Product of interval-valued fuzzy graphs and degree

Abstract

Theoretical concepts of graphs are highly utilized by computer science applications. Especially in research areas of computer science such as data mining, image segmentation, clustering, image capturing and networking. The interval-valued fuzzy graphs are more flexible and compatible than fuzzy graphs due to the fact that they have many applications in networks. In this paper, at first we define three new operations on interval-valued fuzzy graphs namely strong product, tensor product and lexicographic product. Likewise, we study about the degree of a vertex in interval-valued fuzzy graphs which are obtained from two given interval-valued fuzzy graphs using the operations Cartesian product, composition, tensor and strong product of two interval-valued fuzzy graphs. These operations are highly utilized by computer science, geometry, algebra, number theory and operation research. In addition to the existing operations these properties will also be helpful to study large interval-valued fuzzy graph as a combination of small, interval-valued fuzzy graphs and to derive its properties from those of the smaller ones.

Keywords

Cartesian product composition tensor product lexicographic product

1. Introduction

Presently, science and technology is featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models is based on an extension of the ordinary set theory, namely, fuzzy sets. Graph theory has numerous applications to problems in computer science, electrical engineering, system analysis, operations research, economics, networking routing, transportation, etc. In many cases, some aspects of a graph-theoretic problem may be uncertain. For example, the vehicle travel time or vehicle capacity on a road network may not be known exactly. In such cases, it is natural to deal with the uncertainty using the methods of fuzzy sets and fuzzy logic. But, using hypergraphs as the models of various systems (social, economic systems, communication networks and others) leads to difficulties. In 1975, Zadeh [31] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets [32] in which the values of the membership degrees are interval of numbers instead of the numbers. Rosenfeld [17] introduced the notation of fuzzy graphs in 1975. The operation of union, join, Cartesian product and composition on two fuzzy graphs were defined by Mordeson and Peng [9 –14]. Akram and Dudec [1] defined interval-valued fuzzy graphs and give some operation on it. Borzooei et al. [2 –8] studied domination, degree of vertices, and new concepts of vague graphs and bipolar fuzz graphs. Talebi and Rashmanlou [29] studied properties of isomorphism and complement on interval-valued fuzzy graphs. Likewise, they defined isomorphism on vague graphs [30]. Samanta and Pal introduced different types of fuzzy graphs [18 –28]. Pramanick and Pal [16] defined interval-valued planar graph and presented several properties. Nayeem and Pal [15] has designed an algorithm to find the shortest paths between two given vertices on fuzzy graphs.

In this paper, we study about the degree of a vertex in interval-valued fuzzy graphs which are obtained from two given interval-valued fuzzy graphs using the operations Cartesian product, composition, tensor and strong product of two interval-valued fuzzy graphs.

2. Preliminaries

Definition 1. A fuzzy subset of a set V is a mapping σ from V to [0, 1]. A fuzzy graph G is a pair of functions G = (σ, μ) where σ is a fuzzy subset of a non-empty set V and μ is a symmetric fuzzy relation on σ, i.e. μ (uv) ≤ σ (u) ∧ σ (v), where σ (u) ∧ σ (v) = min {σ (u), σ (v)}. The underlying crisp graph of G = (σ, μ) is denoted by G^∗ = (V, E) where E ⊆ V × V.

Definition 2. Let G = (σ, μ) be a fuzzy graph, the degree of a vertex u in G is defined by $d_{G} (u) = \sum_{u \neq v} μ (uv) = \sum_{uv \in E} μ (uv),$ where uv represents an edge of G.

Definition 3. The order of a fuzzy graph G is defined by ∘ (G) = ∑_u∈Vσ (u).

Definition 4. [3] By an interval-valued fuzzy graph of a graph G^∗ = (V, E) we mean a pair G = (A, B), where A = [μ_{A
^-}, μ_{A
⁺}] is an interval-valued fuzzy set on V such that μ_{A
^-} (x) ≤ μ_{A
⁺} (x) for all x ∈ V and B = [μ_{B
^-}, μ_{B
⁺}] is an interval-valued fuzzy relation on E, such that: μ_{B
^-} (xy) ≤ min (μ_{A
^-} (x), μ_{A
^-} (y)), μ_B⁺(xy) ≤ min (μ_{A
⁺} (x), μ_{A
⁺} (y)) for all xy ∈ E. We call A the interval-valued fuzzy vertex set of V, B the interval-valued fuzzy edge set of E, respectively. Note that B is a symmetric interval-valued fuzzy relation on A . Thus, G = (A, B) is an interval-valued fuzzy graph of G^* = (V, E) if

$μ_{B^{-} (xy)} \leq \min (μ_{A^{-}} (x), μ_{A^{-}} (y)),$ and

$μ_{B^{+} (xy)} \leq \min (μ_{A^{+}} (x), μ_{A^{+}} (y)) for all xy \in E .$

Example 1. Consider a graph G^∗ = (V, E) such that V = {x, y, z}, E = {xy, yz, zx}. Let A be an interval-valued fuzzy set of V and let B be an interval-valued fuzzy set of E ⊆ V × V defined by $A = 〈 (\frac{x}{0.4}, \frac{y}{0.5}, \frac{z}{0.6}), (\frac{x}{0.6}, \frac{y}{0.7}, \frac{z}{0.7}) 〉$ , $B = 〈 (\frac{xy}{0.3}, \frac{yz}{0.4}, \frac{zx}{0.3}), (\frac{xy}{0.5}, \frac{yz}{0.6}, \frac{zx}{0.6}) 〉$

By routine computations, it is easy to see that G = (A, B) is an interval-valued fuzzy graph of G^∗ (see Fig. 1).

Fig. 1.

An example of interval-valued fuzzy graph.

Definition 5. Given an interval-valued fuzzy graph G = (A, B), with the underlying set V, the order of G is defined and denoted as O (G) = (∑_x∈Vμ_{A
^-} (x), ∑_x∈Vμ_{A
⁺} (x)). The size of an interval-valued fuzzy graph G is $S (G) = (\sum_{\binom{x \neq y}{x, y \in V}} μ_{B^{-}} (xy), \sum_{\binom{x \neq y}{x, y \in V}} μ_{B^{+}} (xy)) .$

Definition 6. Let G = (A, B) be an interval-valued fuzzy graph on G^∗. The open degree of a vertex u is defined as deg (u) = (d^- (u), d⁺ (u)), where $d^{-} (u) = \sum_{\binom{u \neq v}{v \in V}} μ_{B^{-}} (uv)$ and $d^{+} (u) = \sum_{\binom{u \neq v}{v \in V}} μ_{B^{+}} (uv) .$ If all the vertices have the same open neighborhood degree n, then G is called an n-regular interval-valued fuzzy graph.

Definition 7. [12] The Cartesian product of two interval-valued fuzzy graphs G₁ = (σ₁, μ₁) and G₂ = (σ₂, μ₂) is defined as an interval-valued fuzzy graph G = G₁ × G₂ : (σ₁ × σ₂, μ₁ × μ₂) on G^* = (V, E) where V = V₁ × V₂ and E = {(u₁, u₂) (v₁, v₂) ∣ u₁ = v₁, u₂v₂ ∈ E₂ or u₂ = v₂, u₁v₁ ∈ E₁} with (σ₁ × σ₂) (u₁, u₂) = σ₁ (u₁) ∧ σ₂ (u₂) for all (u₁, u₂) ∈ V₁ × V₂, $and (μ_{1} \times μ_{2}) ((u_{1}, u_{2}) (v_{1}, v_{2})) = {\begin{matrix} σ_{1} (u_{1}) \land μ_{2} (u_{2} v_{2}) & if u_{1} = v_{1} and u_{2} v_{2} \in E_{2}, \\ σ_{2} (u_{2}) \land μ_{1} (u_{1} v_{1}) & if u_{2} = v_{2} and u_{1} v_{1} \in E_{1} . \end{matrix}$

Definition 8. [12] The composition of two fuzzy graphs G₁ = (σ₁, μ₁) and G₂ = (σ₂, μ₂) is defined as a fuzzy graph G = G [G₂] : (σ₁ ∘ σ₂, μ₁ ∘ μ₂) on G^∗ = (V, E), where V = V₁ × V₂ and E = {(u₁, u₂) (v₁, v₂) |u₁ = v₁, u₂v₂ ∈ E₂ or u₂ = v₂, u₁v₁ ∈ E₁} ∪ {(u₁, u₂) (v₁, v₂) |u₁v₁ ∈ E₁, u₂ ≠ v₂} with (σ₁ ∘ σ₂) (u₁, u₂) = σ₁ (u₁) ∧ σ₂ (u₂) for all (u₁, u₂) ∈ V₁ × V₂, $and (μ_{1} \circ μ_{2}) ((u_{1}, u_{2}) (v_{1}, v_{2})) = {\begin{matrix} σ_{1} (u_{1}) \land μ_{2} (u_{2} v_{2}) {ifu}_{1} = v_{1} and u_{2} v_{2} \in E_{2}, \\ σ_{2} (u_{2}) \land μ_{1} (u_{1} v_{1}) if u_{2} = v_{2} and u_{1} v_{1} \in E_{1}, \\ σ_{2} (u_{2}) \land σ_{2} (v_{2}) \land μ_{1} (u_{1} v_{1}) if u_{2} \neq v_{2} and \\ u_{1} v_{1} \in E_{1} . \end{matrix}$

Definition 9. The strong product of two fuzzy graphs (σ_i, μ_i) on G_i = (V_i, E_i), i = 1, 2 is defined as a fuzzy graph (σ₁ ⊠ σ₂, μ₁ ⊠ μ₂) on G = (V, E) where V = V₁ × V₂ and E = {((u, u₂) (u, v₂)) ∣ u ∈ V₁, (u₂v₂) ∈ E₂} ⋃ {((u₁, w) (v₁, w)) ∣ (u₁, v₁) ∈ E₁, w ∈ V₂} ⋃ {(u₁, u₂) (v₁, v₂) ∣ (u₁, v₁) ∈ E₁, (u₂, v₂) ∈ E₂}. Fuzzy sets σ₁ ⊠ σ₂ and μ₁ ⊠ μ₂ are defined as (σ₁ ⊠ σ₂) (u₁, u₂) = (σ₁ (u₁) ∧ σ₂ (u₂)), (μ₁ ⊠ μ₂) ((u, u₂) (u, v₂)) = {σ₁ (u) ∧ μ₂ (u₂v₂)} u ∈ V₁ and u₂v₂ ∈ E₂, (μ₁ ⊠ μ₂) ((u₁, w) (v₁, w)) = {μ₁ (u₁v₁)∧ σ₂ (w)} w ∈ V₂ andu₁v₁ ∈ E₁, (μ₁ ⊠ μ₂) ((u₁, u₂) (v₁, v₂)) = {μ₁ (u₁v₁) ∧ μ₂ (u₂v₂)} u₁u₂ ∈ E₁ and v₁v₂ ∈ E₂.

Definition 10. The tensor product of two fuzzy graphs (σ_i, μ_i) on G_i = (V_i, E_i), i = 1, 2 is defined as a fuzzy graph (σ₁ ⊗ σ₂, μ₁ ⊗ μ₂) on G = (V, E) where V = V₁ × V₂ and E = {(u₁, u₂), (v₁, v₂) ∣ (u₁, v₁) ∈ E₁, (u₂, v₂) ∈ E₂}. Fuzzy sets σ₁ ⊗ σ₂ and μ₁ ⊗ μ₂ are defined as (σ₁ ⊗ σ₂) (u₁, u₂) = σ₁ (u₁) ∧ σ₂ (u₂) for all (u₁, u₂) ∈ V, (μ₁ ⊗ μ₂) ((u₁, u₂), (v₁, v₂)) = μ₁ (u₁, v₁) ∧ μ₂ (u₂, v₂) for all (u₁, v₁) ∈ E₁ and (u₂, v₂) ∈ E₂.

Definition 11. The lexicographic product of two fuzzy graphs (σ_i, μ_i) on G_i = (V_i, E_i), i = 1, 2, is defined as a fuzzy graph G₁ ★ G₂ : (σ₁ ★ σ₂, μ₁ ★ μ₂) on G = (V, E) where E = {(u, v₁) (u, v₂) : u∈ V₁, (v₁, v₂) ∈ E₂} ∪ {(u₁, v₁) (u₂, v₂) : (u₁, u₂) ∈ E₁, (v₁, v₂) E₂}, (σ₁ ★ σ₂) (u, v) = σ₁ (u) ∧ σ₂ (v), for all (u, v) ∈ V₁ × V₂, (μ₁ ★ μ₂) ((u, u₂) (u, v₂)) = σ₁ (u) ∧ μ₂ (u₂, v₂) and (μ₁★ μ₂) ((u₁, u₂) (v₁, v₂)) = μ₁ (u₁, v₁) ∧ μ₂ (u₂, v₂) for all (u₁, v₁) ∈ E₁and (u₂, v₂) ∈ E₂.

Definition 12. [3] The Cartesian product G = G₁ × G₂ of two interval-valued fuzzy graphs G₁ = (A₁, B₁) and G₂ = (A₂, B₂) of the graphs $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ is defined as pair (A₁ × A₂, B₁ × B₂) such that: $(i) {\begin{matrix} (μ_{A_{1}}^{-} \times μ_{A_{2}}^{-}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{-} (u_{1}), μ_{A_{2}}^{-} (u_{2})) \\ (μ_{A_{1}}^{+} \times μ_{A_{2}}^{+}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{+} (u_{1}), μ_{A_{2}}^{+} (u_{2})) \end{matrix} for all (u_{1}, u_{2}) \in V,$ $(ii) {\begin{matrix} (μ_{B_{1}}^{-} \times μ_{B_{2}}^{-}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{-} (u), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} \times μ_{B_{2}}^{+}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{+} (u), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u \in V_{1} and u_{2} v_{2} \in E_{2},$ $(iii) {\begin{matrix} (μ_{B_{1}}^{-} \times μ_{B_{2}}^{-}) ((u_{1}, z) (v_{1}, z)) \\ = min (μ_{B_{1}}^{-} (u_{1} v_{1}), μ_{A_{2}}^{-} (z)) \\ (μ_{B_{1}}^{+} \times μ_{B_{2}}^{+}) ((u_{1}, z) (v_{1}, z)) \\ = min (μ_{B_{1}}^{+} (u_{1} v_{1}), μ_{A_{2}}^{+} (z)) . \end{matrix} for all z \in V_{2} and u_{1} v_{1} \in E_{1},$

Definition 13. [3] The composition G₁ [G₂] = (A₁ ∘ A₂, B₁ ∘ B₂) of two interval-valued fuzzy graphs G₁ = (A₁, B₁) and G₂ = (A₂, B₂) of the graphs $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ is defined as follows: $(i) {\begin{matrix} (μ_{A_{1}}^{-} \circ μ_{A_{2}}^{-}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{-} (u_{1}), μ_{A_{2}}^{-} (u_{2})) \\ (μ_{A_{1}}^{+} \circ μ_{A_{2}}^{+}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{+} (u_{1}), μ_{A_{2}}^{+} (u_{2})) \end{matrix} for all (u_{1}, u_{2}) \in V,$ $(ii) {\begin{matrix} (μ_{B_{1}}^{-} \circ μ_{B_{2}}^{-}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{-} (u), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} \circ μ_{B_{2}}^{+}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{+} (u), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u \in V_{1} and u_{2} v_{2} \in E_{2},$ $(iii) {\begin{matrix} (μ_{B_{1}}^{-} \circ μ_{B_{2}}^{-}) ((u_{1}, z) (v_{1}, z)) \\ = min (μ_{B_{1}}^{-} (u_{1} v_{1}), μ_{A_{2}}^{-} (z)) \\ (μ_{B_{1}}^{+} \circ μ_{B_{2}}^{+}) ((u_{1}, z) (v_{1}, z)) \\ = min (μ_{B_{1}}^{+} (u_{1} v_{1}), μ_{A_{2}}^{+} (z)) \end{matrix} for all z \in V_{2} and u_{1} v_{1} \in E_{1},$ $(iv) {\begin{matrix} (μ_{B_{1}}^{-} \circ μ_{B_{2}}^{-}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{A_{2}}^{-} (u_{2}), μ_{A_{2}}^{-} (v_{2}), μ_{B_{1}}^{-} (u_{1} v_{1})) \\ for all (u_{1}, u_{2}) (v_{1}, v_{2}) \in E^{0} - E, \\ (μ_{B_{1}}^{+} \circ μ_{B_{2}}^{+}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{A_{2}}^{+} (u_{2}), μ_{A_{2}}^{+} (v_{2}), μ_{B_{1}}^{+} (u_{1} v_{2}) \end{matrix}$ where E⁰ = E ∪ {(u₁, u₂) (v₁, v₂) |u₁v₁ ∈ E₁, u₂ ≠v₂}.

Definition 14. The strong product G₁ ⊠ G₂ of two interval-valued fuzzy graphs G₁ = (A₁, B₁) and G₂ = (A₂, B₂) of $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ respectively is defined as a pair (A, B), where A = (μ_{A
^-}, μ_{A
⁺}) and B = (μ_{B
^-}, μ_{B
⁺}) are interval-valued fuzzy sets on V = V₁ × V₂ and E = {(u, u₂) (u, v₂) ∣ u ∈ V₁, (u₂v₂) ∈ E₂} ⋃ {((u₁, w) (v₁, w)) ∣ (u₁v₁) ∈ E₁, w ∈ V₂} ⋃ {(u₁, u₂) (v₁, v₂) ∣ (u₁v₁) ∈ E₁, (u₂v₂) ∈ E₂} respectively which satisfies the followings: $(i) {\begin{matrix} (μ_{A_{1}}^{-} ⊠ μ_{A_{2}}^{-}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{-} (u_{1}), μ_{A_{2}}^{-} (u_{2})) \\ (μ_{A_{1}}^{+} ⊠ μ_{A_{2}}^{+}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{+} (u_{1}), μ_{A_{1}}^{+} (u_{2})) \end{matrix} for all (u_{1}, u_{2}) \in V_{1} \times V_{2},$ $(ii) {\begin{matrix} (μ_{B_{1}}^{-} ⊠ μ_{B_{2}}^{-}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{-} (u), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} ⊠ μ_{B_{2}}^{+}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{+} (u), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u \in V_{1} and u_{2} v_{2} \in E_{2},$ $(iii) {\begin{matrix} (μ_{B_{1}}^{-} ⊠ μ_{B_{2}}^{-}) ((u_{1}, w) (v_{1}, w)) \\ = min (μ_{B_{1}}^{-} (u_{1} v_{1}), μ_{A_{2}}^{-} (w)) \\ (μ_{B_{1}}^{+} ⊠ μ_{B_{2}}^{+}) ((u_{1}, w) (v_{1}, w)) \\ = min (μ_{B_{1}}^{+} (u_{1} v_{1}), μ_{A_{2}}^{+} (w)) \end{matrix} for all w \in V_{2} and u_{1} v_{1} \in E_{1},$ $(iv) {\begin{matrix} (μ_{B_{1}}^{-} ⊠ μ_{B_{2}}^{-}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{B_{1}}^{-} (u_{1} v_{1}), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} ⊠ μ_{B_{2}}^{+}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{B_{1}}^{+} (u_{1} v_{1}), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u_{1} v_{1} \in E_{1} and u_{2} v_{2} \in E_{2} .$

Definition 15. The tensor product G₁ ⊗ G₂ of two interval-valued fuzzy graphs G₁ = (A₁, B₁) and G₂ = (A₂, B₂) of $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ respectively is defined as a pair (A, B), where A = (μ_{A
^-}, μ_{A
⁺}) and B = (μ_{B
^-}, μ_{B
⁺}) areinterval - valuedfuzzysetsonV = V₁ × V₂ and E = {(u₁, u₂), (v₁, v₂) ∣ (u₁, v₁) ∈ E₁, (u₂v₂) ∈ E₂}, respectively which satisfies the followings: $(i) {\begin{matrix} (μ_{A_{1}}^{-} \otimes μ_{A_{2}}^{-}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{-} (u_{1}), μ_{A_{2}}^{-} (u_{2})) \\ (μ_{A_{1}}^{+} \otimes μ_{A_{2}}^{+}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{+} (u_{1}), μ_{A_{2}}^{+} (u_{2})) \end{matrix} for all (u_{1}, u_{2}) \in V_{1} \times V_{2},$ $(ii) {\begin{matrix} (μ_{B_{1}}^{-} \otimes μ_{B_{2}}^{-}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{B_{1}}^{-} (u_{1} v_{1}), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} \otimes μ_{B_{2}}^{+}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{B_{1}}^{+} (u_{1} v_{1}), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u_{1} v_{1} \in E_{1} and u_{2} v_{2} \in E_{2} .$

Definition 16. The lexicographic product G₁ ★ G₂ of two interval-valued fuzzy graphs G₁ = (A₁, B₁) and G₂ = (A₂, B₂) of $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ respectively is defined as a pair (A, B), where A = (μ_{A
^-}, μ_{A
⁺}) and B = (μ_{B
^-}, μ_{B
⁺}) are interval-valued fuzzy sets on V = V₁ × V₂ and E = {(u, u₂) (u, v₂) ∣ u ∈ V₁, (u₂v₂) ∈ E₂} ⋃ {((u₁, u₂) (v₁, v₂)) ∣ (u₁v₁) ∈ E₁, u₂v₂ ∈ E₂} respectively which satisfies the followings: $(i) {\begin{matrix} (μ_{A_{1}}^{-} ★ μ_{A_{2}}^{-}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{-} (u_{1}), μ_{A_{2}}^{-} (u_{2})) \\ (μ_{A_{1}}^{+} ★ μ_{A_{2}}^{+}) (u_{1}, u_{2}) = min (μ_{A_{1}}^{+} (u_{1}), μ_{A_{2}}^{+} (u_{2})) \end{matrix} for all (u_{1}, u_{2}) \in V_{1} \times V_{2},$ $(ii) {\begin{matrix} (μ_{B_{1}}^{-} ★ μ_{B_{2}}^{-}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{-} (u), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} ★ μ_{B_{2}}^{+}) ((u, u_{2}) (u, v_{2})) \\ = min (μ_{A_{1}}^{+} (u), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u \in V_{1} and u_{2} v_{2} \in E_{2},$ $(iii) {\begin{matrix} (μ_{B_{1}}^{-} ★ μ_{B_{2}}^{-}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{B_{1}}^{-} (u_{1} v_{1}), μ_{B_{2}}^{-} (u_{2} v_{2})) \\ (μ_{B_{1}}^{+} ★ μ_{B_{2}}^{+}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = min (μ_{B_{1}}^{+} (u_{1} v_{1}), μ_{B_{2}}^{+} (u_{2} v_{2})) \end{matrix} for all u_{1} v_{1} \in E_{1} and u_{2} v_{2} \in E_{2} .$

3. Degree of a vertex in Cartesian product

By the definition, for any vertex (u₁, u₂)∈ $V_{1} \times V_{2}, d_{G_{1} \times G_{2}}^{-} (u_{1}, u_{2}) = \sum_{(u_{1}, u_{2}) (v_{1}, v_{2}) \in E} (μ_{B_{1}^{-}} \times$ $μ_{B_{2}^{-}}) ((u_{1}, u_{2}) (v_{1}, v_{2})) = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}^{-}} (u_{1}) \land$ $μ_{B_{2}^{-}} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}^{-}} (u_{2}) \land μ_{B_{1}^{-}} (u_{1}$ $v_{1}) . d_{G_{1} \times G_{2}}^{+} (u_{1}, u_{2}) = \sum_{(u_{1}, u_{2}) (v_{1}, v_{2}) \in E} (μ_{B_{1}^{+}} \times μ_{B_{2}^{+}}) ((u_{1}, u_{2}) (v_{1}, v_{2})) = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}^{+}} (u_{1}) \land$ $μ_{B_{2}^{+}} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}^{+}} (u_{2}) \land μ_{B_{1}^{+}} (u_{1} v_{1}) .$

In the following theorem, we find the degree of (u₁, u₂) in G₁ × G₂ in terms of those in G₁ and G₂ in some particular cases.

Theorem 1. Let G₁ = (A₁, B₁) and G₂ = (A₂, B₂) be two interval-valued fuzzy graphs.

If $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{1}}^{+} \geq μ_{B_{2}}^{+}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{A_{2}}^{+} \geq μ_{B_{1}}^{+}$ then d_G₁×G₂ (u₁, u₂) = d_{G
₁} (u₁) + d_{G
₂} (u₂).

Proof. From the definition of a vertex in Cartesian product we have: $d_{G_{1} \times G_{2}}^{-} (u_{1}, u_{2}) = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{-} (u_{1}) \land μ_{B_{2}}^{-}$ $(u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) = \sum_{u_{2} v_{2} \in E_{2}} μ_{B_{2}}^{-} (u_{2} v_{2}) + \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1})$ (Since $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ ) $= d_{G_{1}}^{-} (u_{1}) + d_{G_{2}}^{-} (u_{2})$ .

Similarly, we can show that $d_{G_{1} \times G_{2}}^{+} = d_{G_{1}}^{+} (u_{1}) + d_{G_{2}}^{+} (u_{2}) .$

Example 2. Consider the interval-valued fuzzy graphs G₁, G₂ and G₁ × G₂ shown in Fig. 2.

Since $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{1}}^{+} \geq μ_{B_{2}}^{+}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{A_{2}}^{+} \geq μ_{B_{1}}^{+}$ by Theorem 1 we have $\begin{matrix} d_{G_{1} \times G_{2}}^{-} (u_{1}, u_{2}) & = & d_{G_{1}}^{-} (u_{1}) + d_{G_{2}}^{-} (u_{2}) \\ = & 0.1 + 0.1 = 0.2, \\ d_{G_{1} \times G_{2}}^{+} (u_{1}, u_{2}) & = & d_{G_{1}}^{+} (u_{1}) + d_{G_{2}}^{+} (u_{2}) \\ = & 0.2 + 0.3 = 0.5, \\ d_{G_{1} \times G_{2}}^{-} (u_{1}, v_{2}) & = & d_{G_{1}}^{-} (u_{1}) + d_{G_{2}}^{-} (v_{2}) \\ = & 0.1 + 0.1 = 0.2, \\ d_{G_{1} \times G_{2}}^{+} (u_{1}, v_{2}) & = & d_{G_{1}}^{+} (u_{1}) + d_{G_{2}}^{+} (v_{2}) \\ = & 0.2 + 0.3 = 0.5 . \end{matrix}$

Similarly, we can find the degrees of all the vertices in G₁ × G₂. This can be verified in Fig. 2.

Fig. 2.

Cartesian product (G₁ × G₂) of G₁ and G₂.

4. Degree of a vertex in composition

By the definition, for any vertex (u₁, u₂) ∈ V₁ × V₂, $d_{G_{1} [G_{2}]} (u_{1}, u_{2}) = \sum_{(u_{1}, u_{2}) (v_{1}, v_{2}) \in E} (μ_{B_{1}}^{-}$ $\circ μ_{B_{2}}^{-}) ((u_{1}, u_{2}), (v_{1}, v_{2})) = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{-} (u_{1})$ $\land μ_{B_{2}}^{-} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1})$ $+ \sum_{u_{2} \neq v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) .$ $d_{G_{1} [G_{2}]}^{+}$ $(u_{1}, u_{2}) = \sum_{(u_{1}, u_{2}) (v_{1}, v_{2}) \in E} (μ_{B_{1}}^{+} \circ μ_{B_{2}}^{+}) ((u_{1}, u_{2}) (v_{1}$ , $v_{2})) = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{+} (u_{1}) \land μ_{B_{2}}^{+} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{+} (u_{2}) \land μ_{B_{1}}^{+} (u_{1} v_{1}) + \sum_{u_{2} \neq v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{+} (u_{2}) \land μ_{B_{1}}^{+} (u_{1} v_{1}) .$

Theorem 2. Let G₁ = (A₁, B₁) and G₂ = (A₂, B₂) be two interval-valued fuzzy graphs. If $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{1}}^{+} \geq μ_{B_{2}}^{+}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{A_{2}}^{+} \geq μ_{B_{1}}^{+}$ then d_G₁[G₂] (u₁, u₂) = |V₂|d_{G
₁} (u₁) + d_{G
₂} (u₂).

Proof. $d_{G_{1} [G_{2}]}^{-} (u_{1}, u_{2}) = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{-} (u_{1}) \land μ_{B_{2}}^{-} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1})$ $+ \sum_{u_{2} \neq v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) = \sum_{u_{2} v_{2} \in E_{2}} μ_{B_{2}}^{-} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1})$ $+ \sum_{u_{2} \neq v_{2}, u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1})$ (Since $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ ) $= d_{G_{2}}^{-} (u_{2}) + | V_{2} | \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1})$ $= d_{G_{2}}^{-} (u_{2}) + | V_{2} | d_{G_{1}}^{-} (u_{1}) .$ Similarly, we can show that $d_{G_{1} [G_{2}]}^{+} (u_{1}, u_{2}) = d_{G_{2}}^{+} (u_{2}) + | V_{2} | d_{G_{1}}^{+} (u_{1}) .$

So, d_G₁[G₂] (u₁, u₂) = d_{G
₂} (u₂) + |V₂|d_{G
₁} (u₁).

Example 3. Consider the interval-valued fuzzy graphs G₁, G₂ and G₁ [G₂] in Fig. 3.

Fig. 3.

Composition (G₁ [G₂]) of G₁ and G₂.

Here $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{1}}^{+} \geq μ_{B_{2}}^{+}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{A_{2}}^{+} \geq μ_{B_{1}}^{+}$ . By Theorem 2 we have:

$\begin{matrix} d_{G_{1} [G_{2}]}^{-} (u_{1}, u_{2}) & = & d_{G_{2}}^{-} (u_{2}) + | V_{2} | d_{G_{1}}^{-} (u_{1}) \\ = & 0.1 + 2 (0.2) = 0.5, \\ d_{G_{1} [G_{2}]}^{+} (u_{1}, u_{2}) & = & d_{G_{2}}^{+} (u_{2}) + | V_{2} | d_{G_{1}}^{+} (u_{1}) \\ = & 0.3 + 2 (0.4) = 1.1, \\ d_{G_{1} [G_{2}]}^{-} (u_{1}, v_{2}) & = & d_{G_{2}}^{-} (v_{2}) + | V_{2} | d_{G_{1}}^{-} (u_{1}) \\ = & 0.1 + 2 (0.2) = 0.5, \\ d_{G_{1} [G_{2}]}^{+} (u_{1}, v_{2}) & = & d_{G_{2}}^{+} (v_{2}) + | V_{2} | d_{G_{1}}^{+} (u_{1}) \\ = & 0.3 + 2 (0.4) = 1.1 . \end{matrix}$ Similarly, we can find the degrees of all other vertices in G₁ [G₂].

5. Degree of a vertex in tensor product

By definition for any (u₁, u₂) ∈ V₁ × V₂ $\begin{matrix} d_{G_{1} \otimes G_{2}}^{-} (u_{1}, u_{2}) = \sum (μ_{B_{1}}^{-} \otimes μ_{B_{2}}^{-}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1}) \land μ_{B_{2}}^{-} (u_{2}, v_{2}) \\ d_{G_{1} \otimes G_{2}}^{+} (u_{1}, u_{2}) = \sum (μ_{B_{1}}^{+} \otimes μ_{B_{2}}^{+}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{+} (u_{1} v_{1}) \land μ_{B_{2}}^{+} (u_{2}, v_{2}) \end{matrix}$

Theorem 3. Let G₁ = (A₁, B₁) and G₂ = (A₂, B₂) be two interval-valued fuzzy graphs, if $μ_{B_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{B_{2}}^{+} \geq μ_{B_{1}}^{+}$ then d_G₁⊗G₂ (u₁, u₂) = d_{G
₁} (u₁) and if $μ_{B_{1}}^{-} \geq μ_{B_{2}}^{-}, μ_{B_{1}}^{+} \geq μ_{B_{2}}^{+}$ then d_G₁⊗G₂ (u₁, u₂) = d_{G
₂} (u₂).

Proof. Let $μ_{B_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{B_{2}}^{+} \geq μ_{B_{1}}^{+}$ , so we have: $\begin{matrix} d_{G_{1} \otimes G_{2}}^{-} (u_{1}, u_{2}) = \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1}) \land μ_{B_{2}}^{-} (u_{2} v_{2}) \\ = \sum μ_{B_{1}}^{-} (u_{1} v_{1}) = d_{G_{1}}^{-} (u_{1}) \\ d_{G_{1} \otimes G_{2}}^{+} (u_{1}, u_{2}) = \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{+} (u_{1} v_{1}) \land μ_{B_{2}}^{+} (u_{2} v_{2}) \\ = \sum μ_{B_{1}}^{+} (u_{1} v_{1}) = d_{G_{1}}^{+} (u_{1}) \end{matrix}$

So, d_G₁⊗G₂ (u₁, u₂) = d_{G
₁} (u₁).

Similarly, if $μ_{B_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{B_{1}}^{+} \geq μ_{B_{2}}^{+}$ then d_G₁⊗G₂ (u₁, u₂) = d_{G
₂} (u₂).

Example 4. Consider the interval-valued fuzzy graphs G₁, G₂ and G₁ ⊗ G₂ in Fig. 4.

Fig. 4.

Tensor product (G₁ ⊗ G₂) of G₁ and G₂.

Here $μ_{B_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{B_{2}}^{+} \geq μ_{B_{1}}^{+}$ and by Theorem 3 $\begin{matrix} d_{G_{1} \otimes G_{2}}^{-} (u_{1}, u_{2}) & = & d_{G_{1}}^{-} (u_{1}), \\ = & 0.1, \end{matrix}$ $\begin{matrix} d_{G_{1} \otimes G_{2}}^{+} (u_{1}, u_{2}) & = & d_{G_{1}}^{+} (u_{1}) \\ = & 0.2, \\ d_{G_{1} \otimes G_{2}}^{-} (v_{1}, v_{2}) & = & d_{G_{1}}^{-} (v_{1}) \\ = & 0.1, \\ d_{G_{1} \otimes G_{2}}^{+} (v_{1}, v_{2}) & = & d_{G_{1}}^{+} (v_{1}) \\ = & 0.2 . \end{matrix}$ So, d_G₁⊗G₂ (u₁, u₂) = (0.1, 0.2), d_G₁⊗G₂ (v₁, v₂) = (0.1, 0.2).

This can be verified in Fig. 4.

6. Degree of a vertex in strong product

By definition, for any (u₁, u₂) ∈ V₁ × V₂ $\begin{matrix} d_{G_{1} ⊠ G_{2}}^{-} (u_{1}, u_{2}) = \\ \sum_{(u_{1}, v_{1}) (u_{2}, v_{2}) \in E} (μ_{B_{1}}^{-} ⊠ μ_{B_{2}}^{-}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{-} (u_{1}) \land μ_{B_{2}}^{-} (u_{2} v_{2}) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) \\ + \sum_{u_{2} v_{2} \in E_{2}, u_{1} v_{1} \in E_{1}} μ_{B_{2}}^{-} (u_{2} v_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) \end{matrix}$ $\begin{matrix} d_{G_{1} ⊠ G_{2}}^{+} (u_{1}, u_{2}) = \\ \sum_{(u_{1}, v_{1}) (u_{2}, v_{2}) \in E} (μ_{B_{1}}^{+} ⊠ μ_{B_{2}}^{+}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{+} (u_{1}) \land μ_{B_{2}}^{+} (u_{2} v_{2}) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{+} (u_{2}) \land μ_{B_{1}}^{+} (u_{1} v_{1}) \\ + \sum_{u_{2} v_{2} \in E_{2}, u_{1} v_{1} \in E_{1}} μ_{B_{2}}^{+} (u_{2} v_{2}) \land μ_{B_{1}}^{+} (u_{1} v_{1}) . \end{matrix}$

Theorem 4. Let G₁ = (A₁, B₁) and G₂ = (A₂, B₂) be two interval-valued fuzzy graphs. If $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{1}}^{+} \geq μ_{B_{2}}^{+}$ and $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{A_{2}}^{+} \geq μ_{B_{1}}^{+}$ and $μ_{B_{1}}^{-} \leq μ_{B_{2}}^{-}$ , $μ_{B_{1}}^{+} \leq μ_{B_{2}}^{+}$ , then d_G₁⊠G₂ (u₁, u₂) = |V₂|d_{G
₁} (u₁) + d_{G
₂} (u₂).

Proof. $\begin{matrix} d_{G_{1} ⊠ G_{2}}^{-} (u_{1}, u_{2}) = \\ \sum_{(u_{1}, v_{1}) (u_{2}, v_{2}) \in E} (μ_{B_{1}}^{-} ⊠ μ_{B_{2}}^{-}) ((u_{1} v_{1}) (u_{2} v_{2})) \\ = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} μ_{A_{1}}^{-} (u_{1}) \land μ_{B_{2}}^{-} (u_{2} v_{2}) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{A_{2}}^{-} (u_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) \\ + \sum_{u_{2} v_{2} \in E_{2}} μ_{B_{2}}^{-} (u_{2} v_{2}) \land μ_{B_{1}}^{-} (u_{1} v_{1}) \end{matrix}$ $\begin{matrix} = \sum_{u_{2} v_{2} \in E_{2}} μ_{B_{2}}^{-} (u_{2} v_{2}) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1}) \\ + \sum_{u_{1} v_{1} \in E_{1}} μ_{B_{1}}^{-} (u_{1} v_{1}) \end{matrix}$

Since $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ and $μ_{B_{1}}^{-} \leq μ_{B_{2}}^{-}$ , $d_{G_{1} ⊠ G_{2}}^{-} (u_{1}, u_{2}) = d_{G_{2}}^{-} (u_{2}) + | V_{2} | d_{G_{1}}^{-} (u_{1})$ . Similarly, we can show that $d_{G_{1} ⊠ G_{2}}^{+} (u_{1}, u_{2}) = d_{G_{2}}^{+} (u_{2}) + | V_{2} | d_{G_{1}}^{+} (u_{1}) .$

Hence, d_G₁⊠G₂ (u₁, u₂) = |V₂|d_{G
₁} (u₁) + d_{G
₂} (u₂).

Example 5. Consider the interval-valued fuzzy graphs G₁, G₂ and G₁ ⊠ G₂ in Fig. 5. Here $μ_{A_{1}}^{-} \geq μ_{B_{2}}^{-}$ , $μ_{A_{1}}^{+} \geq μ_{B_{2}}^{+}$ , $μ_{A_{2}}^{-} \geq μ_{B_{1}}^{-}$ , $μ_{A_{2}}^{+} \geq μ_{B_{1}}^{+}$ and $μ_{B_{1}}^{-} \leq μ_{B_{2}}^{-}$ , $μ_{B_{1}}^{+} \leq μ_{B_{2}}^{+}$ . So by Theorem 4

Fig. 5.

Strong product (G₁ ⊠ G₂) of G₁ and G₂.

$\begin{matrix} d_{G_{1} ⊠ G_{2}}^{-} (u_{1}, u_{2}) & = & d_{G_{2}}^{-} (u_{2}) + | V_{2} | d_{G_{1}}^{-} (u_{1}) \\ = & 0.2 + 2 (0.2) = 0.6, \\ d_{G_{1} ⊠ G_{2}}^{+} (u_{1}, u_{2}) & = & d_{G_{2}}^{+} (u_{2}) + | V_{2} | d_{G_{1}}^{+} (u_{1}) \\ = & 0.4 + 2 (0.3) = 1, \end{matrix}$ $\begin{matrix} so, d_{G_{1} ⊠ G_{2}} (u_{1}, u_{2}) & = & (0.6, 1) . \\ d_{G_{1} ⊠ G_{2}}^{-} (v_{1}, v_{2}) & = & d_{G_{2}}^{-} (v_{2}) + | V_{2} | d_{G_{1}}^{-} (v_{1}) \\ = & 0.2 + 2 (0.2) = 0.6, \\ d_{G_{1} ⊠ G_{2}}^{+} (v_{1}, v_{2}) & = & d_{G_{2}}^{+} (v_{2}) + | V_{2} | d_{G_{1}}^{+} (v_{1}) \\ = & 0.4 + 2 (0.3) = 1, \\ so, d_{G_{1} ⊠ G_{2}} (v_{1}, v_{2}) & = & (0.6, 1) . \end{matrix}$

Similarly, we can find the degrees of all other vertices in G₁ ⊠ G₂. This can be verified in Fig. 5.

7. Conclusion

The interval valued models give more precision, flexibility and compatibility to the system as compared to the classical and fuzzy models, in this paper, we have found the degree of vertices in G₁ × G₂, G₁ [G₂], G₁ ⊗ G₂ and G₁ ⊠ G₂, in terms of the degree of vertices in two interval-value fuzzy graphs G₁ and G₂ under some conditions and illustrated them through examples. This will be helpful when the graphs are very large. Also they will be very useful in studying various properties of Cartesian product, composition, tensor product and strong product of two interval-valued fuzzy graphs.

Footnotes

Acknowledgment

This work was supported by the research fund of Hanyang University (HY-2018-F), (Project Number 201800000001370).

The authors are grateful to the reviewers for the valuable comments for improvement of the paper.

References

Akram and

W.A.

Dudec , Interval-valued fuzzy graphs, Computers and Mathematics with Applications 61 (2011), 289–299.

R.A.

Borzooei ,

Rashmanlou ,

Samanta and

Pal , A study on fuzzy labelling graphs, Journal of Intelligent and Fuzzy Systems 30 (2016), 3349–3355.

R.A.

Borzooei ,

Rashmanlou ,

Samanta and

Pal , Regularity of vague graphs, Journal of Intelligent and Fuzzy Systems 30 (2016), 3681–3689.

R.A.

Borzooei and

Rashmanlou , Domination in vague graphs and its applications, Journal of Intelligent and Fuzzy Systems 29 (2015), 1933–1940.

R. A.

Borzooei and

Rashmanlou , New concepts of vague graphs, International Journal of Machine Learning and Cybernetics. DOI 10.1007/s13042-015-0475-x.

R.A.

Borzooei and

Rashmanlou , Degree of vertices in vague graphs, Journal of Applied Mathematics and Informatics 33(5) (2015), 545–557.

R.A.

Borzooei and

Rashmanlou , More results on vague graphs, UPB Sci Bull, Series A 78(1) (2016), 109–122.

R.A.

Borzooei and

Rashmanlou , Semi global domination sets in vague graphs with application, Journal of Intelligent and Fuzzy Systems 30 (2016), 3645–3652.

J.N.

Mordeson and

C.S.

Peng , Operation on fuzzy graphs, Information Sciences 19 (1994), 159–170.

10.

J.N.

Mordeson and

P.S.

Nair , Cycles and cocycles of fuzzy graphs, Information Sciences 90 (1996), 39–49.

11.

J.N.

Mordeson and

P.S.

Nair , Fuzzy Graphs and, Fuzzy Hypergraphs, Physica-Verlag, Heidelberg, 1998. Second edition 2001.

12.

Nagoorgani and

V.T.

Chandrasekaron , Fuzzy graph theory, Allied Publisher Pvt. Ltd.

13.

Nagoorgani and

Basheer Ahmad , Order and size of fuzzy graphs, Bulletion of Pure and Applied Sciences 22(1) (2003), 145–148.

14.

Nagoorgani and

Radha , The degree of vertex in some fuzzy graphs, International Jurnal of Algorithms, Computing and Mathematics 2 (2009), 107–116.

15.

Sk.Md.

Abu Nayeem and

Pal , Shortest path problem on a network with imprecise edge weight, Fuzzy Optimization and Decision Making 4(4) (2005), 293–312.

16.

Pramanik ,

Samanta and

Pal , Interval-valued fuzzy planar graphs, International Journal of Machine Learning and Cybernetics 7(4) (2016), 653–664.

17.

Rosenfeld , Fuzzy graphs, In: Zadeh, L.A. Fu, K.S. Shimura and M. (Eds.), Fuzzy sets and their Applications, Academic Press, New York, pp. 77–95.

18.

Samanta and

Pal , Fuzzy tolerance graphs, International Journal of Latest Trends in Mathematics 1(2) (2011), 57–67.

19.

Samanta and

Pal , Fuzzy threshold graphs, CIITInternational Journal of Fuzzy Systems 3(12) (2011), 360–364.

20.

Samanta and

Pal , Bipolar fuzzy hypergraphs, International Journal of Fuzzy Logic Systems 2(1) (2012), 17–28.

21.

Samanta and

Pal , Irregular bipolar fuzzy graphs, Iner-national Journal of Applications of Fuzzy Sets 2 (2012), 91–102.

22.

Samanta and

Pal , Some more results on bipolar fuzzy sets and bipolar fuzzy intersection graphs, The Journal of Fuzzy Mathematics 22(2) (2014), 253–262.

23.

Samanta and

Pal , Telecommunication system based on fuzzy graphs, Journal of Telecommunication System and Management 3(1) (2013), 1–6.

24.

Samanta and

Pal , Fuzzy k-competition graphs and p- competition fuzzy graphs, Fuzzy Inf Eng 5(2) (2013), 191–204.

25.

Samanta ,

Akram and

Pal , M-step fuzzy competition graphs, Journal of Applied Mathematics and Computing (2014). DOI: 10.1007/s12190-s12014-s10785-s10782

26.

Samanta ,

Pal and

Pal , New concepts of fuzzy planar graph, International Journal of Advanced Research in Artificial Intelligence 3(1) (2014), 52–59.

27.

Samanta ,

Pal and

Pal , Some more results on fuzzy k-competition graphs, International Journal of Advanced Research in Artificial Intelligence 3(1) (2014), 60–67.

28.

Samanta ,

Pal ,

Rashmanlou and

R.A.

Borzooei , Vague graphs and strengths, Journal of Intelligent & Fuzzy Systems 30(6) (2016), 3675–3680.

29.

A.A.

Talebi and

Rashmanlou , Isomorphism on inter-valvalued fuzzy graphs, Annals of Fuzzy Mathematics and Informatics 6(1) (2013), 47–58.

30.

A.A.

Talebi ,

Rashmanlou and

Mehdipoor , Isomor-phismon on vague graphs, Annals of Fuzzy Mathematics and Informatics 6(3) (2013), 575–588.

31.

L.A.

Zadeh , The concept of a linguistic and application to approximat reasoning I, Information Sciences 8 (1975), 199–249.

32.

L.A.

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

33.

W.R.

Zhang , Bipolar fuzzy sets and relation; a computational framework for cognitive modeline and multiagent decision analysis, Proceeding of IEEE Conf, 1994, pp. 305–309.

34.

W.R.

Zhang , Bipolar fuzzy sets, Proceedings of Fuzzy-IEEE, 1998, pp. 835–840.