitionistic fuzzy soft graphs. We illustrate these concepts by describing several examples. We investigate some of their related properties. We introduce the notion of edge regular intuitionistic fuzzy soft graphs and present some of their interesting properties. We also present an application of intuitionistic fuzzy soft graphs in decision-making.

Keywords

Regular intuitionistic fuzzy soft graphs irregular intuitionistic fuzzy soft graph edge regular intuitionistic fuzzy soft graphs decision-making

1 Introduction

Atanassov [7] introduced the concept of intuitionistic fuzzy sets as an extension of Zadeh’s fuzzy set [33]. The concept of an intuitionistic fuzzy set can be viewed as an alternative approach in such case where available information is not sufficient to define the impreciseness by the conventional fuzzy set. In fuzzy sets the degree of acceptance is considered only but intuitionistic fuzzy set is characterized by a membership function and a non-membership function, the only requirement is that the sum of both values is not more than one. Intuitionistic fuzzy sets are being studied and used in different fields of science, including computer science, mathematics, medical and engineering [21].

Mathematical modelling, analysis and computing of problems with uncertainty is one of the hottest areas in interdisciplinary research, involving applied mathematics, computational intelligence and decision-making. It is worth noting that uncertainty arises from various domains has very different nature and cannot be captured within a single mathematical framework. Molodtsov’s soft set theory provide us a new mathematical tool for dealing with uncertainty from the viewpoint of parameterization. Molodtsov [28] introduced the concept of soft set theory. He gave us new technique for dealing with uncertainty from the viewpoint of parameters. It has been revealed that soft sets have potential applications in several fields [28]. Maji et al. [24] introduced fuzzy soft sets, a more generalized notion combining fuzzy sets and soft sets. By using this definition of fuzzy soft sets, many interesting applications of fuzzy soft set theory have been expanded by some researchers. Roy and Maji [30] gave some applications of fuzzy soft sets. Som [32] defined soft relation and fuzzy soft relation on the theory of soft sets. Several researchers applied the concepts of soft sets and fuzzy soft sets to algebraic structures (see [6 , 35–36]). Maji et al. [25] introduced the noble concept of intuitionistic fuzzy soft set and presented some operations on intuitionistic fuzzy soft sets. Ca $\overset{˘}{g}$ man et al. [11 –13] presented applications of fuzzy soft set theory, soft matrix theory and intuitionistic fuzzy soft set theory in decision making.

Fuzzy graph theory is finding an increasing number of applications in modeling real time systems where the level of information inherent in the system varies with different levels of precision. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional numerical models used in engineering and sciences and the symbolic models used in expert systems. Kaufmann’s initial definition of a fuzzy graph [20] was based on Zadeh’s fuzzy relations [33]. Rosenfeld [31] described the structure of fuzzy graphs obtaining analogs of several graph theoretical concepts. Bhattacharya [9] gave some remarks on fuzzy graphs and several concepts were introduced by Mordeson and Nair in [27]. Akram et al. [1 –5] introduced many new concepts, including intuitionistic fuzzy hypergraphs, soft graphs, fuzzy soft graphs and applications of intuitionistic fuzzy digraphs. In this research article, we present a novel frame work for handling intuitionistic fuzzy soft information by combining the theory of intuitionistic fuzzy soft sets with graphs. We introduce the notions of intuitionistic fuzzy soft graphs, regular intuitionistic fuzzy soft graphs, irregular intuitionistic fuzzy soft graphs and edge regular intuitionistic fuzzy soft graph. We illustrate these novel concepts by several examples, and investigate some of their related properties. We also present an application of intuitionistic fuzzy soft graph in decision-making.

2 Preliminaries

In this section, we review some basic definitions that will be used in the sequel.

Definition 2.1. [33, 34] A fuzzy subsetμ on a set X is a mapping μ : X → [0, 1]. A mapping ν : X × X → [0, 1] is called a fuzzy relation on X if ν (x, y)≤ min({μ (x) , μ (y)} ∀ x, y ∈ X.

Definition 2.2. [7] A mapping A = (μ_A, ν_A) : X → [0, 1] × [0, 1] is called an intuitionistic fuzzy set in X if μ_A (x) + ν_A (x) ≤1 ∀ x ∈ X, where the mappings μ_A : X → [0, 1] and ν_A : X → [0, 1] denote the degree of membership (namely μ_A (x)) and the degree of non-membership (namely ν_A (x)) of each element x ∈ X to A, respectively.

Definition 2.3. [7] An intuitionistic fuzzy graph with underlying set V is defined to be a pair G = (A, B) such that

μ_A : V → [0, 1] and ν_A : V → [0, 1] denote the degree of membership and non-member-ship of each element x ∈ V, respectively, such that μ_A (x) + ν_A (x) ≤1,

the functions μ_B : E ⊆ V × V → [0, 1] and ν_B : E ⊆ V × V → [0, 1] are defined byμ_B (x, y) ≤ min {μ_A (x) , μ_A (y)}, ν_B (x, y) ≤ max {ν_A (x) , ν_A (y)} such that 0 ≤ μ_B (x, y) + ν_B (x, y) ≤1 ∀ xy ∈ E.

We call A the intuitionistic fuzzy vertex set of V and B the intuitionistic fuzzy edge set of E. G = (A, B) is an intuitionistic graph on G^∗ = (V, E).

Definition 2.4. [17] Let G be an intuitionistic fuzzy graph. Then the degree of a vertex v_i is defined by d_G (v_i) = (d_μ (v_i, d_ν) (v_i)) , where d_μ (v_i) = ∑_{v_i≠v_j}μ_B (v_i, v_j) and d_ν (v_i) = ∑_{v_i≠v_j}ν_B (v_i, v_j).

Definition 2.5. [19] Let G be an intuitionistic fuzzy graph and e_ij = (v_i, v_j) ∈ E be an edge in G. Then the degree of an edge (v_i, v_j) ∈ E is defined by d_G (v_i, v_j) = (d_μ (v_i, v_j) , d_ν (v_i, v_j)) , where d_μ (v_i, v_j) = d_μ (v_i) + d_μ (v_j) -2μ_B (v_i, v_j) and d_ν (v_i, v_j) = d_ν (v_i) + d_ν (v_j) -2ν_B (v_i, v_j).

Definition 2.6. [19] Let G be an intuitionistic fuzzy graph and e_ij = (v_i, v_j) ∈ E be an edge in G. Then the total edge degree of an edge (v_i, v_j) ∈ E is defined by td_G (v_i, v_j) = (td_μ (v_i, v_j) , td_ν (v_i, v_j)) , where td_μ (v_i, v_j) = d_μ (v_i) + d_μ (v_j) - μ_B (v_i, v_j) and td_ν (v_i, v_j) = d_ν (v_i) + d_ν (v_j) - ν_B (v_i, v_j).

Definition 2.7. [19] Let G be an intuitionistic fuzzy graph. If each edge in G has same degree (l₁, l₂) then G is said to be an edge regular intuitionistic fuzzy graph.

Soft set theory was proposed by Molodtsov [28] in 1999. This theory provides a parameterized point of view for uncertainty modelling and soft computing. Let U be the universe of discourse and P be the universe of all possible parameters related to the objects in U. Each parameter is a word or a sentence. In most cases, parameters are considered to be attributes, characteristics or properties of objects in U. The pair (U, P) is also known as a soft universe. The power set of U is denoted by 𝒫(U).

Definition 2.8. A pair (F, M) is called soft set over U, where M ⊆ P, F is a set-valued function F : M→ 𝒫(U). In other words, a soft set over U is a parameterized family of subsets of U.

By means of parametrization, a soft set produces a series of approximate descriptions of a complicated object being perceived from various points of view. It is apparent that a soft set (F, M) over a universe U can be viewed as a parameterized family of subsets of U. For any parameter ε ∈ M, the subset F (ε) ⊆ U may be interpreted as the set of ε-approximate elements. Maji et al. [25, 30] further introduced the concept of fuzzy soft set and intuitionistic fuzzy soft set.

Definition 2.9. [30] Let 𝒫(U) denotes the collection of all fuzzy sets of U. Let M_i ⊂ P.

A pair (F_i, M_i) is called a fuzzy-soft-set over U, where F_i is a mapping given by F_i : M_i→ 𝒫(U).

Definition 2.10. [25] Let $P (U)$ denotes the collection of all intuitionistic fuzzy sets of U. Let M ⊂ P. A pair (F, M) is called an intuitionistic fuzzy soft set over U, where F is a mapping given by $F : M \to P (U)$ .

Definition 2.11. [25] An intuitionistic fuzzy soft set (F, M) is said to be null intuitionistic fuzzy soft set denoted by ∅, if for all e ∈ M, F (e) = null intuitionistic fuzzy set of U.

Definition 2.12. [25] An intuitionistic fuzzy soft set (F, M) is said to be absolute intuitionistic fuzzy soft set denoted by $\tilde{A}$ , if for all e ∈ M, F (e) = U.

Definition 2.13. [25] The complement of an intuitionistic fuzzy soft set (F, M) is denoted by (F, M) ^c and is defined by (F, M) ^c = (F^c, ¬ M).

3 Regular intuitionistic fuzzy soft graphs

Let U be an initial universe and P the set of all parameters. $P (U)$ denotes the set of all intuitionistic fuzzy sets of U. Let M be a subset of P. A pair (F, M) is called an intuitionistic fuzzy soft set [25] over U, where intuitionistic fuzzy approximation function is given by $F = (F_{μ}, F_{ν}) : M \to P (U)$ .

Let $P (V)$ denotes the set of all intuitionistic fuzzy sets of V and $P (E)$ denotes the set of all intuitionistic fuzzy sets of E.

Definition 3.1. An intuitionistic fuzzy soft graph G = (G^*, Φ, Ψ, M) is an ordered 4-tuple such that

G^* is a simple graph,

M = (V, E) is a non-empty set of parameters,

(Φ, M) is an intuitionistic fuzzy soft set over V,

(Ψ, M) is an intuitionistic fuzzy soft set over E ⊆ V × V,

(Φ (a) , Ψ (a)) is an intuitionistic fuzzy graph for all a ∈ M,

that is,

\begin{matrix} Ψ_{μ} (a) (xy) & \leq & min (Φ_{μ} (a) (x), Φ_{μ} (a) (y)), \\ Ψ_{ν} (a) (xy) & \leq & max (Φ_{ν} (a) (x), Φ_{ν} (a) (y)) \end{matrix}

such that

Ψ_{μ} (a) (xy) + Ψ_{ν} (a) (xy) \leq 1

for all a ∈ M, x, y ∈ V.

The intuitionistic fuzzy graph (Φ (a) , Ψ (a)) is denoted by H (a). An intuitionistic fuzzy soft graph is a parameterized family of intuitionistic fuzzy graphs. Throughout this paper, G^* will be a crisp graph, and G = (Φ, Ψ, M) an intuitionistic fuzzy soft graph.

Example 3.2. Consider a crisp graph G^* = (V, E) such that V = {v₁, v₂, v₃, v₄} and E = {v₁v₂, v₂v₃, v₃v₁, v₁v₄, v₂v₄, v₃v₄}. Let P = {e₁, e₂, e₃, e₄} be a set of all parameters and M = {e₁, e₂} ⊂ P. Let (Φ, M) be an intuitionistic fuzzy soft set over V with intuitionistic fuzzy approximation function $Φ : M \to P (V)$ defined by $\begin{matrix} Φ (e_{1}) & = & {(v_{1}, 0.3, 0.4), (v_{2}, 0.5, 0.2), \\ (v_{3}, 0.7, 0.1), (v_{4}, 0.9, 0.0)}, \\ Φ (e_{2}) & = & {(v_{1}, 0.7, 0.1), (v_{2}, 0.3, 0.4), \\ (v_{3}, 0.4, 0.4), (v_{4}, 0.6, 0.2)} . \end{matrix}$

Let (Ψ, M) be an intuitionistic fuzzy soft set over E with intuitionistic fuzzy approximation function $Ψ : M \to P (E)$ defined by $\begin{matrix} Ψ (e_{1}) & = & {(v_{1} v_{2}, 0.0, 0.0), (v_{2} v_{3}, 0.4, 0.2), \\ (v_{1} v_{3}, 0.3, 0.3), (v_{1} v_{4}, 0.2, 0.4), \\ (v_{2} v_{4}, 0.3, 0.2), (v_{3} v_{4}, 0.5, 0.1)}, \\ Ψ (e_{2}) & = & {(v_{1} v_{2}, 0.3, 0.4), (v_{2} v_{3}, 0.3, 0.4), \\ (v_{1} v_{3}, 0.4, 0.4), (v_{1} v_{4}, 0.2, 0.2), \\ (v_{2} v_{4}, 0.3, 0.3), v_{3} v_{4}, 0.4, 0.1)} . \end{matrix}$

Thus intuitionistic fuzzy graphs H (e₁) = (Φ (e₁) , Ψ (e₁)) and H (e₂) = (Φ (e₂) , Ψ (e₂)) corresponding to the parameters e₁ and e₂, respectively are shown in Fig. 1.

Tabular representation of intuitionistic fuzzy soft graph G is given in Table 1.

Hence G = {H (e₁) , H (e₂)} is an intuitionistic fuzzy soft graph on M.

Definition 3.3. The order of an intuitionistic soft graph is $\begin{matrix} O (G) & = & (\sum_{e_{i} \in M} (\sum_{u \in V} Φ_{μ} (e_{i}) (u)), \\ \sum_{e_{i} \in M} (\sum_{v \in V} Φ_{ν} (e_{i}) (u))) . \end{matrix}$

The size of an intuitionistic soft graph is $\begin{matrix} S (G) & = & (\sum_{e_{i} \in M} (\sum_{uv \in E} Ψ_{μ} (e_{i}) (uv)), \\ \sum_{e_{i} \in L} (\sum_{uv \in E} Ψ_{ν} (e_{i}) (uv))) . \end{matrix}$

Example 3.4. Consider a crisp graph G^* = (V, E)such that V = {v₁, v₂, v₃, v₄, v₅} and E = {v₁v₂, v₂v₃, v₂v₄, v₄v₅, v₁v₅, v₃v₅}. Let M = {e₁, e₂, e₃} be a parameter set and let (Φ, M) be an intuitionistic fuzzy soft set over V with intuitionistic fuzzy approximation function $Φ : A \to P (V)$ defined by $\begin{matrix} Φ (e_{1}) & = & {(v_{1}, 0.3, 0.4), (v_{2}, 0.5, 0.3), (v_{3}, 0.7, 0.1), \\ (v_{4}, 0.9, 0.0), (v_{5}, 0.0, 0.1)}, \\ Φ (e_{2}) & = & {(v_{1}, 0.7, 0.1), (v_{2}, 0.6, 0.1), (v_{3}, 0.4, 0.3), \\ (v_{4}, 0.0, 0.0), (v_{5}, 0.6, 0.2)} . \\ Φ (e_{3}) & = & {(v_{1}, 0.6, 0.1), (v_{2}, 0.2, 0.5), (v_{3}, 0.1, 0.3), \\ (v_{4}, 0.4, 0.2), (v_{5}, 0.5, 0.1)} . \end{matrix}$

Let (Ψ, M) be an intuitionistic fuzzy soft set over E with intuitionistic fuzzy approximation function $Ψ : A \to P (E)$ defined by $\begin{matrix} Ψ (e_{1}) & = & {(v_{1} v_{2}, 0.2, 0.3), (v_{2} v_{3}, 0.0, 0.3), \\ (v_{2} v_{4}, 0.5, 0.3), (v_{4} v_{5}, 0.0, 0.0), \\ (v_{1} v_{5}, 0.0, 0.4), (v_{3} v_{5}, 0.0, 0.0)} \\ Ψ (e_{2}) & = & {(v_{1} v_{2}, 0.2, 0.1), (v_{2} v_{3}, 0.4, 0.2), \\ (v_{2} v_{4}, 0.0, 0.1), (v_{4} v_{5}, 0.0, 0.2), \\ (v_{1} v_{5}, 0.6, 0.1), (v_{3} v_{5}, 0.3, 0.3)}, \\ Ψ (e_{3}) & = & {(v_{1} v_{2}, 0.2, 0.3), (v_{2} v_{3}, 0.0, 0.3), \\ (v_{2} v_{4}, 0.2, 0.4), (v_{4} v_{5}, 0.0, 0.0), \\ (v_{1} v_{5}, 0.4, 0.1), (v_{3} v_{5}, 0.1, 0.2)} . \end{matrix}$

Tabular representation of an intuitionistic fuzzy soft graph is given in Table 2.

The intuitionistic fuzzy graphs of G are H (e₁) = (Φ (e₁) , Ψ (e₁)), H (e₂) = (Φ (e₂) , Ψ (e₂)) and H (e₄) = (Φ (e₄) , Ψ (e₃)) corresponding to parameters e₁, e₂ and e₃, respectively. Hence G = {H (e₁) , H (e₂) , H (e₃)} is an intuitionistic fuzzy soft graph on M.

In this example, the order of intuitionistic fuzzy soft graph is (∑_{e_i∈M} (∑_u∈VΦ_μ (e_i) (u)) , ∑_{e_i∈M} (∑_v∈VΦ_ν (e_i) (u)) ) = ((0.3 + 0.5 + 0.9 + 0.7) + (0.7 + 0.6 + 0.4 + 0.6) + (0.6 + 0.2 + 0.1 + 0.4 + 0.5) , (0.4 + 0.3 + 0.1 + 0.1) + (0.1 + 0.1 + 0.3 + 0.2) + (0.1 + 0.5 + 0.3 + 0.2 + 0.1)) = (6.5, 2.8).

The size of an intuitionistic fuzzy soft graph is (∑_{e_i∈M} (∑_uv∈EΨ_μ (e_i) (uv)) , ∑_{e_i∈L} (∑_uv∈EΨ_ν (e_i) (uv)) ) = ((0.2 + 0.5) + (0.2 + 0.4 + 0.6 + 0.3) + (0.2 + 0.2 + 0.4 + 0.1) , (0.3 + 0.3 + 0.3 + 0.4) + (0.1 + 0.2 + 0.1 + 0.2 + 0.1 + 0.3) + (0.3 + 0.3 + 0.4 + 0.1 + 0.2)) = (3.1, 3.6) .

Definition 3.5. An intuitionistic fuzzy soft graph G is a strong intuitionistic fuzzy soft graph if H (a) is a strong intuitionistic fuzzy graph for all a ∈ M. An intuitionistic fuzzy soft graph G is a complete intuitionistic fuzzy soft graph if H (a) is a complete intuitionistic fuzzy graph for all a ∈ M.

Definition 3.6. Let G be an intuitionistic fuzzy soft graph of G^*. G is called a regular intuitionistic fuzzy soft graph if H (e) is a regular intuitionistic fuzzy graph for all e ∈ M. If H (e) is a regular intuitionistic fuzzy graph of degree $(r, \overset{´}{r})$ for all e ∈ M, then G is a $(r, \overset{´}{r}) -$ regular intuitionistic fuzzy soft graph.

Example 3.7. Consider a simple graph G^* = (V, E), where V = {v₁, v₂, v₃, v₄} and E = {v₁v₂, v₂v₃, v₃v₄, v₄v₁}. Let M = {e₁, e₂, e₃, e₄} be a set of parameters. Let G = (H, M) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁), H (e₂), H (e₃) and H (e₄) corresponding to the parameters e₁, e₂, e₃ and e₄, respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(v_{1}, 0.4, 0.5), (v_{2}, 0.5, 0.4), \\ (v_{3}, 0.4, 0.6), (v_{4}, 0.5, 0.3)}, \\ {(v_{1} v_{2}, 0.3, 0.5), (v_{2} v_{3}, 0.2, 0.3), \\ (v_{3} v_{4}, 0.3, 0.5), (v_{4} v_{1}, 0.2, 0.3)}), \\ H (e_{2}) & = & ({(v_{1}, 0.5, 0.4), (v_{2}, 0.5, 0.2), \\ (v_{3}, 0.6, 0.3), (v_{4}, 0.5, 0.5)}, \\ {(v_{1} v_{2}, 0.4, 0.2), (v_{2} v_{3}, 0.5, 0.1), \\ (v_{3} v_{4}, 0.4, 0.2), (v_{4} v_{1}, 0.5, 0.1)}), \\ H (e_{3}) & = & ({(v_{1}, 0.4, 0.5), (v_{2}, 0.2, 0.7), \\ (v_{4}, 0.3, 0.6), (v_{3}, 0.4, 0.5)}, \\ {(v_{1} v_{2}, 0.2, 0.6), (v_{2} v_{3}, 0.1, 0.3), \\ (v_{3} v_{4}, 0.2, 0.6), (v_{4} v_{1}, 0.1, 0.3)}) \\ H (e_{4}) & = & ({(v_{1}, 0.4, 0.4), (v_{2}, 0.5, 0.2), \\ (v_{3}, 0.6, 0.3), (v_{4}, 0.4, 0.5)}, \\ {(v_{1} v_{2}, 0.4, 0.3), (v_{2} v_{3}, 0.2, 0.1), \\ (v_{3} v_{4}, 0.4, 0.3), (v_{4} v_{1}, 0.2, 0.1)}) . \end{matrix}$

Clearly, intuitionistic fuzzy graphs H (e₁), H (e₂), H (e₃) and H (e₄) of G^* corresponding to the parameter e₁, e₂, e₃ and e₄, respectively are regular intuitionistic fuzzy graphs. Hence G is a regular intuitionistic fuzzy soft graph.

Definition 3.8. Let G be an intuitionistic fuzzy soft graph of G^*. G is said to be a totally regular intuitionistic fuzzy soft graph if H (e) is a totally regular intuitionistic fuzzy graph for all e ∈ M. If H (e) is a totally regular intuitionistic fuzzy graph of total degree $(t, \overset{´}{t})$ for all e ∈ M, then G is a $(t, \overset{´}{t}) -$ totally regular intuitionistic fuzzy soft graph.

Example 3.9. Consider a simple graph G^* = (V, E), where V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₂u₃, u₃u₄, u₁u₃}. Let M = {e₁, e₂} be a set of parameters. Let G = (H, M) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁) = (Φ₁ (e₁) , Ψ₁ (e₁)) and H (e₂) = (Φ₂ (e₂) , Ψ₂ (e₂)) corresponding to the parameters e₁ and e₂, respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(u_{1}, 0.3, 0.6), (u_{2}, 0.2, 0.4), \\ (u_{3}, 0.3, 0.3), (u_{4}, 0.4, 0.5)}, \\ {(u_{1} u_{2}, 0.2, 0.3), (u_{2} u_{3}, 0.1, 0.2), \\ (u_{3} u_{4}, 0.1, 0.4)}), \\ H (e_{2}) & = & ({(u_{1}, 0.4, 0.4), (u_{2}, 0.3, 0.4), \\ (u_{3}, 0.2, 0.3), (u_{4}, 0.5, 0.5)}, \\ {(u_{1} u_{2}, 0.1, 0.2), (u_{2} u_{3}, 0.2, 0.1), \\ (u_{1} u_{3}, 0.1, 0.1), (u_{3} u_{4}, 0.1, 0.2)}) . \end{matrix}$

Clearly, in intuitionistic fuzzy graphs H (e₁) = (Φ₁ (e₁) , Ψ₁ (e₁)), tdeg (u₁) = (0.5, 0.9) = tdeg (u₂), tdeg (u₃) = (0.5, 0.9) = tdeg (u₄), so H (e₁) is atotally regular intuitionistic fuzzy graph. Also, in intuitionistic fuzzy graphs H (e₂) = (Φ₁ (e₂) , Ψ₁ (e₂)), tdeg (u₁) = (0.6, 0.7) = tdeg (u₂), tdeg (u₃) = (0.6, 0.7) = tdeg (u₄), so H (e₂) is a totally regular intuitionistic fuzzy graph. Hence G = {H (e₁) , H (e₂)} is a totally regular intuitionistic fuzzy soft graph.

We state the following propositions without their proofs.

Proposition 3.10. Let G be an intuitionistic fuzzy soft graph of G^*. If G is a regular (totally regular) intuitionistic fuzzy soft graph and Φ is a constant function in intuitionistic fuzzy graph H (e_i) of G^* for all e_i ∈ M for i = 1, 2, ⋯ , n. Then G is a totally regular (regular) intuitionistic fuzzy soft graph.

Proposition 3.11. Let G be an intuitionistic fuzzy soft graph over an odd cycle G^* = (V, E). Then G is regular intuitionistic fuzzy soft graph if and only if Ψ is a constant function in intuitionistic fuzzy graph H (e_i) over H^* (e_i) is an odd cycle for all e_i ∈ M for i = 1, 2, 3, ⋯ , n.

Proposition 3.12. A regular intuitionistic fuzzy soft graph on an odd cycle does not have an intuitionistic fuzzy bridge and intuitionistic fuzzy cut vertex.

Proposition 3.13. Let G be an intuitionistic fuzzy soft graph over an even cycle G^* = (V, E). Then G is regular intuitionistic fuzzy soft graph if and only if Ψ is a constant function or alternate edges have same membership degree and non-membership degrees in intuitionistic fuzzy graph H (e_i) over H^* (e_i) is an even cycle for all e_i ∈ M for i = 1, 2, 3, ⋯ , n.

Proposition 3.14. If G is a regular intuitionistic fuzzy soft graph and Φ is a constant function then G^c is a regular fuzzy soft graph.

Proposition 3.15. If G is a totally regular intuitionistic fuzzy soft graph and Φ is a constant function then G^c is a totally regular fuzzy soft graph.

Proposition 3.16. A $(r, \overset{´}{r}) -$ regular intuitionistic fuzzy soft graph G on G^* with |V|≥3 have no end node.

Definition 3.17. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^* = (V, E). G is called irregular intuitionistic fuzzy soft graph if H (e) is irregular intuitionistic fuzzy graph of G^* for all e ∈ M. Equivalently, an intuitionistic fuzzy soft graph G is called an irregular intuitionistic fuzzy soft graph if there is a vertex which is adjacent to the vertices with distinct degrees in H (e) for all e ∈ M.

Example 3.18. Consider a simple graph G^* = (V, E), where V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₂u₃, u₃u₄, u₄u₁, u₁u₃, u₂u₄}. Let M = {e₁, e₂, e₃, e₄} be a set of parameters. G = (H, M) be an irregular intuitionistic fuzzy soft graph, where irregular intuitionistic fuzzy graphs H (e₁) , H (e₂) and H (e₃) corresponding to the parameters e₁, e₂ and e₃, respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(u_{1}, 0.4, 0.5), (u_{2}, 0.5, 0.4), \\ (u_{3}, 0.4, 0.6), (u_{4}, 0.5, 0.3)}, \\ {(u_{1} u_{2}, 0.3, 0.4), (u_{2} u_{3}, 0.2, 0.4), \\ (u_{3} u_{4}, 0.1, 0.6), (u_{4} u_{1}, 0.2, 0.5)}), \\ H (e_{2}) & = & ({(u_{1}, 0.4, 0.3), (u_{2}, 0.5, 0.2), \\ (u_{3}, 0.4, 0.3), (u_{4}, 0.4, 0.6)}, \\ {(u_{1} u_{2}, 0.1, 0.2), (u_{2} u_{3}, 0.2, 0.3), \\ (u_{3} u_{4}, 0.1, 0.4), (u_{4} u_{1}, 0.4, 0.5), \\ (u_{2} u_{4}, 0.2, 0.6)}), \\ H (e_{3}) & = & ({(u_{1}, 0.3, 0.5), (u_{2}, 0.2, 0.7), \\ (u_{3}, 0.4, 0.5), (u_{4}, 0.3, 0.6)}, \\ {(u_{1} u_{2}, 0.2, 0.4), (u_{2} u_{3}, 0.2, 0.7), \\ (u_{3} u_{4}, 0.1, 0.6), (u_{4} u_{1}, 0.3, 0.4), \\ (u_{1} u_{3}, 0.1, 0.5)}) . \end{matrix}$

Hence G = {H (e₁) , H (e₂) , H (e₃)} is an irregular intuitionistic fuzzy soft graph.

Definition 3.19. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^* = (V, E). G is called neighborly irregular intuitionistic fuzzy soft graph if H (e) is neighborly irregular intuitionistic fuzzy graph of G^* for all e ∈ M. Equivalently, an intuitionistic fuzzy soft graph G is called a neighborly irregular intuitionistic fuzzy soft graph if every two adjacent vertices have distinct degrees in H (e) for all e ∈ M.

Definition 3.20. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^* = (V, E). G is called highly irregular intuitionistic fuzzy soft graph if H (e) is highly irregular intuitionistic fuzzy graph of G^* for all e ∈ M. Equivalently, an intuitionistic fuzzy soft graph G is called a highly irregular intuitionistic fuzzy soft graph if every vertex is adjacent to the vertices of distinct degrees in H (e) for all e ∈ M.

Remark 3.21. A highly irregular intuitionistic fuzzy soft graph may not be neighborly irregular intuitionistic fuzzy soft graph.

Example 3.22. Consider a simple graph G^* = (V, E), where V = {u₁, u₂, u₃, u₄, u₅} and E = {u₁u₂, u₂u₃, u₃u₄, u₄u₁, u₂u₅}. Let M = {e₁, e₂} be a set of parameters. G = (H, M) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁) and H (e₂) corresponding to the parameters e₁ and e₂, respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(u_{1}, 0.4, 0.4), (u_{2}, 0.4, 0.5), \\ (u_{3}, 0.5, 0.3), (u_{4}, 0.4, 0.4), \\ (u_{5}, 0.2, 0.1)}, {(u_{1} u_{2}, 0.1, 0.0), \\ (u_{2} u_{3}, 0.2, 0.1), (u_{3} u_{4}, 0.3, 0.3), \\ (u_{4} u_{1}, 0.3, 0.2), (u_{2} u_{5}, 0.1, 0.1)}), \\ H (e_{2}) & = & ({(u_{1}, 0.3, 0.4), (u_{2}, 0.4, 0.5), \\ (u_{3}, 0.4, 0.3), (u_{4}, 0.4, 0.5), \\ (u_{5}, 0.3, 0.1)}, {(u_{1} u_{2}, 0.3, 0.4), \\ (u_{2} u_{3}, 0.2, 0.5), (u_{3} u_{4}, 0.3, 0.4), \\ (u_{4} u_{1}, 0.3, 0.4), (u_{2} u_{5}, 0.1, 0.2)}) . \end{matrix}$

Since each vertex is adjacent to the vertices with distinct degrees in intuitionistic fuzzy graphs H (e₁) and H (e₂).

Hence G = {H (e₁) , H (e₂)} is a highly irregular intuitionistic fuzzy soft graph. As deg (u₁) = (0.4, 0.2) = deg (u₂) in H (e₁) and deg (u₁) = (0.6, 0.8) = deg (u₄) in H (e₂), so H (e₁) and H (e₂) are not neighborly irregular intuitionistic fuzzy graphs. Hence G = {H (e₁) , H (e₂)} is not neighborly irregular intuitionistic fuzzy soft graph.

Remark 3.23. A neighborly irregular intuitionistic fuzzy soft graph may not be a highly irregular intuitionistic fuzzy soft graph.

Theorem 3.24. An intuitionistic fuzzy soft graph G = (H, M) is highly irregular intuitionistic fuzzy graph and neighborly irregular intuitionistic fuzzy graph if and only if the degrees of all vertices of H (e) for all e ∈ M are distinct.

Example 3.25. Consider a simple graph G^* = (V, E), where V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₂u₃, u₃u₄, u₄u₁, u₁u₃, u₂u₄}. Let M = {e₁, e₂} be a set of parameters. let G = (H, M) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁) and H (e₂) corresponding to the parameters e₁ and e₂, respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(u_{1}, 0.4, 0.5), (u_{2}, 0.3, 0.7), \\ (u_{3}, 0.4, 0.5), (u_{4}, 0.3, 0.6)}, \\ {(u_{1} u_{2}, 0.3, 0.6), (u_{2} u_{4}, 0.2, 0.4), \\ (u_{4} u_{3}, 0.3, 0.5), (u_{1} u_{3}, 0.3, 0.4), \\ (u_{1} u_{4}, 0.3, 0.2)}), \\ H (e_{2}) & = & ({(u_{1}, 0.4, 0.4), (u_{2}, 0.5, 0.4), \\ (u_{3}, 0.6, 0.3), (u_{4}, 0.4, 0.5)}, \\ {(u_{1} u_{2}, 0.1, 0.2), (u_{2} u_{4}, 0.1, 0.1), \\ (u_{4} u_{3}, 0.1, 0.2), (u_{1} u_{4}, 0.1, 0.2), \\ (u_{2} u_{3}, 0.2, 0.4)}) . \end{matrix}$

Clearly, H (e₁) and H (e₂) are highly irregular and neighborly irregular intuitionistic fuzzy graphs, all the vertices of intuitionistic fuzzy graphs H (e₁) and H (e₂) have distinct degrees.

Definition 3.26. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^* = (V, E). G is called totally irregular intuitionistic fuzzy soft graph if H (e) is totally irregular intuitionistic fuzzy graph of G^* for all e ∈ M.

Example 3.27. Consider a simple graph G^* = (V, E), where V = {v₁, v₂, v₃, v₄} and E = {v₁v₂, v₁v₃, v₁v₄, v₃v₄}. Let M = {e₁, e₂} be a set of parameters. G = (M, H) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁) and H (e₂) corresponding to the parameters e₁ and e₂, respectively are defined as follows:

$\begin{matrix} H (e_{1}) & = & ({(v_{1}, 0.3, 0.3), (v_{2}, 0.3, 0.2), \\ (v_{3}, 0.5, 0.4), (v_{4}, 0.2, 0.5)}, \\ {(v_{1} v_{2}, 0.3, 0.2), (v_{1} v_{3}, 0.2, 0.1), \\ (v_{1} v_{4}, 0.1, 0.0), (v_{3} v_{4}, 0.2, 0.1)}), \\ H (e_{2}) & = & ({(v_{1}, 0.3, 0.5), (v_{2}, 0.4, 0.4), \\ (v_{3}, 0.4, 0.3), (v_{4}, 0.4, 0.4)}, \\ {(v_{1} v_{2}, 0.3, 0.2), (v_{1} v_{3}, 0.2, 0.4), \\ (v_{1} v_{4}, 0.2, 0.2), (v_{3} v_{4}, 0.3, 0.4)}) . \end{matrix}$ tdeg (v₁) = (0.9, 0.6), tdeg (v₃) = (0.9, 0.6) and tdeg (v₄) = (0.5, 0.6) in H (e₁), vertex v₄ is adjacent to the vertices v₁ and v₃ which have distinct degree. So H (e₁) is totally irregular intuitionistic fuzzy graph. Similarly, by routine calculation it is easy to see that H (e₂) is totally irregular intuitionistic fuzzy of G^* graph. Hence G = {H (e₁) , H (e₂)} is a totally irregular intuitionistic fuzzy soft graph.

Definition 3.28. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^* = (V, E). G is called neighborly totally irregular intuitionistic fuzzy soft graph if H (e) is neighborly totally irregular intuitionistic fuzzy graph of G for all e ∈ M.

Proposition 3.29. If G is neighborly irregular intuitionistic fuzzy soft graph then G^c is not a neighborly irregular intuitionistic fuzzy soft graph.

Proposition 3.30. Let G be an intuitionistic fuzzy soft graph. If G is a neighborly irregular intuitionistic fuzzy soft graph and Φ is a constant function then G is a neighborly totally irregular intuitionistic fuzzy soft graph.

Proof. Suppose that G is a neighborly irregular intuitionistic fuzzy soft graph and Φ is a constant function, then $Φ (e_{i}) (v) = (m_{i}, {\overset{´}{m}}_{i})$ , where $m_{i}, {\overset{´}{m}}_{i} \in [0, 1]$ , for all v ∈ V, e_i ∈ M for i = 1, 2, 3, ⋯ , n. We have $\deg (v_{1}) = (r_{i}, {\overset{´}{r}}_{i})$ and $\deg (v_{2}) = (s_{i}, {\overset{´}{s}}_{i})$ in intuitionistic fuzzy graph H (e_i), for all e_i ∈ M, v₁, v₂ ∈ V for i = 1, 2, 3, ⋯ , n. Since, tdeg (v₁) = deg (v₁) + Φ (e_i) (v₁) and tdeg (v₂) = deg (v₂) + Φ (e_i) (v₂) in H (e_i), for all e_i ∈ M, v ∈ V for i = 1, 2, 3, ⋯ , n. We claims that tdeg (v₁) ≠ tdeg (v₂) in intuitionistic fuzzy graph H (e_i), for all e_i ∈ M, v ∈ V for i = 1, 2, 3, ⋯ , n. On contrary, suppose that tdeg (v₁) = tdeg (v₂) $(m_{i} + r_{i}, {\overset{´}{m}}_{i} + {\overset{´}{r}}_{i}) = (m_{i} + s_{i}, {\overset{´}{m}}_{i} + {\overset{´}{s}}_{i})$ $\Rightarrow (r_{i}, {\overset{´}{r}}_{i}) = (s_{i}, {\overset{´}{s}}_{i}) .$

This implies deg (v₁) = deg (v₂), which is a contradiction to the fact that G is a neighborly irregular intuitionistic fuzzy soft graph. Hence G is a neighborly totally irregular intuitionistic fuzzy soft graph. □

Proposition 3.31. Let G be an intuitionistic fuzzy soft graph. If G is a neighborly totally irregular intuitionistic fuzzy soft graph and Φ is a constant function then G is a neighborly irregular intuitionistic fuzzy soft graph.

Proof. Suppose that G is a neighborly totally irregular intuitionistic fuzzy soft graph and Φ is a constant function, then $Φ (e_{i}) (v) = (m_{i}, {\overset{´}{m}}_{i})$ ,where $m_{i}, {\overset{´}{m}}_{i} \in [0, 1]$ , for all v ∈ V, e_i ∈ M for i = 1, 2, 3, ⋯ , n. We have $tdeg (v_{1}) = (s_{i}, {\overset{´}{s}}_{i})$ and $tdeg (v_{2}) = (t_{i}, {\overset{´}{t}}_{i})$ in intuitionistic fuzzy graphH (e_i) for all e_i ∈ M, v₁, v₂ ∈ V for i = 1, 2, 3, ⋯ , n. Since tdeg (v₁) = deg (v₁) + Φ (e_i) (v₁) and tdeg (v₂) = deg (v₂) + Φ (e_i) (v₂) in H (e_i) for alle_i ∈ M, v ∈ V for i = 1, 2, 3, ⋯ , n. Thisimplies $\deg (v_{1}) = (s_{i} - m_{i}, {\overset{´}{s}}_{i} - {\overset{´}{m}}_{i})$ and $\deg (v_{2}) = (t_{i} - m_{i}, {\overset{´}{t}}_{i} - {\overset{´}{m}}_{i})$ in intuitionistic fuzzy graph H (e_i) for all e_i ∈ M, v₁, v₂ ∈ V for i = 1, 2, 3, ⋯ , n. We claims that deg (v₁) ≠ deg (v₂) in intuitionistic fuzzy graph H (e_i) for all e_i ∈ M, v ∈ V for i = 1, 2, 3, ⋯ , n. On contrary, suppose that G is a regular intuitionistic fuzzy soft graph, then deg (v₁) = deg (v₂) $(s_{i} - m_{i}, {\overset{´}{s}}_{i} - {\overset{´}{m}}_{i}) = (t_{i} - m_{i}, {\overset{´}{t}}_{i} - {\overset{´}{m}}_{i})$ $\Rightarrow (s_{i}, {\overset{´}{s}}_{i}) = (t_{i}, {\overset{´}{t}}_{i})$ ⇒tdeg (v₁) = tdeg (v₂), which is a contradiction to the fact that G is a neighborly totally irregular intuitionistic fuzzy soft graph.

Hence G is a neighborly irregular intuitionistic fuzzy soft graph. □

Proposition 3.32. Let G be an intuitionistic fuzzy soft graph. If G is both neighborly irregular intuitionistic fuzzy soft graph and neighborly totally irregular intuitionistic fuzzy soft graph, then Φ need not be a constant function.

Remark 3.33. If G₁ is neighborly irregular intuitionistic fuzzy soft graph, then intuitionistic fuzzy soft subgraph G₂ of G₁ may not be a neighborly irregular intuitionistic fuzzy soft graph.

Example 3.34. Consider a simple graph G^* = (V, E), where V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₂u₃, u₃u₄, u₄u₁}. Let M = {e₁, e₂} be a set of parameters. Let G = (H, M) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁) and H (e₂) corresponding to the parameters e₁ and e₂ respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(u_{1}, 0.4, 0.3), (u_{2}, 0.2, 0.4), \\ (u_{3}, 0.4, 0.2), (u_{4}, 0.5, 0.4)}, \\ {(u_{1} u_{2}, 0.2, 0.3), (u_{2} u_{4}, 0.2, 0.2), \\ (u_{4} u_{3}, 0.4, 0.3), (u_{1} u_{3}, 0.1, 0.1)}), \\ H (e_{2}) & = & ({(u_{1}, 0.3, 0.5), (u_{2}, 0.4, 0.4), \\ (u_{3}, 0.2, 0.5), (u_{4}, 0.3, 0.4)}, \\ {(u_{1} u_{2}, 0.1, 0.3), (u_{2} u_{4}, 0.1, 0.2), \\ (u_{4} u_{3}, 0.2, 0.4), (u_{1} u_{3}, 0.1, 0.3)}) . \end{matrix}$

In intuitionistic fuzzy graphs H (e₂) and H (e₂), every two adjacent vertices have distinct degree. So, G = {H (e₁) , H (e₂)} is neighborly irregular intuitionistic fuzzy soft graph.

$\overset{´}{G} = (N, \overset{´}{H})$ be an intuitionistic fuzzy soft subgraph of G = (M, H), where intuitionistic fuzzy graphs $\overset{´}{H} (e_{1})$ and $\overset{´}{H} (e_{2})$ are subgraph of H (e₁) and H (e₂) corresponding to the parameters e₁ and e₂ respectively.

Clearly, two adjacent vertices have same degree in intuitionistic fuzzy graphs $\overset{´}{H} (e_{1})$ and $\overset{´}{H} (e_{2})$ corresponding to the parameters e₁ and e₂ respectively. Thus intuitionistic fuzzy soft subgraph $\overset{´}{G} = {\overset{´}{H} (e_{1}), \overset{´}{H} (e_{2})}$ of G is not neighborly irregular intuitionistic fuzzy soft graph.

Remark 3.35.

A complete intuitionistic fuzzy soft graph may not be a neighborly irregular intuitionistic fuzzy soft graph.

A neighborly irregular intuitionistic fuzzy soft graph may not be a neighborly totally irregular intuitionistic fuzzy soft graph.

4 Edge regular intuitionistic fuzzy soft graph

Definition 4.1. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^*. G is said to be an edge regular intuitionistic fuzzy soft graph if H (e) is an edge regular intuitionistic fuzzy graph for all e ∈ M. If H (e) is an edge regular intuitionistic fuzzy graph of degree $(l, \overset{´}{l})$ for all e ∈ M, then G is a $(l, \overset{´}{l}) -$ edge regular intuitionistic fuzzy soft graph.

Definition 4.2. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^*. G is said to be a totally edge regular intuitionistic fuzzy soft graph if H (e) is a totally edge regular intuitionistic fuzzy graph for all e ∈ M. If H (e) is a totally edge regular intuitionistic fuzzy graph of total edge degree $(t_{1}, {\overset{´}{t}}_{1})$ for all e ∈ M, then G is a $(t_{1}, {\overset{´}{t}}_{1}) -$ totally edge regular intuitionistic fuzzy soft graph.

Example 4.3. Consider a simple graph G^* = (V, E) such that V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₁u₃, u₁u₄, u₂u₃, u₂u₄, u₃u₄}. Let M = {e₁, e₂, e₃} be set of parameters.

Let G = (H, M) be an intuitionistic fuzzy soft graph, where intuitionistic fuzzy graphs H (e₁), H (e₂) and H (e₃) corresponding to the parameters e₁, e₂ and e₃, respectively are defined as follows: $\begin{matrix} H (e_{1}) & = & ({(u_{1}, 0.4, 0.5), (u_{2}, 0.3, 0.4), \\ (u_{3}, 0.2, 0.4), (u_{4}, 0.4, 0.3)}, \\ {(u_{1} u_{2}, 0.3, 0.5) (u_{1} u_{3}, 0.2, 0.3), \\ (u_{3} u_{4}, 0.2, 0.2), (u_{2} u_{4}, 0.3, 0.4)}), \\ H (e_{2}) & = & ({(u_{1}, 0.4, 0.5), (u_{2}, 0.4, 0.4), \\ (u_{3}, 0.6, 0.3), (u_{4}, 0.5, 0.2)}, \\ {(u_{1} u_{2}, 0.3, 0.3), (u_{1} u_{3}, 0.4, 0.3), \\ (u_{1} u_{4}, 0.3, 0.4), (u_{2} u_{4}, 0.3, 0.3), \\ (u_{3} u_{4}, 0.4, 0.1), (u_{2} u_{3}, 0.4, 0.2)}), \\ H (e_{3}) & = & ({(u_{1}, 0.5, 0.1), (u_{2}, 0.4, 0.4), \\ (u_{3}, 0.6, 0.2), (u_{4}, 0.3, 0.4)}, \\ {(u_{1} u_{3}, 0.5, 0.1) (u_{1} u_{4}, 0.3, 0.2), \\ (u_{2} u_{4}, 0.2, 0.4), (u_{2} u_{3}, 0.4, 0.3)}) . \end{matrix}$

As deg (u₁u₂) = deg (u₁u₃) = deg (u₂u₄) = deg (u₃u₄) = (0.5, 0.7) in H (e₁) , deg (u₁u₂) = deg (u₁u₃) = deg (u₁u₄) = deg (u₂u₃) = deg (u₂u₄) = deg (u₃u₄) = (1.4, 1.2) in H (e₂) and deg (u₁u₃) = deg (u₁u₄) = deg (u₂u₃) = deg (u₂u₄) = (0.7, 0.5)in H (e₃), so H (e₁), H (e₂) and H (e₃) are edge regular intuitionistic fuzzy graphs. Hence G = {H (e₁) , H (e₂) , H (e₃)} is an edge regular intuitionistic fuzzy soft graph.

Definition 4.4. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph of G^*. The maximum edge degree of G is Δ_E (G) = (Δ_μ (G) , Δ_ν (G)), where Δ_μ (G) = ∨ {deg_μ (v_iv_j) (e_k) | v_iv_j ∈ E, e_k ∈ M} and Δ_ν (G) = ∧ {deg_ν (v_iv_j) (e_k) | v_iv_j ∈ E, e_k ∈ M}.

Definition 4.5. The minimum edge degree of intuitionistic fuzzy soft graph G is δ_E (G) = (δ_μ (G) , δ_ν (G)), where δ_μ (G) = ∧ {deg_μ (v_iv_j) (e_k) |v_iv_j ∈ E, e_k ∈ M} and δ_ν (G) = ∨ {deg_ν (v_iv_j) (e_k) |v_iv_j ∈ E, e_k ∈ M}.

Theorem 4.6. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph on a cycle G^* = (V, E). Then $\sum_{e_{k} \in M} \sum_{v_{i} \in V} \deg_{G} (u_{i}) (e_{k}) = \sum_{e_{k} \in M} \sum_{u_{i} u_{j} \in E} \deg_{G} (u_{i} u_{j}) (e_{k}) .$

Proof. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph over a cycle u₁u₂, u₂u₃, ⋯ , u_nu_n+1, where u_n+1 = u₁ and M = {e₁, e₂, ⋯ , e_n} be a set of parameters. In intuitionistic fuzzy soft graph G, ∑_{e_k∈M}∑_{u_iu_j∈E}deg_G (u_iu_j) (e_k) = (∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) , ∑_{e_k∈M}∑_{u_iu_j∈E}deg_ν (u_iu_j) (e_k) ). Consider, ∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) = deg_μ (u₁u₂) (e₁) + … + deg_μ (u_nu₁) (e₁) + μ (u₁u₂) (e₂) + deg_μ (u₂u₃) (e₂) + … + deg_μ (u_nu₁) (e₂) + … + deg_μ (u₁u₂) (e_n) + … + deg_μ (u_nu₁) (e_n) = deg_μ (u₁) (e₁) + deg_μ (u₂) (e₁) -2Ψ_μ (u₁u₂) + deg_μ (u₂) (e₁) + deg_μ (u₃) (e₁) -2Ψ_μ (u₂u₃) (e₁) + … + deg_μ (u_n) (e₁) + deg_μ (u₁) (e₁) -2Ψ_μ (u_nu₁) (e₁) + … + deg_μ (u₁) (e_n) + deg_μ (u₂) (e_n) -2Ψ_μ (u₁u₂) (e_n) + deg_μ (u₂) (e_n) + deg_μ (u₃) (e_n) -2Ψ_μ (u₂u₃) (e_n) + … + deg_μ (u_n) (e_n) + deg_μ (u₁) (e_n) -2Ψ_μ (u_nu₁) =2deg_μ (u₁) (e₁) +2deg_μ (u₂) (e₁) + ⋯ +2deg_μ (u_n) (e₁) +2deg_μ (u₁) (e₂) +2deg_μ (u₂) (e₂) + ⋯ +2deg_μ (u₁) (e_n) +2deg_μ (u₂) (e_n) + ⋯ +2deg_μ (u_n) (e_n) -2 (Ψ_μ (u₁u₂) (e₁) + Ψ_μ (u₂u₃) (e₁) + ⋯ + Ψ_μ (u_nu₁) (e₁) + Ψ_μ (u₁u₂) (e₂) + Ψ_μ (u₂u₃) (e₂) + ⋯ + Ψ_μ (u₁u₂) (e_n) + Ψ_μ (u₂u₃) (e_n) + ⋯ + Ψ_μ (u_nu₁) (e_n) ) ∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) =2∑_{e_k∈M}∑_{v_i∈V}deg_μ (v_i) (e_k) -2∑_{e_k∈M}∑_{u_iu_j∈E}Ψ_μ (u_iu_j) (e_k) . Since ∑_{e_k∈M}∑_{v_i∈V}deg_μ (v_i) (e_k) =2∑_{e_k∈M}∑_{u_iu_j∈E}Ψ_μ (u_iu_j) (e_k), ∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) = ∑_{e_k∈M}∑_{v_i∈V}deg_μ (v_i) (e_k) , ∑_{e_k∈M}∑_{u_iu_j∈E}deg_ν (u_iu_j) (e_k) = ∑_{e_k∈M}∑_{v_i∈V}deg_ν (v_i) (e_k).

Hence ∑_{e_k∈M}∑_{u_iu_j∈E}deg_G (u_iu_j) (e_k) = ( ∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) , ∑_{e_k∈M}∑_{u_iu_j∈E}deg_ν (u_iu_j) (e_k) ) . □

Theorem 4.7. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph on G^*, then ∑_{e_k∈M}∑_{u_iu_j∈E}deg_G (u_iu_j) (e_k) = (∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_μ (u_iu_j) (e_k) , ∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_ν (u_iu_j) (e_k) ), where deg_{G
^*} (u_iu_j) = deg_{G
^*} (u_i) + deg_{G
^*} (u_j) -2 for all u_iu_j ∈ E, e_k ∈ M.

Proof. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph on G^* = (V, E). For each e_k ∈ M and u_iu_j ∈ E, deg_G (u_iu_j) (e_k) = (deg_μ (u_iu_j) (e_k) , deg_ν (u_iu_j) (e_k)).

In ∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) every edge contributes its membership strength exactly the degree of edge in the corresponding simple graph G^* for each e_k ∈ M, then ∑_{e_k∈M}∑_{u_iu_j∈E}deg_μ (u_iu_j) (e_k) = ∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_μ (u_iu_j) (e_k) and ∑_{e_k∈M}∑_{u_iu_j∈E}deg_ν (u_iu_j) (e_k) = ∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_ν (u_iu_j) (e_k) .

Hence ∑_{e_k∈M}∑_{u_iu_j∈E}deg_G (u_iu_j) (e_k) = (∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_μ (u_iu_j) (e_k) , ∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_ν (u_iu_j) (e_k) ) . □

Proposition 4.8. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph on k-regular crisp graph G^*. Then ∑_{u_iu_j∈E}deg_G (u_iu_j) (e_k) = ((k - 1) ∑_{u_i∈V}deg_μ (u_i) (e_k) , (k - 1) ∑_{v_i∈V}deg_ν (v_i) (e_k) ) ∀ e_k ∈ M .

Proposition 4.9. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph on k-regular crisp graph G^*. Then ∑_{e_k∈M}∑_{u_iu_j∈E}tdeg_G (u_iu_j) (e_k) = (∑_{e_k∈M}∑_{u_iu_j∈E}deg_{G
^*} (u_iu_j) Ψ_μ (u_iu_j) (e_k) + ∑_{e_k∈M}∑_{u_iu_j∈E}Ψ_μ (u_iu_j) (e_k) , ∑_{e_k∈M}∑_{u_iu_j∈E}Ψ_νdeg_{G
^*} (u_iu_j) Ψ_νu_iu_j (e_k) + ∑_{e_k∈M}∑_{u_iu_j∈E}Ψ_ν (u_iu_j) (e_k) ) .

Theorem 4.10. If an intuitionistic fuzzy soft graph is both edge regular and totally edge regular, then Ψ is a constant function.

Proof. Let G = (Φ, Ψ, M) be an edge regular and totally edge regular intuitionistic fuzzy soft graph on M. For each u_iu_j ∈ E, e_k ∈ M, $\deg_{G} (u_{i} u_{j}) (e_{k}) = (l_{k}, {\overset{´}{l}}_{k})$ , ${tdeg}_{G} (u_{i} u_{j}) (e_{k}) = (t_{k}, {\overset{´}{t}}_{k})$ and tdeg_G (u_iu_j) (e_k) = deg_G (u_iu_j) (e_k) + Ψ (u_iu_j) (e_k). Ψ_μ (u_iu_j) (e_k) = t_k - l_k and $Ψ_{ν} (u_{i} u_{j}) (e_{k}) = {\overset{´}{t}}_{k} - {\overset{´}{l}}_{k}$ , then $Ψ (u_{i} u_{j}) (e_{k}) = (t_{k} - l_{k}, {\overset{´}{t}}_{k} - {\overset{´}{l}}_{k})$ for all u_iu_j ∈ E, e_k ∈ M. Hence Ψ = (Ψ_μ, Ψ_ν) is a constant function. □

Theorem 4.11. Let G = (Φ, Ψ, M) be an intuitionistic fuzzy soft graph on a crisp graph G^*. If Ψ is a constant function. Then G is edge regular intuitionistic fuzzy soft graph if and only if G^* is edge regular.

Proof. Given that Φ is a constant function, i.e., Ψ_μ (u_iu_j) = c and $Ψ_{ν} (u_{i} u_{j}) = \overset{´}{c}$ , where c, $\overset{´}{c}$ are constants. Assume that intuitionistic fuzzy soft graph G is edge regular. On contrary suppose that simple graph G^* is not edge regular, then there exists at least one pair u_iu_j, u_su_t ∈ E such that deg_{G
^*} (u_iu_j) ≠ deg_{G
^*} (u_su_t). By definition of degree of an edge in intuitionistic fuzzy soft graph, for all u_iu_j ∈ E, e_k ∈ M $\deg_{G} (u_{i} u_{j}) (e_{k}) = (\deg_{μ} (u_{i} u_{j}) (e_{k}), \deg_{ν} (u_{i} u_{j}) (e_{k})), where \deg_{μ} (u_{i} u_{j}) (e_{k}) = \sum_{u_{i} u_{l} \in E i \neq l} Ψ_{μ} (u_{i} u_{l}) (e_{k}) + \sum_{u_{l} u_{j} \in E l \neq j} Ψ_{μ} (u_{l} u_{j}) (e_{k}) = \sum_{u_{i} u_{l} \in E i \neq l} c + \sum_{u_{l} u_{j} \in E l \neq j} c \deg_{μ} (u_{i} u_{j}) (e_{k}) = c (\deg_{G^{*}} (u_{i}) - 1) + c (\deg_{G^{*}} (u_{j}) - 1) = c (\deg_{G^{*}} (u_{i} u_{j})), \deg_{ν} (u_{i} u_{j}) (e_{k}) = \overset{´}{c} (\deg_{G^{*}} (u_{i} u_{j})) . G (u_{i} u_{j}) (e_{k}) = (c (\deg_{G^{*}} (u_{i} u_{j})), \overset{´}{c} (\deg_{G^{*}} (u_{i} u_{j}))) and G (u_{j} u_{l}) (e_{k}) = (c (\deg_{G^{*}} (u_{j} u_{l})), \overset{´}{c} (\deg_{G^{*}} (u_{j} u_{l}))) .$ Since simple graph G^* is not edge regular, this implies deg_{G
^*} (u_ju_l) ≠ deg_{G
^*} (u_iu_j), deg_G (u_ju_l) (e_k) ≠ deg_G (u_iu_i) (e_k) ∀ u_iu_j ∈ E, e_k ∈ M. This implies G is not edge regular intuitionistic fuzzy soft graph. Its contradict to our supposition. Hence G^* is an edge regular graph.

Conversely, assume that Ψ is constant function and G^* is edge regular graph. On contrary, suppose that G is not edge regular intuitionistic fuzzy soft graph, then there exists at least one pair u_iu_j, u_su_t ∈ E such that (deg_μ (u_iu_j) (e_k) , deg_ν (u_iu_j) (e_k) ) ≠ (deg_μ (u_su_t) (e_k) , deg_ν (u_su_t) (e_k) ) ∀ u_iu_j ∈ E, e_k ∈ M . By definition of degree of an edge in intuitionistic fuzzy soft graph for all u_iu_j ∈ E, e_k ∈ M, deg_μ (u_iu_j) (e_k) ≠ deg_ν (u_su_t) (e_k) , ∑_{u_iu_l∈Ei≠l}Ψ_μ (u_iu_l) (e_k) + ∑_{u_lu_j∈El≠j}Ψ_μ (u_lu_j) (e_k) ≠ ∑_{u_su_l∈Ei≠l}Ψ_μ (u_su_l) (e_k) + ∑_{u_lu_t∈El≠t}Ψ_μ (u_lu_t) (e_k).

Since Ψ is a constant function, this implies deg_{G
^*} (u_iu_j) ≠ deg_{G
^*} (u_su_t) implies G^* is not edge regular graph. Its contradicts to our supposition. Hence G is an edge regular graph intuitionistic fuzzy soft graph. □

Theorem 4.12. Let G = (Φ, Ψ, M) be a regular intuitionistic fuzzy soft graph, then G is edge regular intuitionistic fuzzy soft graph if and only if Ψ is a constant function.

Proof. Let G = (Φ, Ψ, M) be a $(r_{k}, {\overset{´}{r}}_{k})$ regular intuitionistic fuzzy soft graph, then $\deg (v) (e_{k}) = (r_{k}, {\overset{´}{r}}_{k})$ for all v ∈ V, e_k ∈ M. Suppose that Ψ is a constant function, i.e., Ψ_μ (u_iu_j) (e_k) = c and $Ψ_{ν} (u_{i} u_{j}) (e_{k}) = \overset{´}{c}$ for all u_iu_j ∈ E, e_k ∈ M, where c, $\overset{´}{c}$ are constants. By definition of degree of an edge in intuitionistic fuzzy soft graph deg_G (u_iu_j) (e_k) = (deg_μ (u_iu_j) (e_k) , deg_ν (u_iu_j) (e_k)) for all u_iu_j ∈ E, e_k ∈ M, where $\deg_{μ} (u_{i} u_{j}) (e_{k}) = 2 (r_{k} - c), \deg_{ν} (u_{i} u_{j}) (e_{k}) = 2 ({\overset{´}{r}}_{k} - c)$ . Hence G is edge regular intuitionistic fuzzy soft graph.

Conversly, suppose that G is an edge regular intuitionistic fuzzy soft graph, then $\deg_{G} (u_{i} u_{j}) (e_{k}) = (l_{k}, {\overset{´}{l}}_{k})$ , deg_G (u_iu_j) (e_k) = (deg_μ (u_iu_j) (e_k) , deg_ν (u_iu_j) (e_k)), where deg_μ (u_iu_j) (e_k) = deg_μ (u_i) (e_k) + deg_μ (u_j) (e_k) -2Ψ_μ (u_iu_j) (e_k) and deg_μ (u_iu_j) (e_k) = deg_ν (u_i) (e_k) + deg_ν (u_j) (e_k) -2Ψ_ν (u_iu_j) (e_k). We have $Ψ_{μ} (u_{i} u_{j}) (e_{k}) = \frac{(2 r_{k} - l_{k})}{2}$ and $Ψ_{ν} (u_{i} u_{j}) (e_{k}) = \frac{(2 {\overset{´}{r}}_{k} - {\overset{´}{l}}_{k})}{2} \forall u_{i} u_{j} \in E, e_{k} \in M .$ Hence Ψ is a constant function. □

5 Application

We present an application of intuitionistic fuzzy soft graph in a decision-making problem. The problem of object recognition has received paramount importance in recent times. The recognition problem may be viewed as a decision making-problem, where the final identification of the object is based upon the available set of information. We use the technique to calculate the score for the selected objects on the basis of k number of parameters (e₁, e₂, …, e_k) out of n number of objects (A₁, A₂, …, A_n). Our aim is to find out the eligible candidate for any post with the choice of parameters e₁ = “technical knowledge and skill assessment” and e₂ = “competency based interview” for the hiring manager of a trade center. Suppose that V = {A₁, A₂, A₃, A₄, A₅, A₆} is the set of shortlisted candidates to pass the assessment stage of the recruitment process. We consider an intuitionistic fuzzy soft graph G = (Φ, Ψ, M), where (Φ, M) is an intuitionistic fuzzy soft set over V which describes the membership and non-membership values of the applicants based upon the given parameters e₁ and e₂, (Ψ, M) is an intuitionistic fuzzy soft set over E = {A₁A₂, A₁A₄, A₁A₅, A₂A₃, A₂A₄, A₂A₆, A₃A₄, A₃A₅, A₄A₅, A₄A₆, A₅A₆} ⊆ V × V describes the membership and non-membership values of the relationship between two applicants corresponding to the given parameters e₁ and e₂ . An intuitionistic fuzzy soft graph G = {H (e₁) , H (e₂)} is given in Table 3.

The intuitionistic fuzzy graphs H (e₁) and H (e₂) of intuitionistic fuzzy soft graph G = {H (e₁) , H (e₂)} corresponding to the parameters “technical knowledge and skill assessment” and “competency based interview” are shown in Fig. 8. The adjacency matrix of intuitionistic fuzzy graph H (e₁) is $\begin{matrix} H (e_{1}) \\ = (\begin{matrix} (0, 0) & (0.2, 0.3) & (0, 0) & (0.3, 0.3) & (0.1, 0.5) & (0, 0) \\ (0.2, 0.3) & (0, 0) & (0.4, 0.2) & (0, 0) & (0, 0) & (0.1, 0.3) \\ (0, 0) & (0.4, 0.2) & (0, 0) & (0, 0) & (0.3, 0.4) & (0, 0) \\ (0.3, 0.3) & (0, 0) & (0, 0) & (0, 0) & (0.2, 0.5) & (0, 0) \\ (0.1, 0.5) & (0, 0) & (0.3, 0.4) & (0.2, 0.5) & (0, 0) & (0, 0) \\ (0, 0) & (0.1, 0.3) & (0, 0) & (0, 0) & (0, 0) & (0, 0) \end{matrix}) . \end{matrix}$

The final score values of intuitionistic fuzzy graph H (e₁) is computed with the score function $S_{ij} = \sqrt{μ_{j}^{2} + (1 - ν_{j})^{2}}$ and the choice values $C_{i}^{k} = \sum_{j} S_{ij}$ for all i, j are given in Table 4.

The adjacency matrix of intuitionistic fuzzy graph H (e₂) is $\begin{matrix} H (e_{2}) \\ = (\begin{matrix} (0, 0) & (0, 0) & (0, 0) & (0.3, 0.4) & (0.5, 0.3) & (0, 0) \\ (0, 0) & (0, 0) & (0, 0) & (0.4, 0.2) & (0, 0) & (0, 0) \\ (0, 0) & (0, 0) & (0, 0) & (0.3, 0.2) & (0.4, 0.2) & (0, 0) \\ (0.3, 0.4) & (0.4, 0.2) & (0.3, 0.2) & (0, 0) & (0.2, 0.3) & (0.4, 0.4) \\ (0.5, 0.3) & (0, 0) & (0.4, 0.2) & (0.2, 0.3) & (0, 0) & (0.5, 0.2) \\ (0, 0) & (0, 0) & (0, 0) & (0.4, 0.4) & (0.5, 0.2) & (0, 0) \end{matrix}) . \end{matrix}$

The final score values of intuitionistic fuzzy graph H (e₂) is computed with the score function $S_{ij} = \sqrt{μ_{j}^{2} + (1 - ν_{j})^{2}}$ and the choice values are given in Table 5.

The decision value is $S_{i} = ⋁_{i}^{6} (⋀_{k} C_{i}^{k}) = ⋁_{i = 1}^{6} {5.4 \land 4.9, 5.4 \land 5.9, 5.5 \land 5.8, 5.2 \land 4.8, 4.6 \land 4.2, 5.7 \land 5.5} = 5.5,$ from the choice value $C_{i}^{k}$ of intuitionistic fuzzy graphs H (e_k) for k = 1, 2. Clearly, the eligible candidate is A₃ or A₆.

We present an algorithm for most appropriate selection of an object of our decision-making problem.

Algorithm

Input the choice parameters e₁, e₂, . . . , e_p for the selection of objects.

Input adjacency matrices H(e₁), H(e₂), . . . , H(e_p) with respect to parameters.

Calculate score function of each adjacency matrix as $S_{i j} = \sqrt{M_{j}^{2} + (1 - r_{j})}$ , j = 1, 2, . . . , n, i = 1, 2, . . . , P.

Calculate choice values of objects with respect to choice parameters $C_{i}^{k} = \sum_{j} S_{i j}$ .

The decision is S_i if $S_{i} = ⋁_{i} (⋀_{k} C_{i}^{k})$ .

If i has more than one value then any one of S_i may be chosen.

6 Conclusions and future work

Graph theory has various applications in different areas including mathematics, science and technology, biochemistry (genomics), electrical engineering (communication networks and coding theory), computer science (algorithms and computation) and operations research (scheduling). Soft set theory plays a significant role as a mathematical tool for mathematical modeling, system analysis and computing of decision making problems with uncertainty. An intuitionistic fuzzy soft model is a generalization of the fuzzy soft model which gives more precision, flexibility, and compatibility to a system when compared with the fuzzy soft model. We have applied the concept of intuitionistic fuzzy soft sets to graphs in this paper. We have presented certain types of intuitionistic fuzzy soft graphs. We extend our research of fuzzification to (1) Interval-valued fuzzy soft graphs; (2) Bipolar fuzzy soft hypergraphs, (3) Fuzzy rough soft graphs, and (4) Application of intuitionistic fuzzy soft graphs in decision support systems.

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