A study on cubic graphs with novel application

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 cubic graphs are more flexible and compatible than fuzzy graphs due to the fact that they have many applications in networks. In this paper, we define the direct product, strong product, and degree of a vertex in cubic graphs and investigate some of their properties. Likewise, we introduce the notion of complete cubic graphs and present some properties of self complementary cubic graphs. Finally, We present fuzzy cubic organizational model as an example of cubic digraph in decision support system.

Keywords

Cubic set cubic graphs self complementary strong cubic graphs

1 Introduction

Zadeh introduced the concept of fuzzy set in his definitive paper [38] of 1965. A fuzzy set of a universe X is a function from X into the unit closed interval [0, 1] of real number. In [39] Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set, i.e., a fuzzy set with an interval-valued membership function. Interval-valued fuzzy sets have been actively used in real-life applications. For example, Sambuc [26] in medical diagnosis in thyroidian pathology, Kohout [10] also in medicine, Turksen in preferences modelling [37], etc. These works and others show the importance of these sets. Jun et al. [6] introduced cubic sets. Later on, Muhiuddin et al. applied the notion cubic sets on different aspects (see for e.g., [7 –9]). The fuzzy graph theory as a generalization of Euler’s graph theory was first introduced by Rosenfeld [25] in 1975.

After Rosenfeld [25] the fuzzy graph theory increases for several years with its different branches, such as - fuzzy threshold graph [28], fuzzy tolerance graph [29], bipolar fuzzy graphs [21, 22], isometry on interval-valued fuzzy graphs [24], highly irregular interval valued fuzzy graphs [19], balanced interval-valued fuzzy graphs [17, 23], fuzzy planar graphs [27], etc. In 2015, Samanta et al. have introduced fuzzy colouring of fuzzy graph [30]. Colouring of fuzzy graphs have studied in [18]. Novel concept of bipolar fuzzy competition graphs is introduced by Sarwar and Akram [32]. Akram et al. [2] defined strong intuitionistic fuzzy graphs, also they discuss intuitionistic fuzzy hypergraphs with applications [3]. Balanced intuitionistic fuzzy graphs is discuss by Karunambigai et al. [11]. The novel concepts of intuitionistic fuzzy competition graph are given by Sahoo et al. [34]. They also discussed intuitionistic fuzzy tolerance graph with application [35], various types of products on intuitionistic fuzzy graphs [33] and product of intuitionistic fuzzy graphs and their degree [36]. Later, Bhattacharya [4] gave some remarks on fuzzy graphs and some operations on fuzzy graphs were introduced by Mordeson and Peng [14]. The complement of a fuzzy graph was defined by Mordeson [15] and further studied by Sunitha and Vijayakumar [31]. Hongmei and Lianhua gave the definition of interval-valued fuzzy graphs [5]. Akram and Dudek defined some operations on interval-valued fuzzy graphs [1]. Neutrosophic graph have been studied in [16]. Recently a lot of work related to fuzzy graph have done [12 , 41].

Rashmanlou et al. [20 –22] introduced some properties of highly irregular interval-valued fuzzy graphs, and new concepts of bipolar fuzzy graphs. In this paper, we define the the direct product, semi-strong product, strong product, union, and join on cubic graphs and investigate some of their properties. Likewise, we introduce the notion of complete cubic graphs and present some properties of self complementary cubic graphs.

2 Preliminaries

We introduce below necessary notions and present a few auxiliary results that will be used throughout the paper.

A map λ : X → [0, 1] is called a fuzzy subset of X. For any two fuzzy subsets λ and μ of X, λ ≤ μ means that, for all x ∈ X, λ (x) ≤ μ (x). The symbols λ ∧ μ, and λ ∨ μ will mean the following fuzzy subsets of X $\begin{matrix} (λ \land μ) (x) & = λ (x) \land μ (x) and (λ \lor μ) (x) \\ = λ (x) \lor μ (x), \end{matrix}$ for all x ∈ X.

Let X be a non-empty set. A function A : X → [I] is called an interval-valued fuzzy set (shortly, an IVF set) in X. Let, [I] ^X stands for the set of all IVF sets in X. For every A ∈ [I] ^X and x ∈ X, A (x) = [A^- (x) , A⁺ (x)] is called the degree of membership of an element x to A, where A^- : X → I and A⁺ : X → I are fuzzy sets in X which are called a lower fuzzy set and an upper fuzzy set in X, respectively. For simplicity, we denote A = [A^-, A⁺]. For every A, B ∈ [I] ^X, we define A ⊆ B if and only if A (x) ≤ B (x), for all x ∈ X.

Definition 1. [39] Let A = [A^-, A⁺], and B = [B^-, B⁺] be two interval-valued fuzzy sets in X (briefly, IVF sets). Then we define r min {A (x) , B (x)} = [min {A^- (x) , B^- (x)} , min {A⁺ (x) , B⁺ (x)}], r max {A (x) , B (x)} = [max {A^- (x) , B^- (x)} , max {A⁺ (x) , B⁺ (x)}].

Definition 2. [6] Let X be a non-empty set. By a cubic set in X, we mean a structure $A = {〈 x, A (x), λ (x) 〉 : x \in X}$ in which A is an interval-valued fuzzy sets in X (briefly, IVF set) and λ is a fuzzy set in X. A cubic set $A = {〈 x, A (x), λ (x) 〉 : x \in X}$ is simply denoted by $A = 〈 A, λ 〉$ . The collection of all cubic sets in X is denoted by CP (X).

3 Cubic graphs

Definition 3. A cubic graph is a triplet G = (G^*, P, Q) where G^* = (V, E) is a graph, $P = (\tilde{μ_{P}}, λ_{P})$ is a cubic set on V and $Q = (\tilde{μ_{Q}}, λ_{Q})$ is a cubic set on V × V such that: $\begin{matrix} \tilde{μ_{Q}} (xy) & \leq r min {\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)}, λ_{Q} (xy) \\ \geq max {λ_{P} (x), λ_{P} (y)} . \end{matrix}$

Example 1. Consider the graph G^* = (V, E) where V = {u, v, x, w} and E = {uv, vx, xw, uw, vw}. Define $P = (\tilde{μ_{P}}, λ_{P})$ a cubic set on V and $Q = (\tilde{μ_{Q}}, λ_{Q})$ a cubic set on E as follows:

Table 1
Weight of vertices and edges, first row represents name of vertices, second row represents the degree of acceptance of the vertices and third row represents the degree of confidence of the ’degree of acceptance’ and other rows are similar for vertices”

V u v x w

$\tilde{μ_{P}}$ [0.2, 0.3] [0.27, 0.4] [0.6, 0.7] [0.5, 0.6]

λ _P 0.3 0.4 0.5 0.2

E uv vx xw uw vw

$\tilde{μ_{Q}}$ [0.2, 0.3] [0.2, 0.2] [0.4, 0.5] [0.1, 0.25] [0.1, 0.3]

λ _Q 0.5 0.6 0.7 0.4 0.5

V	u	v	x	w
$\tilde{μ_{P}}$	[0.2, 0.3]	[0.27, 0.4]	[0.6, 0.7]	[0.5, 0.6]
λ _P	0.3	0.4	0.5	0.2
E	uv	vx	xw	uw	vw
$\tilde{μ_{Q}}$	[0.2, 0.3]	[0.2, 0.2]	[0.4, 0.5]	[0.1, 0.25]	[0.1, 0.3]
λ _Q	0.5	0.6	0.7	0.4	0.5

Then, simple calculation show that G = (G^*, P, Q) is a cubic graph of G^*.

Definition 4. A cubic graph G = (G^*, P, Q) is called (i) complete if $\tilde{μ_{Q}} (xy) = r min {\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)}$ and λ_Q (xy) = max {λ_P (x) , λ_P (y)}, for all x, y ∈ V. (ii) Strong if $\tilde{μ_{Q}} (xy) = r min {\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)}$ and λ_Q (xy) = max {λ_P (x) , λ_P (y)}, for all xy ∈ E.

Example 2. Consider a cubic graph G = (G^*, P, Q), where V = {u, v, x, w}, E = {uv, uw, vx, xw, vw}, $P = (\tilde{μ_{P}}, λ_{P})$ and $Q = (\tilde{μ_{Q}}, λ_{Q})$ are defined as follows

Table 2

Weight of vertices and edges

	u	v
$\tilde{μ_{P}}$	[0.2, 0.3]	[0.2, 0.27]
λ _P	0.3	0.4
	x	w
$\tilde{μ_{P}}$	[0.4, 0.5]	[0.5, 0.6]
λ _P	0.5	0.2
	uv	uw	vx
$\tilde{μ_{Q}}$	[0.2, 0.27]	[0.2, 0.3]	[0.2, 0.27]
λ _Q	0.4	0.3	0.5
	xw	vw
$\tilde{μ_{Q}}$	[0.4, 0.5]	[0.2, 0.27]
λ _Q	0.5	0.4

Proposition 1. Let $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ be two cubic graphs, where $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ . Then direct product of G₁ and G₂ is defined by $G_{1} ⊓ G_{2} = (G_{1}^{*} ⊓ G_{2}^{*}, P_{1} ⊓ P_{2}, Q_{1} ⊓ Q_{2})$ , where $G_{1}^{*} ⊓ G_{2}^{*} = (V_{1} \times V_{2}, E)$ , E = {(u₁, v₁) (u₂, v₂) | (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂} and for all (u, v) , (u₁, v₁) , (u₂, v₂) ∈ V₁ × V₂ we have $\begin{matrix} (\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u, v) \\ = r min (\tilde{μ_{P_{1}}} (u), \tilde{μ_{P_{2}}} (v)), (λ_{P_{1}} ⊓ λ_{P_{2}}) (u, v) \\ = max (λ_{P_{1}} (u), λ_{P_{2}} (v)), \end{matrix}$ $\begin{matrix} (\tilde{μ_{Q_{1}}} ⊓ \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = r min (\tilde{μ_{Q_{1}}} (u_{1} u_{2}), \tilde{μ_{Q_{2}}} (v_{1} v_{2})), \\ (λ_{Q_{1}} ⊓ λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = max (λ_{Q_{1}} (u_{1} u_{2}), λ_{Q_{2}} (v_{1} v_{2})) \end{matrix}$ is a cubic graph.

Proof. The proof is straightforward.

Example 3. Let $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ be two crisp graphs such that V₁ = {u, v}, V₂ = {x, w}, E₁ = {uv}, and E₂ = {xw}. Consider cubic graphs $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ as Fig. 3. Then G₁ ⊓ G₂ is as follows.

Fig.1

Cubic graph G.

Fig.2

Complete cubic graph G.

Fig.3

G₁, G₂ and G₁ ⊓ G₂.

It is easy to show that G₁ ⊓ G₂ is a cubic graph.

Proposition 2. If $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ are two strong cubic graphs, then G₁ ⊓ G₂ is a strong cubic graph too.

Proof. If (u₁, v₁) (u₂, v₂) ∈ E, since G₁ and G₂ are strong, then we have $\begin{matrix} (\tilde{μ_{Q_{1}}} ⊓ \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) = r min (\tilde{μ_{Q_{1}}} (u_{1} u_{2}), \tilde{μ_{Q_{2}}} (v_{1} v_{2})) \\ = r min (r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{1}}} (u_{2})), r min (\tilde{μ_{P_{2}}} (v_{1}), \tilde{μ_{P_{2}}} (v_{2}))) \\ = r min ((\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u_{1}, v_{1}), (\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u_{2}, v_{2})) . \end{matrix}$ $\begin{matrix} (λ_{Q_{1}} ⊓ λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) = max (λ_{Q_{1}} (u_{1} u_{2}), λ_{Q_{2}} (v_{1} v_{2})) \\ = max (max (λ_{P_{1}} (u_{1}), λ_{P_{1}} (u_{2})), max (λ_{P_{2}} (v_{1}), λ_{P_{2}} (v_{2}))) \\ = max ((λ_{P_{1}} ⊓ λ_{P_{2}}) (u_{1}, v_{1}), (λ_{P_{1}} ⊓ λ_{P_{2}}) (u_{2}, v_{2})) . \end{matrix}$ Hence, G₁ ⊓ G₂ is a strong cubic graph.

The strong product between cubic graph is an important construction of cubic graph as it contains an edge between every pair of vertices.

Proposition 3. Let $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ be two cubic graphs where $G_{1}^{*} = (V_{1}, E_{1})$ and $G_{2}^{*} = (V_{2}, E_{2})$ . Then strong product of G₁ and G₂ is defined by $G_{1} \otimes G_{2} = (G_{1}^{*} \otimes G_{2}^{*}, P_{1} \otimes P_{2}, Q_{1} \otimes Q_{2})$ , where $G_{1}^{*} \otimes G_{2}^{*} = (V_{1} \times V_{2}, E)$ , E = {(u, v₁) (u, v₂) | u ∈ V₁, (v₁, v₂) ∈ E₂} ∪ {(u₁, w) (u₂, w) | w ∈ V₂, (u₁, u₂) ∈ E₁} ∪ {(u₁, v₁) (u₂, v₂) | (u₁, u₂) ∈ E₁, (v₁, v₂) ∈ E₂}, and for all u, v, u₁, u₂, v₁, v₂, w ∈ V, we have $\begin{matrix} (\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u, v) & = & r min (\tilde{μ_{P_{1}}} (u), \tilde{μ_{P_{2}}} (v)), \\ (λ_{P_{1}} \otimes λ_{P_{2}}) (u, v) & = & max (λ_{P_{1}} (u), λ_{P_{2}} (v)), \end{matrix}$ $(i) {\begin{matrix} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u, v_{1}) (u, v_{2})) = r min (\tilde{μ_{P_{1}}} (u) \land \tilde{μ_{Q_{2}}} (v_{1} v_{2})), \\ (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) = max (λ_{P_{1}} (u_{1}), λ_{Q_{2}} (v_{1} v_{2})), \end{matrix}$ $(ii) {\begin{matrix} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) = r min (\tilde{μ_{P_{2}}} (v_{1}), \tilde{μ_{Q_{1}}} (u_{1} u_{2})), \\ (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) = max (λ_{P_{2}} (w), λ_{Q_{1}} (u_{1} u_{2})), \end{matrix}$ $(iii) {\begin{matrix} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = r min (\tilde{μ_{Q_{1}}} (u_{1} u_{2}), & \tilde{μ_{Q_{2}}} (v_{1} v_{2})), \\ (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = max (λ_{Q_{1}} (u_{1} u_{2}), & λ_{Q_{2}} (v_{1} v_{2})) \end{matrix}$ is a cubic graph.

Proof. The proof is straightforward.

The following example illustrates that strong product of cubic graphs is another cubic graph where between every pair of vertices, there exist an edge.

Example 4. In this example, we consider cubic graphs $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ as Fig. 4. Then G₁ ⊗ G₂ is as follows.

Fig.4

G₁, G₂ and G₁ ⊗ G₂.

By a routine computation, it is easy to see that G₁ ⊗ G₂ is a cubic graph.

Proposition 4. If $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ are two complete cubic graphs, then G₁ ⊗ G₂ is also complete.

Proof. For all (u, v₁) (u, v₂) ∈ E, since G₂ is complete, we have

$\begin{matrix} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u, v_{1}) (u, v_{2})) \\ = r min (\tilde{μ_{P_{1}}} (u), \tilde{μ_{Q_{2}}} (v_{1} v_{2})) \\ = r min (\tilde{μ_{P_{1}}} (u), r min (\tilde{μ_{P_{2}}} (v_{1}), \tilde{μ_{P_{2}}} (v_{2}))) \\ = r min ((\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u, v_{1}), (\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u, v_{2})) . \end{matrix}$

$\begin{matrix} (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u, v_{1}) (u, v_{2})) \\ = max (λ_{P_{1}} (u), λ_{Q_{2}} (v_{1} v_{2})) \\ = max (λ_{P_{1}} (u), max (λ_{P_{2}} (v_{1}), λ_{P_{2}} (v_{2}))) \\ = max ((λ_{P_{1}} \otimes λ_{P_{2}}) (u, v_{1}), (λ_{P_{1}} \otimes λ_{P_{2}}) (u, v_{2})) . \end{matrix}$

Now, if (u₁, w) (u₂, w) ∈ E, then $\begin{matrix} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u_{1}, w) (u_{2}, w)) \\ = r min (\tilde{μ_{P_{2}}} (w), \tilde{μ_{Q_{1}}} (u_{1} u_{2})) \\ = r min (\tilde{μ_{P_{2}}} (w), r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{1}}} (u_{2}))) \\ = r min ((\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u_{1}, w), (\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u_{2}, w)) . \end{matrix}$

Similarly, we can show that $\begin{matrix} (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u_{1}, w) (u_{2}, w)) \\ = max ((λ_{P_{1}} \otimes λ_{P_{2}}) (u_{1}, w), (λ_{P_{1}} \otimes λ_{P_{2}}) (u_{2}, w)) . \end{matrix}$ Now, if (u₁, v₁) (u₂, v₂) ∈ E, then since G₁ and G₂ are complete, we get $\begin{matrix} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = r min (\tilde{μ_{Q_{1}}} (u_{1} u_{2}), \tilde{μ_{Q_{2}}} (v_{1} v_{2})) \\ = r min (r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{1}}} (u_{2})), \end{matrix}$ $\begin{matrix} r min (\tilde{μ_{P_{2}}} (u_{2}), \tilde{μ_{P_{2}}} (v_{2}))) \\ = r min ((\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u_{1}, v_{1}), \\ (\tilde{μ_{P_{1}}} \otimes \tilde{μ_{P_{2}}}) (u_{2}, v_{2})) . \end{matrix}$

$\begin{matrix} (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = max (λ_{Q_{1}} (u_{1} u_{2}), λ_{Q_{2}} (v_{1} v_{2})) \\ = max (max (λ_{P_{1}} (u_{1}), λ_{P_{1}} (u_{2})), \\ max (λ_{P_{2}} (v_{1}), λ_{P_{2}} (v_{2}))) \\ = max ((λ_{P_{1}} \otimes λ_{P_{2}}) (u_{1}, v_{1}), (λ_{P_{1}} \otimes λ_{P_{2}}) (u_{2}, v_{2})) . \end{matrix}$ Hence, G₁ ⊗ G₂ is complete.

Theorem 1. If $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ are two cubic graphs, such that G₁ ⊓ G₂ is strong, then at least one of G₁ or G₂ must be strong.

Proof. Suppose that G₁ and G₂ are not strong. Then there exists (u₁, v₁) ∈ E₁ and (u₂, v₂) ∈ E₂ such that $\tilde{μ_{Q_{1}}} (u_{1} v_{1}) < r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{1}}} (v_{1})),$

$λ_{Q_{1}} (u_{1} v_{1}) > max (λ_{P_{1}} (u_{1}), λ_{P_{1}} (v_{1}))$ and $\tilde{μ_{Q_{2}}} (u_{2} v_{2}) < r min (\tilde{μ_{P_{2}}} (u_{2}), \tilde{μ_{P_{2}}} (v_{2})),$ $λ_{Q_{2}} (u_{2} v_{2}) > max (λ_{P_{2}} (u_{2}), λ_{P_{2}} (v_{2})) .$ Hence $\begin{matrix} (\tilde{μ_{Q_{1}}} ⊓ \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) = \\ = r min (\tilde{μ_{Q_{1}}} (u_{1} u_{2}), \tilde{μ_{Q_{2}}} (v_{1} v_{2})) \\ < r min (r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{1}}} (u_{2})), \\ r min (\tilde{μ_{P_{2}}} (v_{1}), \tilde{μ_{P_{2}}} (v_{2}))) . \end{matrix}$ and $\begin{matrix} (λ_{Q_{1}} ⊓ λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ > max (max (λ_{P_{1}} (u_{1}), \\ λ_{P_{1}} (u_{2})), max (λ_{P_{2}} (v_{1}), λ_{P_{2}} (v_{2}))) . \end{matrix}$ Now, since $(\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u_{1}, v_{1}) = r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{2}}} (v_{1}))$ and $(\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u_{2}, v_{2}) = r min (\tilde{μ_{P_{1}}} (u_{1}),$ $\tilde{μ_{P_{2}}} (v_{2}))$ . Thus, $\begin{matrix} r min ((\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u_{1}, v_{1}), (\tilde{μ_{P_{1}}} ⊓ \tilde{μ_{P_{2}}}) (u_{2}, v_{2})) = \\ r min (r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{P_{1}}} (u_{2})), r min (\tilde{μ_{P_{2}}} (v_{1}), \tilde{μ_{P_{2}}} (v_{2}))) \\ > (\tilde{μ_{Q_{1}}} ⊓ \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) . \end{matrix}$ Similarly, we can show that $\begin{matrix} max ((λ_{P_{1}} ⊓ λ_{P_{2}}) (u_{1}, v_{1}), (λ_{P_{1}} ⊓ λ_{P_{2}}) (u_{2}, v_{2})) \\ < (λ_{Q_{1}} ⊓ λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) . \end{matrix}$ Hence, G₁ ⊓ G₂ is not strong, which is a contradiction.

Definition 5. The open degree of a vertex u in a cubic graph G = (G^*, P, Q) is defined as d (u) = (d^μ (u) , d^λ (u)), where $d^{μ} (u) = \sum_{\begin{array}{l} u \neq v u \\ v \in V \end{array}} \tilde{μ_{Q}} (u v)$ and $d^{λ} (u) = \sum_{\begin{array}{l} u \neq v \\ u, v \in V \end{array}} λ_{Q} (u v)$ . If all the vertices have the same open neighborhood degree n, then G is called an n-regular cubic graph.

Example 5. In Fig. 3, we have d^μ (u) = [0.3, 0.55], d^λ (u) =0.9, d^μ (x) = [0.6, 0.7], and d^λ (x) =1.3.

Herein, we derive the degree of a vertex in direct product.

By definition of direct product for any (u₁, u₂) ∈ V₁ × V₂ we have:

$\begin{matrix} d_{G_{1} ⊓ G_{2}}^{μ} (u_{1}, u_{2}) & = \sum (\tilde{μ_{Q_{1}}} ⊓ \tilde{μ_{Q_{2}}}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = \sum r min (\tilde{μ_{Q_{1}}} (u_{1} v_{1}), \tilde{μ_{Q_{2}}} (u_{2} v_{2})) \end{matrix}$ $\begin{matrix} d_{G_{1} ⊓ G_{2}}^{λ} (u_{1}, u_{2}) & = \sum (λ_{Q_{1}} ⊓ λ_{Q_{2}}) ((u_{1}, u_{2}) (v_{1}, v_{2})) \\ = \sum max (λ_{Q_{1}} (u_{1} v_{1}), λ_{Q_{2}} (u_{2} v_{2})) . \end{matrix}$

Theorem 2. Let $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ be two cubic graphs. If $\tilde{μ_{Q_{2}}} \geq \tilde{μ_{Q_{1}}}$ and λ_{Q
₂} ≤ λ_{Q
₁}, then d_G₁⊓G₂ (u₁, u₂) = d_{G
₁} (u₁). Also, if $\tilde{μ_{Q_{1}}} \geq \tilde{μ_{Q_{2}}}$ and λ_{Q
₁} ≤ λ_{Q
₂}, then d_G₁⊓G₂ (u₁, u₂) = d_{G
₂} (u₂).

Proof. Let $\tilde{μ_{Q_{2}}} \geq \tilde{μ_{Q_{1}}}$ and λ_{Q
₂} ≤ λ_{Q
₁}. Then

$\begin{matrix} d_{G_{1} ⊓ G_{2}}^{μ} (u_{1}, u_{2}) \\ = \sum_{u_{1} v_{1} \in E_{1}} r min (\tilde{μ_{Q_{1}}} (u_{1} v_{1}), \tilde{μ_{Q_{2}}} (u_{2} v_{2})) \\ = \sum \tilde{μ_{Q_{1}}} (u_{1} v_{1}) = d_{G_{1}}^{μ} (u_{1}), d_{G_{1} ⊓ G_{2}}^{λ} (u_{1}, u_{2}) \\ = \sum_{u_{1} v_{1} \in E_{1}} max (λ_{Q_{1}} (u_{1} v_{1}), λ_{Q_{2}} (u_{2} v_{2})) \\ = \sum λ_{Q_{1}} (u_{1} v_{1}) = d_{G_{1}}^{λ} (u_{1}) . \end{matrix}$ Hence, d_G₁⊓G₂ (u₁, u₂) = d_{G
₁} (u₁). Similarly, if μ_{Q
₁} ≥ μ_{Q
₂} and λ_{Q
₁} ≤ λ_{Q
₂}, then d_G₁⊓G₂ (u₁, u₂) = d_{G
₂} (u₂).

Now we derive the degree of a vertex in strong product.

By the definition of strong product for any (u₁, u₂) ∈ V₁ × V₂ we have: $\begin{matrix} d_{G_{1} \otimes G_{2}}^{μ} (u_{1}, u_{2}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{Q_{2}}} (u_{2} v_{2})) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} r min (\tilde{μ_{P_{2}}} (u_{2}), \tilde{μ_{Q_{1}}} (u_{1} v_{1})) \\ + \sum_{u_{2} v_{2} \in E_{2}, u_{1} v_{1} \in E_{1}} r min (\tilde{μ_{Q_{2}}} (u_{2} v_{2}), \tilde{μ_{Q_{1}}} (u_{1} v_{1})) \\ d_{G_{1} \otimes G_{2}}^{λ} (u_{1}, u_{2}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (λ_{Q_{1}} \otimes λ_{Q_{2}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} max (λ_{P_{1}} (u_{1}), λ_{Q_{2}} (u_{2} v_{2})) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} max (λ_{P_{2}} (u_{2}), λ_{Q_{1}} (u_{1} v_{1})) \\ = \sum_{u_{2} v_{2} \in E_{2}, u_{1} v_{1} \in E_{1}} max (λ_{Q_{2}} (u_{2} v_{2}), λ_{Q_{1}} (u_{1} v_{1})) . \end{matrix}$

Theorem 3. Let $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ be two cubic graphs. If $\tilde{μ_{P_{1}}} \geq \tilde{μ_{Q_{2}}}$ , λ_{P
₁} ≤ λ_{Q
₂}, $\tilde{μ_{P_{2}}} \geq \tilde{μ_{Q_{1}}}$ , λ_{P
₂} ≤ λ_{Q
₁}, $\tilde{μ_{Q_{1}}} \leq \tilde{μ_{Q_{2}}}$ and λ_{Q
₁} ≥ λ_{Q
₂} then, d_G₁⊗G₂ (u₁, u₂) = |V₂|d_{G
₁} (u₁) + d_{G
₂} (u₂).

Proof. Let $\tilde{μ_{P_{1}}} \geq \tilde{μ_{Q_{2}}}$ , λ_{P
₁} ≤ λ_{Q
₂}, $\tilde{μ_{P_{2}}} \geq \tilde{μ_{Q_{1}}}$ , λ_{P
₂} ≤ λ_{Q
₁}, $\tilde{μ_{Q_{1}}} \leq \tilde{μ_{Q_{2}}}$ , and λ_{Q
₁} ≥ λ_{Q
₂}. Then $\begin{matrix} d_{G_{1} \otimes G_{2}}^{μ} (u_{1}, u_{2}) \\ = \sum_{((u_{1}, v_{1}) (u_{2}, v_{2})) \in E} (\tilde{μ_{Q_{1}}} \otimes \tilde{μ_{Q_{2}}}) ((u_{1}, v_{1}) (u_{2}, v_{2})) \\ = \sum_{u_{1} = v_{1}, u_{2} v_{2} \in E_{2}} r min (\tilde{μ_{P_{1}}} (u_{1}), \tilde{μ_{Q_{2}}} (u_{2} v_{2})) \\ + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} r min (\tilde{μ_{P_{2}}} (u_{2}), \tilde{μ_{Q_{1}}} (u_{1} v_{1})) \\ + \sum_{u_{2} v_{2} \in E_{2}, u_{1} v_{1} \in E_{1}} r min (\tilde{μ_{Q_{2}}} (u_{2} v_{2}), \tilde{μ_{Q_{1}}} (u_{1} v_{1})) \\ = \sum_{u_{2} v_{2} \in E_{2}} \tilde{μ_{Q_{2}}} (u_{2} v_{2}) + \sum_{u_{2} = v_{2}, u_{1} v_{1} \in E_{1}} \tilde{μ_{Q_{1}}} (u_{1} v_{1}) \\ + \sum_{u_{1} v_{1} \in E_{1}} \tilde{μ_{Q_{1}}} (u_{1} v_{1}) \\ = d_{G_{2}}^{μ} (u_{2}) + | V_{2} | d_{G_{1}}^{μ} (u_{1}) . \end{matrix}$ In the same way we can show that $d_{G_{1} \otimes G_{2}}^{λ} (u_{1}, u_{2}) = d_{G_{2}}^{λ} (u_{2}) + | V_{2} | d_{G_{1}}^{λ} (u_{1})$ . Hence, d_G₁⊗G₂ (u₁, u₂) = d_{G
₂} (u₂) + |V₂|d_{G
₁} (u₁).

Definition 6. Let $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ be two cubic graphs, where $G_{1}^{*} = (V_{1}, E_{1})$ , $G_{2}^{*} = (V_{2}, E_{2})$ and V₁∩ V₂ = ∅. Then their join $G_{1} + G_{2} = (G_{1}^{*} + G_{2}^{*}, P_{1} + P_{2}, Q_{1} + Q_{2})$ , where $P_{1} + P_{2} = (\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}, λ_{P_{1}} + λ_{P_{2}})$ and $Q_{1} + Q_{2} = (\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}, λ_{Q_{1}} + λ_{Q_{2}})$ are defined as follows: $\begin{matrix} (i) {\begin{matrix} (\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}) (x) = \tilde{μ_{P_{1}}} (x) if x \in V_{1} \\ (\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}) (x) = \tilde{μ_{P_{2}}} (x) if x \in V_{2} \end{matrix} \\ (ii) {\begin{matrix} (λ_{P_{1}} + λ_{P_{2}}) (x) = λ_{P_{1}} (x) if x \in V_{1} \\ (λ_{P_{1}} + λ_{P_{2}}) (x) = λ_{P_{2}} (x) if x \in V_{2} \end{matrix} \\ (iii) {\begin{matrix} (\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}) (xy) = \tilde{μ_{Q_{1}}} (xy) if xy \in E_{1} \\ (\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}) (xy) = \tilde{μ_{Q_{2}}} (xy) if xy \in E_{2} \\ (\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}) (xy) = r min (\tilde{μ_{P_{1}}} (x), \tilde{μ_{P_{2}}} (y)) if xy \in E^{'} \end{matrix} \\ (iv) {\begin{matrix} (λ_{Q_{1}} + λ_{Q_{2}}) (xy) = λ_{Q_{1}} (x) if xy \in E_{1} \\ (λ_{Q_{1}} + λ_{Q_{2}}) (xy) = λ_{Q_{2}} (x) if xy \in E_{2} \\ (λ_{Q_{1}} + λ_{Q_{2}}) (xy) = max (λ_{P_{1}} (x), λ_{P_{2}} (y)) if xy \in E^{'} \end{matrix} \end{matrix}$ where E′ is the set of all edges joining the nodes V₁ and V₂.

Proposition 5. The join of two cubic graphs is a cubic graph too.

Proof. Proof is obvious.

4 Automorphic cubic graphs

In this section, we define homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism between two cubic graphs and show that isomorphism between cubic graphs is an equivalence relation.

Definition 7. Let G₁ and G₂ be two cubic graphs.

(1) A homomorphism f : G₁ → G₂ is a mapping f : V₁ → V₂ which satisfies the following conditions:

$(a) \tilde{μ_{P_{1}}} (x_{1}) \leq \tilde{μ_{P_{2}}} (f (x_{1})), λ_{P_{1}} (x_{1}) \geq λ_{P_{2}} (f (x_{1}))$ , $(b) \tilde{μ_{Q_{1}}} (x_{1} y_{1}) \leq \tilde{μ_{Q_{2}}} (f (x_{1}) f (y_{1})), λ_{Q_{1}} (x_{1} y_{1}) \geq λ_{Q_{2}} (f (x_{1}) f (y_{1}))$ , for all x₁ ∈ V₁, x₁y₁ ∈ E₁.

(2) An isomorphism f : G₁ → G₂ is a bijective mapping f : V₁ → V₂ which satisfies the following conditions: $(c) \tilde{μ_{P_{1}}} (x_{1}) = \tilde{μ_{P_{2}}} (f (x_{1})), λ_{P_{1}} (x_{1}) = λ_{P_{2}} (f (x_{1}))$ , $(d) \tilde{μ_{Q_{1}}} (x_{1} y_{1}) = \tilde{μ_{Q_{2}}} (f (x_{1}) f (y_{1})), λ_{Q_{1}} (x_{1} y_{1}) = λ_{Q_{2}} (f (x_{1}) f (y_{1}))$ , for all x₁ ∈ V₁ and x₁y₁ ∈ E₁.

If f : G₁ → G₂ is an isomorphism, then we say that G₁ and G₂ are isomorphic and we write G₁ ≅ G₂.

(3) A weak isomorphism f : G₁ → G₂ is a bijective mapping f : V₁ → V₂ which satisfies the following conditions:

(e) f is homomorphism.

$(f) \tilde{μ_{P_{1}}} (x_{1}) = \tilde{μ_{P_{2}}} (f (x_{1})), λ_{P_{1}} (x_{1}) = λ_{P_{2}} (f (x_{1}))$ , for all x₁ ∈ V₁.

Example 6. Consider cubic graphs G₁ and G₂ as follows:

Let f : V₁ → V₂ is defined by f (u) = w and f (v) = x. Then we see that:

$\tilde{μ_{P_{1}}} (u) = \tilde{μ_{P_{2}}} (w)$ , λ_{P
₁} (u) = λ_{P
₂} (w), $\tilde{μ_{P_{1}}} (v) = \tilde{μ_{P_{2}}} (x)$ , λ_{P
₁} (v) = λ_{P
₂} (x), λ_{Q
₁} (uv) = λ_{Q
₂} (xw), but $\tilde{μ_{Q_{1}}} (uv) \neq \tilde{μ_{Q_{2}}} (f (u) f (v)) = \tilde{μ_{Q_{2}}} (xw)$ . Hence, the map is a weak isomorphism but it is not an isomorphism.

Definition 8. Let G₁ and G₂ be two cubic graphs. A co-weak isomorphism f : G₁ → G₂ is a bijective mapping f : V₁ → V₂ which satisfies:

(g) f is homomorphism,

$(h) \tilde{μ_{Q_{1}}} (x_{1} y_{1}) = \tilde{μ_{Q_{2}}} (f (x_{1}) f (y_{1}))$ , λ_{Q
₁} (x₁y₁) = λ_{Q
₂} (f (x₁) f (y₁)), for all x₁, y₁ ∈ V₁.

Example 7. Consider cubic graphs G₁ and G₂ as follows:

Let f : V₁ → V₂ is defined by f (u) = x, and f (v) = w. By routine computations, we can show that the map f is a co-weak isomorphism but it is not an isomorphism. Since $\tilde{μ_{P_{1}}} (u) \neq \tilde{μ_{P_{2}}} (w)$ and λ_{P
₁} (u) ≠ λ_{P
₂} (w).

Proposition 6. Isomorphism relation between cubic graphs, is an equivalence relation.

Definition 9. The complement of a cubic graph G = (G^*, P, Q), where G^* = (V, E), is a cubic graph $\bar{G} = (\bar{G^{*}}, \bar{P}, \bar{Q})$ , where $\bar{P} = P = (\tilde{μ_{P}}, λ_{P})$ , $\bar{Q} = (\bar{\tilde{μ_{Q}}}, \bar{λ_{Q}})$ and $\bar{\tilde{μ_{Q}}} (xy) = r min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)) - \tilde{μ_{Q}} (xy)$ , $\bar{λ_{Q}} (xy) = λ_{Q} (xy) - max (λ_{P} (x), λ_{P} (y))$ , for all x, y ∈ V.

Now, a cubic graph G is said to be self complementary if $G ≅ \bar{G}$ .

Example 8. Consider cubic graph G as follows.

Here G is self complementary, since bijective mapping $f : G \to \bar{G}$ which is defined by v → v, u → w, w → u is an isomorphism.

Theorem 4. Let G be a self complementary cubic graph. Then $\sum_{x \neq y} \tilde{μ_{Q}} (xy) = \frac{1}{2} \sum_{x \neq y} r min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y))$ and $\sum_{x \neq y} λ_{Q} (xy) = \frac{1}{2} \sum_{x \neq y} max (λ_{P} (x), λ_{P} (y))$ .

Proof. Let G = (G^*, P, Q) be a self complementary cubic graph of graph G^* = (V, E). Then, there exists a weak isomorphism g from G to $\bar{G}$ such that for every x, y ∈ V, we have $\tilde{μ_{P}} (x) = \bar{\tilde{μ_{P}}} (g (x))$ , $λ_{P} (x) = \bar{λ_{P}} (g (x))$ , for all x ∈ V and $\tilde{μ_{Q}} (xy) = \bar{\tilde{μ_{Q}}} (g (x) g (y))$ , $λ_{Q} (xy) = \bar{λ_{Q}} (g (x) g (y))$ , for all x, y ∈ V. Now by definition of $\bar{G}$ , for every x, y ∈ V, we have $\bar{\tilde{μ_{Q}}} (g (x) g (y)) = r min (\tilde{μ_{P}} (g (x)), \tilde{μ_{P}} (g (y))) - \tilde{μ_{Q}} (g (x) g (y))$ , $\bar{λ_{Q}} (g (x) g (y)) = λ_{Q} (g (x) g (y)) - max (λ_{P} (g (x)), λ_{P} (g (y)))$ . It means that $\tilde{μ_{Q}} (xy) = r min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)) - \tilde{μ_{Q}} (g (x) g (y))$ and λ_Q (xy) = λ_Q (g (x) g (y)) - max(λ_P (x) , λ_P (y)) $\begin{matrix} i . e ., {\begin{matrix} \sum_{x \neq y} \tilde{μ_{Q}} (xy) + \sum \tilde{μ_{Q}} (g (x) g (y)) = \sum r min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)) \\ \sum_{x \neq y} λ_{Q} (xy) + \sum λ_{Q} (g (x) g (y)) = \sum max (λ_{P} (x), λ_{P} (y)), \end{matrix} \\ i . e ., {\begin{matrix} 2 \sum \tilde{μ_{Q}} (xy) = \sum min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)) \\ 2 \sum λ_{Q} (xy) = \sum max (λ_{P} (x), λ_{P} (y)) . \end{matrix} \end{matrix}$ Hence, $\sum \tilde{μ_{Q}} (xy) = \frac{1}{2} \sum r min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y))$ and $\sum λ_{Q} (xy) = \frac{1}{2} \sum_{x \neq y} max (λ_{P} (x), λ_{P} (y))$ .

Theorem 5. Let G be a cubic graph of a graph G^*. Then, the automorphism group G and $\bar{G}$ are identical.

Proof. We show that for any injective map h : V → V, h ∈ Aut (G) if and only if $h \in Aut (\bar{G})$ . We have: $\begin{matrix} \bar{\tilde{μ_{P}}} (h (x)) = \tilde{μ_{P}} (h (x)) = \tilde{μ_{P}} (x) \\ = & \bar{\tilde{μ_{P}}} (x), for all x \in V \\ \bar{λ_{P}} (h (x)) = λ_{P} (h (x)) = λ_{P} (x) \\ = & \bar{λ_{P}} (x), for all x \in V . \end{matrix}$ Also, for all x, y ∈ V, $\begin{matrix} \bar{\tilde{μ_{Q}}} (h (x) h (y)) \\ = \bar{\tilde{μ_{Q}}} (xy) \\ \Leftrightarrow r min (\tilde{μ_{P}} (h (x)), \tilde{μ_{P}} (h (y))) - \tilde{μ_{Q}} (h (x) h (y)) \\ = r min (\tilde{μ_{P}} (x), \tilde{μ_{P}} (y)) - \tilde{μ_{Q}} (xy) \\ \Leftrightarrow \tilde{μ_{Q}} (h (x) h (y)) = \tilde{μ_{Q}} (xy), \end{matrix}$ and $\begin{matrix} \bar{λ_{Q}} (h (x) h (y)) \\ = \bar{λ_{Q}} (xy) \\ \Leftrightarrow max (λ_{P} (h (x)), λ_{P} (h (y))) - λ_{Q} (h (x) h (y)) \\ = min (λ_{P} (x), λ_{P} (y)) - λ_{Q} (xy) \\ \Leftrightarrow λ_{Q} (h (x) h (y)) = λ_{Q} (xy) . \end{matrix}$ This complete the proof.

Theorem 6. Let $G_{1} = (G_{1}^{*}, P_{1}, Q_{1})$ and $G_{2} = (G_{2}^{*}, P_{2}, Q_{2})$ be two cubic graphs such that V₁∩ V₂ = ∅. Then, $\bar{G_{1} + G_{2}} ≅ \bar{G_{1}} \cup \bar{G_{2}}$ .

Proof. We shall prove that the identity map is the required isomorphism. Let I : V₁ ∪ V₂ → V₁ ∪ V₂ be the identity map. We prove that for all x, y ∈ V, $(\bar{\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}}) (x) = (\bar{\tilde{μ_{P_{1}}}} \cup \bar{\tilde{μ_{P_{2}}}}) (x)$ ,

$(\bar{λ_{P_{1}} + λ_{P_{2}}}) (x) = (\bar{λ_{P_{1}}} \cup \bar{λ_{P_{2}}}) (x)$

and $(\bar{\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}}) (xy) = (\bar{\tilde{μ_{Q_{1}}}} \cup \bar{\tilde{μ_{Q_{2}}}}) (xy)$ ,

$(\bar{λ_{Q_{1}} + λ_{Q_{2}}}) (xy) = (\bar{λ_{Q_{1}}} \cup \bar{λ_{Q_{2}}}) (xy)$ . Let x, y ∈ V. Then, $\begin{matrix} (\bar{\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}}) (x) & = (\tilde{μ_{P_{1}}} \cup \tilde{μ_{P_{2}}}) (x) \\ = {\begin{matrix} \tilde{μ_{P_{1}}} (x), if x \in V_{1} \\ \tilde{μ_{P_{2}}} (x), if x \in V_{2} \end{matrix} \\ = {\begin{matrix} \bar{\tilde{μ_{P_{1}}}} (x), if x \in V_{1} \\ \bar{\tilde{μ_{P_{2}}}} (x), if x \in V_{2} \end{matrix} \\ = (\bar{\tilde{μ_{P_{1}}}} \cup \bar{\tilde{μ_{P_{2}}}}) (x), \end{matrix}$

$\begin{matrix} (\bar{λ_{P_{1}} + λ_{P_{2}}}) (xy) & = (λ_{P_{1}} \cup λ_{P_{2}}) (xy) \\ = {\begin{matrix} λ_{P_{1}} (x), if x \in V_{1} \\ λ_{P_{2}} (x), if x \in V_{2} \end{matrix} \\ = {\begin{matrix} \bar{λ_{P_{1}}} (x), if x \in V_{1} \\ \bar{λ_{P_{2}}} (x), if x \in V_{2} \end{matrix} \\ = (\bar{λ_{P_{1}}} \cup \bar{λ_{P_{2}}}) (x) . \end{matrix}$

Also,

$\begin{matrix} (\bar{\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}}) (xy) = r min ((\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}) (x), (\tilde{μ_{P_{1}}} + \tilde{μ_{P_{2}}}) (y)) \\ - (\tilde{μ_{Q_{1}}} + \tilde{μ_{Q_{2}}}) (xy) \\ = {\begin{matrix} r min ((\tilde{μ_{P_{1}}} \cup \tilde{μ_{P_{2}}}) (x), (\tilde{μ_{P_{1}}} \cup \tilde{μ_{P_{2}}}) (y)) \\ - (\tilde{μ_{Q_{1}}} \cup \tilde{μ_{Q_{2}}}) (y), & if xy \in E_{1} \cup E_{2} \\ r min ((\tilde{μ_{P_{1}}} \cup \tilde{μ_{P_{2}}}) (x), (\tilde{μ_{P_{1}}} \cup \tilde{μ_{P_{2}}}) (y)) \\ - r min (\tilde{μ_{P_{1}}} (x), \tilde{μ_{P_{2}}} (y)), & if xy \in E^{'} \end{matrix} \\ = {\begin{matrix} r min (\tilde{μ_{P_{1}}} (x), \tilde{μ_{P_{2}}} (y)) - \tilde{μ_{Q_{1}}} (xy), & if xy \in E_{1} \\ r min (\tilde{μ_{P_{1}}} (x), \tilde{μ_{P_{2}}} (y)) - \tilde{μ_{Q_{2}}} (xy), & if xy \in E_{2} \\ r min (\tilde{μ_{P_{1}}} (x), \tilde{μ_{P_{2}}} (y)) - r min (\tilde{μ_{P_{1}}} (x), \tilde{μ_{P_{2}}} (y)), & if xy \in E^{'} \end{matrix} \\ = {\begin{matrix} \bar{\tilde{μ_{Q_{1}}}} (xy), & if xy \in E_{1} \\ \bar{\tilde{μ_{Q_{2}}}} (xy), & if xy \in E_{2} \\ 0 & if xy \in E^{'} \end{matrix} \\ = (\bar{\tilde{μ_{Q_{1}}}} \cup \bar{\tilde{μ_{Q_{2}}}}) (xy) . \end{matrix}$ Similarly, we can show that $(\bar{λ_{Q_{1}} + λ_{Q_{2}}}) (xy) = (\bar{λ_{Q_{1}}} \cup \bar{λ_{Q_{2}}}) (xy)$ .

5 Fuzzy cubic organizational model

Graphs can be used to model many types of relations and processes in physical, biological, social and information systems. In studies of behaviour of group persons, it is noticed that certain persons have influence thinking of others. An influence graph is a directed graph which can be used to model this behaviour. In fuzzy influence graph, if vertices represent the persons and its membership degree the authority of persons and edges represent the influence of a person on another person in the social group/industry. Then we can find the most influential person within the group/industry. Now we discuss a cubic model to find out the most influential person in the department of industry with respect to different attributes of employees including conflict, cooperative, industrious, performance, etc. We consider a bank having employees and their designations as shown in Table 3. Here, we consider only the parameter cooperation and the degree of cooperation is taken as an interval valued fuzzy membership value and its degree of acceptance.

Table 3
Names of employees in a bank and their designations

Name Designation

ALi Bank president

Jafar CEO

Reza CTO

Bashir Financial Officer

Mohammad Vice Chairman

Saleem Director of Currencies

Hassan Supporting Staff

Imran Deputy information officer

Name	Designation
ALi	Bank president
Jafar	CEO
Reza	CTO
Bashir	Financial Officer
Mohammad	Vice Chairman
Saleem	Director of Currencies
Hassan	Supporting Staff
Imran	Deputy information officer

A survey conducted on bank producted the following results.

Jafar is such a person of a bank who is active in every crucial decision and responds calmly in stressful situations.

Jafar and Reza have a good relationship and have worked together. Reza values Jafar.

Jafar is a right hand of the Bank president.

Saleem has a great influence in the development team.

Bashir’s ability to communicate effectively and politely enables him to perform his job effectively. Also, he has good-natured and pleasant to others.

Consider the directed graph G^* with vertex set {Ali, Jafar, Reza, Bashir, Mohammad, Saleem, Hassan, Imran} as shown in Fig. 8. The vertices represent employees and directed edges represent any cooperation/relationship between them.

Fig.5

Cubic graphs G₁ and G₂.

Fig.6

Cubic graphs G₁ and G₂.

Fig.7

Self complementary cubic graph G.

Fig.8

Directed graph.

Now using the results conducted by the bank we can draw fuzzy influence digraph of G^* corresponding to attributes cooperation and conflict.

Fuzzy influence digraph is shown in Fig. 9 and tabular representation of this graph is given in Table 4. The vertices represent the employees and the membership degrees of the vertices represent the power of cooperation of employees in the bank. The edges represent dependence in their cooperation on one another. If there is no edge between any two employees, it does mean the corresponding employees are cooperative. The degree of membership can be interpreted as a percentage of dependence. We can easily seen in Fig. 9, that cooperation power of Ali, Jafar, and Saleem are more than all other employees. If we talk about Saleem, he depends on Jafar and Hassan. The degree of power of cooperation of Mahammad is [0.1, 0.3] , 0.5 i.e., 10% to 30% with guarantee 50%. It is vivid that Jafar is the more important and cooperative person in the bank because many people depend on him.

Fig.9

Fuzzy influence graph w.r.t employees cooperation.

Table 4

Vertices and their membership values

Ali	Jafar	Bashir	Reza
[0.4,0.6],0.8	[0.1,0.2],0.7	[0.3,0.5],0.9	[0.1,0.2],0.5
Mohammad	Saleem	Hassan	Imran
[0.1,0.3],0.5	[0.2,0.5],0.6	[0.1,0.5],0.6	[0.3,0.5],0.7

Fuzzy influence digraph is shown in Fig. 10 and tabular representation of this digraph is given in the Tables 4 and 5. Table 4 indicates the vertices and their membership values and Table 5 represents the edges and their membership values. The vertices represent the employees and membership degree of vertices (Table 4) represent their goodness (behaviour). The edges represent cooperation between two employees. If there is no edge between any two employees, it means that the employees have no cooperation. We can seen in Fig. 10 that, the membership degree of Ali and Imran is [0.2, 0.3] , 0.8 i.e., the degree of cooperation between the employees is 20% to 30% with guarantee 80%. Figure 10 shows that, Ali and Imran have more membership degree of their goodness than all other employees.

Fig.10

Fuzzy influence graph wrt attributes: conflict.

Table 5

Edges and their membership values

$\vec{(Ali, Jafar)}$	$\vec{(Reza, Jafar)}$
[0.1,0.2],0.8	[0.1,0.2],0.7
$\vec{(Bashir, Reja)}$	$\vec{(Mohammad, Bashir)}$
[0.1,0.2],0.9	[0.1,0.3],0.9
$\vec{(Saleem, Mahammad)}$	$\vec{(Saleem, Hassan)}$
[0.1,0.3],0.6	[0.1,0.4],0.7
$\vec{(Imran, Hassan)}$	$\vec{(Ali, Imran)}$
[0.1,0.2],0.8	[0.2,0.3],0.8

$\vec{(Mohammad, Jafar)}$	$\vec{(Mohammad, Ali)}$
[0.1,0.2],0.9	[0.1,0.2],0.8
$\vec{(Jafar, Hassan)}$	$\vec{(Reza, Imran)}$
[0.1,0.2],0.7	[0.1,0.2],0.7
$\vec{(Reza, Hassan)}$	$\vec{(Reza, Saleem)}$
[0.1,0.2],0.6	[0.1,0.2],0.7

6 Conclusion

Graph theory is an extremely useful tool in solving the combinatorial problems in different areas including geometry, algebra, topology, optimization, and computer sciences. So, we have introduced cubic graphs and have presented several properties in this paper. The future study of cubic graphs may also be extended with an application of cubic graph in expert system, neural networks, and geographical information system. This new extension provides a significant addition to existing theories for handling uncertainties and leads to potential areas of further field research. The proposed concepts can be used in system analysis, circuit clustering and pattern recognition, etc.

Footnotes

Acknowledgments

The second author was supported by the research grant S-0198-1440, Deanship of Scientific Research, University of Tabuk, Tabuk-71491, Saudi Arabia.

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A study on cubic graphs with novel application

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Cubic graphs

5 Fuzzy cubic organizational model

Table 3 Names of employees in a bank and their designations Name Designation ALi Bank president Jafar CEO Reza CTO Bashir Financial Officer Mohammad Vice Chairman Saleem Director of Currencies Hassan Supporting Staff Imran Deputy information officer

Footnotes

Acknowledgments

References

Table 3
Names of employees in a bank and their designations

Name Designation

ALi Bank president

Jafar CEO

Reza CTO

Bashir Financial Officer

Mohammad Vice Chairman

Saleem Director of Currencies

Hassan Supporting Staff

Imran Deputy information officer