Product of intuitionistic fuzzy graphs and degree

Abstract

An intuitionistic fuzzy graph is an extended structure of a fuzzy graph that give more precision, flexibility and compatibility to a system when compared with the system that designed using fuzzy graphs. In this paper, we define two new operations on intuitionistic fuzzy graphs namely normal product and tensor product. Also, we calculated the degree of the intuitionistic fuzzy graphs obtained by the operations Cartesian product, tensor product, normal product and composition of intuitionistic fuzzy graphs. These operations are highly used in computer science, geometry, algebra, operations research, etc.

Keywords

Intuitionistic fuzzy graphs cartesian product tensor product normal product

1 Introduction

Concept of graph theory have applications in many areas of computer science, including data mining, image segmentation, clustering, image capturing, networking, etc. An intuitionistic fuzzy set is a generalization of the notion of a fuzzy set. Intuitionistic fuzzy models gives more precision, flexibility and compatibility to the system as compared to the fuzzy models.

In 1983, Atanassov [3, 4] introduced the concept of intuitionistic fuzzy set as a generalization of fuzzy sets. Atanassov added a new components which determines the degree of non-membership in the definition of fuzzy set. The fuzzy sets give the degree of membership, while intuitionistic fuzzy sets give both the degree of membership and the degree of non-membership, which are more or less independent from each other, the only requirement is that the sum of these two degrees is not greater than one. Intuitionistic fuzzy sets have been applied in a wide variety of fields including computer science, engineering, mathematics, medicine, chemistry, economics, etc.

In 1975, Rosenfeld [19] discussed the concept of fuzzy graph whose basic idea was introduced by Kauffman [9] in 1973. The fuzzy relations between fuzzy sets were also considered by Rosenfeld and he developed the structure of fuzzy graphs, obtained analogs of several graphs theoretical concepts. Atanassov introduced the concept of intuitionistic fuzzy relation. Different types of intuitionistic fuzzy graphs and their applications can be found in several papers. Also, Sahoo and Pal [21] discussed the concept of intuitionistic fuzzy competition graph. In this paper, we define two new operations on intuitionistic fuzzy graphs namely Cartesian product and tensor product and study about the degree of the vertex in intuitionistic fuzzy graph, which are obtained from two intuitionistic fuzzy graphs G′ and G^′′ using the operations Cartesian product, composition, tensor product and normal product.

1.1 Review of literature

After Rosenfeld [19] the fuzzy graph theory increases with its various types of branches, such as – fuzzy tolerance graph [27], fuzzy threshold graph [26], bipolar fuzzy graphs [17 , 32], highly irregular interval valued fuzzy graphs [13, 14], isometry on interval-valued fuzzy graphs [16], balanced interval-valued fuzzy graphs [11, 15], fuzzy k-competition graphs and p-competition fuzzy graphs [30], fuzzy planar graphs [25, 33], bipolar fuzzy hypergraphs [28, 29], m-step fuzzy compitition graphs [24], etc. Fuzzy graph is used in telecommunication system [31]. A new concept of fuzzy colouring of fuzzy graph is given [34]. Ghorai and Pal introduced operations on m-polar fuzzy graphs [6], m-polar fuzzy planar graphs [7] and Ceratin Types of Product Bipolar Fuzzy Graphs [5]. Nayeem and Pal introduced shortest path problem on a network with imprecise edge weight [10].

Akram and Al-Shehrie [1] defined intuitionistic fuzzy cycles and intuitionistic fuzzy trees and intuitionistic fuzzy planar graphs [2]. Balanced intuitionistic fuzzy graphs is discuss by Karunambigai et al. [8]. Also, Parvathi and Karunambigai [12] defined intuitionistic fuzzy graphs. Sahoo and Pal [21] discussed the concept of intuitionistic fuzzy competition graph. They also discuss intuitionistic fuzzy tolerance graph [23], different types of products on intuitionistic fuzzy graphs [20] and intuitionistic fuzzy labeling graphs [22].

2 Preliminaries

A graph is an ordered pair G = (V, E), where V is the set of all vertices of G, which is non empty and E is the set of all edges of G. Two vertices x, y in a graph G are said to be adjacent in G if (x, y) is an edge of G. A simple graph is a graph without loops and multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is connected by an edge. The complete graph on n vertices has $\frac{n (n - 1)}{2}$ edges.

An isomorphism of graphs G₁ and G₂ is a bijection between the vertex sets of G₁ and G₂ such that any two vertices v₁ and v₂ of G₁ are adjacent in G₁ if and only if f (v₁) and f (v₂) are adjacent in G₂. An isomorphic graphs are denoted by G₁ ≅ G₂.

Let G₁ = (V₁, E₁) and G₂ = (V₂, E₂) be two simple graphs. Then the direct product of G₁ and G₂ is a graph G₁ ⊓ G₂ = (V, E) with V = V₁ × V₂ and E = {((u₁, v₁) , (u₂, v₂)) ∣ (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂}. The union of graphs G₁ and G₂ is defined as G₁ ∪ G₂ = (V₁ ∪ V₂, E₁ ∪ E₂) and join is the simple graph G₁ + G₂ = (V₁ ∪ V₂, E₁ ∪ E₂ ∪ E′), where E′ is the set of all edges joining the vertices of V₁ and V₂ and assume thatV₁ ∩ V₂ = φ .

2.1 Intuitionistic fuzzy graphs

An intuitionistic fuzzy set A on the set X is characterized by a mapping m : X → [0, 1], which is called as a membership function and n : X → [0, 1], which is called as a non-membership function. An intuitionistic fuzzy set is denoted by A = (X, m_A, n_A). The membership function of the intersection of two intuitionistic fuzzy sets A = (X, m_A, n_A) and B = (X, m_B, n_B) is defined as m_A∩B = min {m_A, m_B} and the non-membership function n_A∩B = max {n_A, n_B}. We write A = (X, m_A, n_A) ⊆ B = (X, m_B, n_B) (intuitionistic fuzzy subset) if m_A (x) ≤ m_B (x) and n_A (x) ≥ n_A (x) for all x ∈ X.

Here, an intuitionistic fuzzy graph is defined below:

Definition 1. [12] An intuitionistic fuzzy graph is of the form G = (V, E, σ, μ) where σ = (σ₁, σ₂), μ = (μ₁, μ₂) and

V = {v₀, v₁, …, v_n} such that σ₁ : V → [0, 1] and σ₂ : V → [0, 1], denote the degree of membership and non-membership of the vertex v_i ∈ V respectively and 0 ≤ σ₁ (v_i) + σ₂ (v_i) ≤1 for every v_i ∈ V (i = 1, 2, …, n).

μ₁ : V × V → [0, 1] and μ₂ : V × V → [0, 1], where μ₁ (v_i, v_j) and μ₂ (v_i, v_j) denote the degree of membership and non-membership value of the edge (v_i, v_j) respectively such that μ₁ (v_i, v_j) ≤ min {σ₁ (v_i) , σ₁ (v_j)} and μ₂ (v_i, v_j) ≤ max {σ₂ (v_i) , σ₂ (v_j)}, 0 ≤ μ₁ (v_i, v_j) + μ₂ (v_i, v_j) ≤1 for every (v_i, v_j).

Now, we give an example of intuitionistic fuzzy graph:

Example 1. Let G = (V, E, σ, μ) be an intuitionistic fuzzy graph, where σ (v) = (σ₁ (v) , σ₂ (v)), μ (u, v) = (μ₁ (u, v) , μ₂ (u, v)). Let the vertex set be V = {v₁, v₂, v₃, v₄} and σ (v₁) = (0.3, 0.6), σ (v₂) = (0.8, 0.2), σ (v₃) = (0.2, 0.8), σ (v₄) = (0.5, 0.4); μ (v₁, v₂) = (0.25, 0.45), μ (v₂, v₃) = (0.18, 0.75), μ (v₃, v₄) = (0.15, 0.52), μ (v₄, v₁) = (0.3, 0.25), μ (v₁, v₃) = (0.2, 0.8), μ (v₂, v₄) = (0.4, 0.35). The corresponding intuitionistic fuzzy graph is shown in Fig. 1.

Definition 2. An intuitionistic fuzzy graph G = (V, E, σ, μ) is said to be complete if μ₁ (v_i, v_j) = σ₁ (v_i) ∧ σ₁ (v_j) and μ₂ (v_i, v_j) = σ₂ (v_i) ∨ σ₂ (v_j) for all v_i, v_j ∈ V .

Definition 3. Let G = (V, E, σ, μ) be an intuitionistic fuzzy graph. The degree of a vertex u in G is denoted by d_G (u) = (d_1G (u) , d_2G (u)) and defined by $d_{1 G} (u) = \sum_{u \neq v} μ_{1} (u, v) = \sum_{(u, v) \in E} μ_{1} (u, v)$ and $d_{2 G} (u) = \sum_{u \neq v} μ_{2} (u, v) = \sum_{(u, v) \in E} μ_{2} (u, v)$ .

3 Degree of vertex of different types of product on intuitionistic fuzzy graphs

We define degree of vertex of different types of product on intuitionistic fuzzy graphs. First we defined the degree of vertex in Cartesian product of two intuitionistic fuzzy graphs.

3.1 Degree of a vertex in Cartesian product

Definition 4. The Cartesian product of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) is defined as an intuitionistic fuzzy graph G = G′ × G^′′ = (V, E, σ′ × σ^′′, μ′ × μ^′′) where V = V′ × V^′′ and E = {((u₁, v₁) , (u₂, v₂)) ∣ u₁ = u₂, (v₁, v₂) ∈ E^′′ or v₁ = v₂, (u₁, u₂) ∈ E′} with $\begin{matrix} (σ_{1}^{'} \times σ_{1}^{″}) (u_{1}, v_{1}) = σ_{1}^{'} (u_{1}) \land σ_{1}^{″} (v_{1}) \\ (σ_{2}^{'} \times σ_{2}^{″}) (u_{1}, v_{1}) = σ_{2}^{'} (u_{1}) \lor σ_{2}^{″} (v_{1}) \end{matrix}$ for all (u₁, v₁) ∈ V′ × V^′′ and $\begin{matrix} (μ_{1}^{'} \times μ_{1}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = {\begin{matrix} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}), & if u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″} \\ μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}), & if v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'} \end{matrix} \\ (μ_{2}^{'} \times μ_{2}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = {\begin{matrix} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}), & if u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″} \\ μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}), & if v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'} \end{matrix} \end{matrix}$

Definition 5. Let G = G′ × G^′′ = (V, E, σ′ × σ^′′, μ′ × μ^′′) be the Cartesian product of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′). Then the degree of the vertex (u₁, v₁) in V is denoted by d_{G′×G^′′} (u₁, v₁) = (d_{1G′×G^′′} (u₁, v₁) , d_{2G′×G^′′} (u₁, v₁)) and defined by $\begin{matrix} d_{1 G^{'} \times G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} \times μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}) \\ d_{2 G^{'} \times G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} \times μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}) \end{matrix}$

Theorem 1. Let G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) be two intuitionistic fuzzy graphs. If $σ_{1}^{'} \geq μ_{1}^{″}$ , $σ_{2}^{'} \leq μ_{2}^{″}$ and $σ_{1}^{″} \geq μ_{1}^{'}$ , $σ_{2}^{″} \leq μ_{2}^{'}$ , then d_{G′×G^′′} (u₁, v₁) = d_G′ (u₁) + d_{G
^′′} (v₁).

Proof. From the definition of degree of the vertex in Cartesian product, we have $\begin{matrix} d_{1 G^{'} \times G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} \times μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}) \\ = \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{1}^{″} (v_{1}, v_{2}) + \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ [since σ_{1}^{'} \geq μ_{1}^{″} and σ_{1}^{″} \geq μ_{1}^{'}] \\ = \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) + \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{1}^{″} (v_{1}, v_{2}) \\ = d_{1 G^{'}} (u_{1}) + d_{1 G^{″}} (v_{1}) . \end{matrix}$ and $\begin{matrix} d_{2 G^{'} \times G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} \times μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}) \\ = \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{2}^{″} (v_{1}, v_{2}) + \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ [since σ_{2}^{'} \leq μ_{2}^{″} and σ_{2}^{″} \leq μ_{2}^{'}] \\ = \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) + \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{2}^{″} (v_{1}, v_{2}) \\ = d_{2 G^{'}} (u_{1}) + d_{2 G^{″}} (v_{1}) . \end{matrix}$

Hence, d_{G′×G^′′} (u₁, v₁) = d_G′ (u₁) + d_{G
^′′} (v₁).

Example 2. Here, $σ_{1}^{'} \geq μ_{1}^{″}$ , $σ_{2}^{'} \leq μ_{2}^{″}$ and $σ_{1}^{″} \geq μ_{1}^{'}$ , $σ_{2}^{″} \leq μ_{2}^{'}$ . Then by Theorem 1, we have $\begin{matrix} d_{1 G^{'} \times G^{″}} (u_{1}, v_{1}) & = & d_{1 G^{'}} (u_{1}) + d_{1 G^{″}} (v_{1}) \\ = & 0.2 + 0.1 = 0.3 . \\ d_{2 G^{'} \times G^{″}} (u_{1}, v_{1}) & = & d_{2 G^{'}} (u_{1}) + d_{2 G^{″}} (v_{1}) \\ = & 0.4 + 0.4 = 0.8 . \end{matrix}$

So, d_{G′×G^′′} (u₁, v₁) = (0.3, 0.8). Similarly, we can find the degree of all vertices in the Cartesian product G′ × G^′′. This is verified in the Fig. 2.

3.2 Degree of vertex in composition

Next, we defined the degree of a vertex in composition of two intuitionistic fuzzy graphs.

Definition 6. The compostion of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) is defined as an intuitionistic fuzzy graph G = G′ [G^′′] = (V, E, σ′ ∘ σ^′′, μ′ ∘ μ^′′) where V = V′ × V^′′ and E = {((u₁, v₁) , (u₂, v₂)) ∣ u₁ = u₂, (v₁, v₂) ∈ E^′′ or v₁ = v₂, (u₁, u₂) ∈ E′} ∪ {((u₁, v₁) , (u₂, v₂)) ∣ v₁ ≠ v₂, (u₁, u₂) ∈ E′} with $(σ_{1}^{'} \circ σ_{1}^{″}) (u_{1}, v_{1}) = σ_{1}^{'} (u_{1}) \land σ_{1}^{″} (v_{1}) (σ_{2}^{'} \circ σ_{2}^{″}) (u_{1}, v_{1}) = σ_{2}^{'} (u_{1}) \lor σ_{2}^{″} (v_{1})$ for all (u₁, v₁) ∈ V′ × V^′′ and

$\begin{matrix} (μ_{1}^{'} \circ μ_{1}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = {= 2 pt \begin{matrix} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}), & if u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″} \\ μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}), & if v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'} \\ μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{'} (v_{1}) \land σ_{1}^{″} (v_{2}), & if v_{1} \neq v_{2} and \\ (u_{1}, u_{2}) \in E^{'} \end{matrix} \\ (μ_{2}^{'} \circ μ_{2}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = {= 2 pt \begin{matrix} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}), & if u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″} \\ μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}), & if v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'} \\ μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{'} (v_{1}) \lor σ_{1}^{″} (v_{2}), & if v_{1} \neq v_{2} and \\ (u_{1}, u_{2}) \in E^{'} \end{matrix} \end{matrix}$

Definition 7. Let G = G′ [G^′′] = (V, E, σ′ ∘ σ^′′, μ′ ∘ μ^′′) be the composition of two intuitionistic fuzzy graph G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′). Then the degree of the vertex (u₁, v₁) in V is denoted by d_{G′[G^′′]} (u₁, v₁) = (d_{1G′[G^′′]} (u₁, v₁) , d_{2G′[G^′′]} (u₁, v₁)) and defined by $\begin{matrix} d_{1 G^{'} [G^{″}]} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} \circ μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{'} (v_{1}) \land σ_{1}^{″} (v_{2}) . \\ d_{2 G^{'} [G^{″}]} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} \circ μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{'} (v_{1}) \lor σ_{1}^{″} (v_{2}) . \end{matrix}$

Theorem 2. Let G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) be two intuitionistic fuzzy graphs. If $σ_{1}^{'} \geq μ_{1}^{″}$ , $σ_{2}^{'} \leq μ_{2}^{″}$ and $σ_{1}^{″} \geq μ_{1}^{'}$ , $σ_{2}^{″} \leq μ_{2}^{'}$ , then d_{G′[G^′′]} (u₁, v₁) = |V^′′|d_G′ (u₁) + d_{G
^′′} (v₁).

Proof. From the definition of vertex degree in composition of two intuitionistic fuzzy graphs, we have $\begin{matrix} d_{1 G^{'} [G^{″}]} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} \circ μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{'} (v_{1}) \land σ_{1}^{″} (v_{2}) \\ = \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{1}^{″} (v_{1}, v_{2}) + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ = \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) + \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{1}^{″} (v_{1}, v_{2}) \\ = | V^{″} | d_{1 G^{'}} (u_{1}) + d_{1 G^{″}} (v_{1}) . \end{matrix}$ and $\begin{matrix} d_{2 G^{'} [G^{″}]} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} \circ μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{1}^{″} (v_{1}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{'} (v_{1}) \lor σ_{2}^{″} (v_{2}) \\ = \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{2}^{″} (v_{1}, v_{2}) + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ = \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ + \sum_{v_{1} \neq v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) + \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{2}^{″} (v_{1}, v_{2}) \\ = | V^{″} | d_{2 G^{'}} (u_{1}) + d_{2 G^{″}} (v_{1}) . \end{matrix}$

Hence, d_{G′[G^′′]} (u₁, v₁) = |V^′′|d_G′ (u₁) + d_{G
^′′} (v₁).

Example 3. Consider the intuitionistic fuzzy graphs G′, G^′′ and G′ [G^′′] as follows.

Here, $σ_{1}^{'} \geq μ_{1}^{″}$ , $σ_{2}^{'} \leq μ_{2}^{″}$ and $σ_{1}^{″} \geq μ_{1}^{'}$ , $σ_{2}^{″} \leq μ_{2}^{'}$ , then by Theorem 2, we have $\begin{matrix} d_{1 G^{'} [G^{″}]} (u_{1}, v_{1}) & = & | V^{″} | d_{1 G^{'}} (u_{1}) + d_{1 G^{″}} (v_{1}) \\ = & 2 \times 0.2 + 0.1 = 0.5 . \\ d_{2 G^{'} [G^{″}]} (u_{1}, v_{1}) & = & | V^{″} | d_{2 G^{'}} (u_{1}) + d_{2 G^{″}} (v_{1}) \\ = & 2 \times 0.4 + 0.4 = 1.2 . \end{matrix}$

So, d_{G′[G^′′]} (u₁, v₁) = (0.5, 1.2). Similarly, we can find the degree of all vertices in the composition G′ [G^′′]. This is verified in the Fig. 3.

3.3 Degree of a vertex in tensor product

Next, we defined degree of a vertex in tensor product of two intuitionistic fuzzy graphs.

Definition 8. The tensor product of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) is defined as an intuitionistic fuzzy graph G = G′ ⊗ G^′′ = (V, E, σ′ ⊗ σ^′′, μ′ ⊗ μ^′′) where V = V′ × V^′′ and E = {((u₁, v₁) , (u₂, v₂)) ∣ (u₁, u₂) ∈ E′, (v₁, v₂) ∈ E^′′} with $\begin{matrix} (σ_{1}^{'} \otimes σ_{1}^{″}) (u_{1}, v_{1}) = σ_{1}^{'} (u_{1}) \land σ_{1}^{″} (v_{1}) \\ (σ_{2}^{'} \otimes σ_{2}^{″}) (u_{1}, v_{1}) = σ_{2}^{'} (u_{1}) \lor σ_{2}^{″} (v_{1}) \end{matrix}$ for all (u₁, v₁) ∈ V′ × V^′′ and $\begin{matrix} (μ_{1}^{'} \otimes μ_{1}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = μ_{1}^{'} (u_{1}, u_{2}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ (μ_{2}^{'} \circ μ_{2}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = μ_{2}^{'} (u_{1}, u_{2}) \lor μ_{2}^{″} (v_{1}, v_{2}) \end{matrix}$ for all (u₁, u₂) ∈ E′, (v₁, v₂) ∈ E^′′ .

Definition 9. Let G = G′ ⊗ G^′′ = (V, E, σ′ ⊗ σ^′′, μ′ ⊗ μ^′′) be the tensor product of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′). Then the degree of the vertex (u₁, v₁) in V is denoted by d_{G′⊗G^′′} (u₁, v₁) = (d_{1G′⊗G^′′} (u₁, v₁) , d_{2G′⊗G^′′} (u₁, v₁)) and definedby $\begin{matrix} d_{1 G^{'} \otimes G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} \otimes μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land μ_{1}^{″} (v_{1}, v_{2}), \\ d_{2 G^{'} \otimes G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} \otimes μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor μ_{2}^{″} (v_{1}, v_{2}) . \end{matrix}$

Theorem 3. Let G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) be two intuitionistic fuzzy graphs. If $μ_{1}^{″} \geq μ_{1}^{'}$ , $μ_{2}^{″} \leq μ_{2}^{'}$ , then d_{G′⊗G^′′} (u₁, v₁) = d_G′ (u₁) and if $μ_{1}^{'} \geq μ_{1}^{″}$ , $μ_{2}^{'} \leq μ_{2}^{″}$ , then d_{G′⊗G^′′} (u₁, v₁) = d_{G
^′′} (v₁).

Proof. Let, $μ_{1}^{″} \geq μ_{1}^{'}$ , $μ_{2}^{″} \leq μ_{2}^{'}$ , then $\begin{matrix} d_{1 G^{'} \otimes G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} \otimes μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ = \sum μ_{1}^{'} (u_{1}, u_{2}) \\ = d_{1 G^{'}} (u_{1}) \\ and d_{2 G^{'} \otimes G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} \otimes μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ = \sum μ_{2}^{'} (u_{1}, u_{2}) \\ = d_{2 G^{'}} (u_{1}) \end{matrix}$

Hence, d_{G′⊗G^′′} (u₁, v₁) = d_G′ (u₁).

Similarly, if $μ_{1}^{'} \geq μ_{1}^{″}$ , $μ_{2}^{'} \leq μ_{2}^{″}$ , then d_{G′⊗G^′′} (u₁, v₁) = d_{G
^′′} (v₁).

Example 4. In this example, we obtain the degree of the vertex of G′ ⊗ G^′′ by the Theorem 3.

Consider the intuitionistic fuzzy graphs G′, G^′′ and G′ [G^′′] as in Fig. 4. Here, $μ_{1}^{″} \geq μ_{1}^{'}$ , $μ_{2}^{″} \leq μ_{2}^{'}$ , then d_{1G′⊗G^′′} (u₁, v₁) = d_1G′ (u₁) =0.2 and d_{2G′⊗G^′′} (u₁, v₁) = d_2G′ (u₁) =0.4.

Hence d_{G′⊗G^′′} (u₁, v₁) = (0.2, 0.4).

3.4 Degree of a vertex in normal product

Finally, we define degree of a vertex in normal product of two intuitionistic fuzzy graphs.

Definition 10. The normal product of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) is defined as an intuitionistic fuzzy graph G = G′ • G^′′ = (V, E, σ′ • σ^′′, μ′ • μ^′′) where V = V′ × V^′′ and E = {((u₁, v₁) , (u₂, v₂)) ∣ u₁ = u₂, (v₁, v₂) ∈ E^′′ or v₁ = v₂, (u₁, u₂) ∈ E′} ∪ {((u₁, v₁) , (u₂, v₂)) ∣ (u₁, u₂) ∈ E′, (v₁, v₂) ∈ E^′′} with $\begin{matrix} (σ_{1}^{'} • σ_{1}^{″}) (u_{1}, v_{1}) = σ_{1}^{'} (u_{1}) \land σ_{1}^{″} (v_{1}) \\ (σ_{2}^{'} • σ_{2}^{″}) (u_{1}, v_{1}) = σ_{2}^{'} (u_{1}) \lor σ_{2}^{″} (v_{1}) \end{matrix}$ for all (u₁, v₁) ∈ V′ × V^′′ and $\begin{matrix} (μ_{1}^{'} • μ_{1}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = {\begin{matrix} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}), & if u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″} \\ μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}), & if v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'} \\ μ_{1}^{'} (u_{1}, u_{2}) \land μ_{1}^{″} (v_{1}, v_{2}), & if (u_{1}, u_{2}) \in E^{'} and \\ (v_{1}, v_{2}) \in E^{″} \end{matrix} \\ (μ_{2}^{'} • μ_{2}^{″}) ((u_{1}, v_{1}), (u_{2}, v_{2})) \\ = {\begin{matrix} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}), & if u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″} \\ μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}), & if v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'} \\ μ_{2}^{'} (u_{1}, u_{2}) \lor μ_{2}^{″} (v_{1}, v_{2}), & if (u_{1}, u_{2}) \in E^{'} and \\ (v_{1}, v_{2}) \in E^{″} \end{matrix} \end{matrix}$

Definition 11. Let G = G′ • G^′′ = (V, E, σ′ • σ^′′, μ′ • μ^′′) be the normal product of two intuitionistic fuzzy graphs G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′). Then the degree of the vertex (u₁, v₁) in V is denoted by d_{G′•G^′′} (u₁, v₁) = (d_{1G′•G^′′} (u₁, v₁) , d_{2G′•G^′′} (u₁, v₁)) and defined by $\begin{matrix} d_{1 G^{'} • G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} • μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}, (v_{1}, v_{2}) \in E^{″}} μ_{1}^{'} (u_{1}, u_{2}) \land μ_{1}^{″} (v_{1}, v_{2}) \end{matrix}$ $\begin{matrix} d_{2 G^{'} • G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} • μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}, (v_{1}, v_{2}) \in E^{″}} μ_{2}^{'} (u_{1}, u_{2}) \lor μ_{2}^{″} (v_{1}, v_{2}) . \end{matrix}$

Theorem 4. Let G′ = (V′, E′, σ′, μ′) and G^′′ = (V^′′, E^′′, σ^′′, μ^′′) be two intuitionistic fuzzy graphs. If $σ_{1}^{'} \geq μ_{1}^{″}$ , $σ_{2}^{'} \leq μ_{2}^{″}$ , $σ_{1}^{″} \geq μ_{1}^{'}$ , $σ_{2}^{″} \leq μ_{2}^{'}$ , $μ_{1}^{'} \leq μ_{1}^{″}$ and $μ_{2}^{'} \geq μ_{2}^{″}$ , then d_{G′•G^′′} (u₁, v₁) = |V^′′|d_G′ (u₁) + d_{G
^′′} (v₁).

Proof. Let $σ_{1}^{'} \geq μ_{1}^{″}$ , $σ_{2}^{'} \leq μ_{2}^{″}$ , $σ_{1}^{″} \geq μ_{1}^{'}$ , $σ_{2}^{″} \leq μ_{2}^{'}$ , $μ_{1}^{'} \leq μ_{1}^{″}$ and $μ_{2}^{'} \geq μ_{2}^{″}$ , then we have $\begin{matrix} d_{1 G^{'} • G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{1}^{'} • μ_{1}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{1}^{'} (u_{1}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \land σ_{1}^{″} (v_{1}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}, (v_{1}, v_{2}) \in E^{″}} μ_{1}^{'} (u_{1}, u_{2}) \land μ_{1}^{″} (v_{1}, v_{2}) \\ = \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{1}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ = \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{1}^{'} (u_{1}, u_{2}) \\ + \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{1}^{″} (v_{1}, v_{2}) \\ = | V^{″} | d_{1 G^{'}} (u_{1}) + d_{1 G^{″}} (v_{1}) \end{matrix}$ and $\begin{matrix} d_{2 G^{'} • G^{″}} (u_{1}, v_{1}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (μ_{2}^{'} • μ_{2}^{″}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = u_{2}, (v_{1}, v_{2}) \in E^{″}} σ_{2}^{'} (u_{1}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \lor σ_{2}^{″} (v_{1}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}, (v_{1}, v_{2}) \in E^{″}} μ_{2}^{'} (u_{1}, u_{2}) \lor μ_{2}^{″} (v_{1}, v_{2}) \\ = \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{2}^{″} (v_{1}, v_{2}) \\ + \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ = \sum_{v_{1} = v_{2}, (u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ + \sum_{(u_{1}, u_{2}) \in E^{'}} μ_{2}^{'} (u_{1}, u_{2}) \\ + \sum_{(v_{1}, v_{2}) \in E^{″}} μ_{2}^{″} (v_{1}, v_{2}) \\ = | V^{″} | d_{2 G^{'}} (u_{1}) + d_{2 G^{″}} (v_{1}) . \end{matrix}$

Hence, d_{G′•G^′′} (u₁, v₁) = |V^′′|d_G′ (u₁) + d_{G
^′′} (v₁).

4 Conclusion

In this paper, we determined the degree of the vertices of the graphs G′ × G^′′, G′ [G^′′], G′ ⊗ G^′′ and G′ • G^′′ in terms of the degree of the vertices of an intuitionistic fuzzy graphs G′ and G^′′ under some conditions and illustrated them through examples. The vertices and their degree of any graph are very important parameters. This study is very useful to analyse various properties of Cartesian product, composition, tensor product and normal product of two intuitionistic fuzzy graphs.

Footnotes

Acknowledgments

Financial support of first author offered by Council of Scientific and Industrial Research, New Delhi, India (Sanction no. 09/599(0057)/2014-EMR-I) is thankfully acknowledged.

The authors are thankful to the reviewers for their valuable comments and suggestions to improve the presentation of the paper.

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