f-morphism on bipolar fuzzy graphs

Abstract

The main purpose of this paper is to introduce the notion of bipolar f-morphism on bipolar fuzzy graphs and regular bipolar fuzzy graphs. We study the action of bipolar f-morphism on bipolar fuzzy graphs and derive some elegant results on weak and co-weak isomorphism. Also, we define d₂-degree and total d₂-degree of a vertex in bipolar fuzzy graphs and study (2, k)-regularity and totally (2, k)-regularity of bipolar fuzzy graphs. d₂-degree and total d₂-degree of a vertex in bipolar fuzzy graphs are highly utilized by computer science, geometry, algebra, number theory and operation research. In addition these properties will also be helpful to study large bipolar fuzzy graph as a combination of small, bipolar fuzzy graphs and to derive its properties from those of the smaller ones.

Keywords

Total degree totally regular bipolar fuzzy graph

1 Introduction

In 1965, Zadeh [23] first proposed the theory of fuzzy sets. The most important feature of a fuzzy set is that a fuzzy set A is a class of objects that satisfy a certain property. Each object x has a membership degree of A, denoted as μ_A (x). Zhang [24, 25] initiated the concept of bipolar fuzzy sets as a generalization of fuzzy sets. Although bipolar fuzzy sets and intuitionistic fuzzy sets look similar to each other, they are essentially different sets [6]. Bipolar fuzzy sets are an extension of fuzzy sets whose membership degree range is [−1,1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [-1, 0] of an element indicates that the element somewhat satisfies the implicit counter property. In 1975, Rosenfeld [8] introduced the notion of fuzzy graphs. Gani and Radha [4, 5] introduced regular fuzzy graphs and irregular fuzzy graphs. Sunitha and Vijayakumar [21] studied some properties of complement on fuzzy graphs.Rashmanlou et al. [7 , 12–14] studied irregular interval-valued fuzzy graphs, antipodal interval-valued fuzzy graphs, balanced interval-valued fuzzy graphs, some properties of highly irregular interval-valued fuzzy graphs and bipolar fuzzy graphs with categorical properties. Rashmanlou and Yong Bae Jun [11] investigated complete interval-valued fuzzy graphs. Samanta and Pal introduced fuzzy tolerance graphs [15], fuzzy threshold graphs [16], fuzzy planar graphs [17], fuzzy k-competition graphs and p-competition fuzzy graphs [19], irregular bipolar fuzzy graphs [18]. Seethalakshmi [20] investigated regularity conditions on an intuitionistic fuzzy graph. Akram et al. [1 –3] introduced bipolar fuzzy graphs, regular bipolar fuzzy graphs and strong intuitionistic fuzzy graphs. Yang et al. [22] introduced notes on bipolar fuzzy graphs. In this paper, we introduce the notion of bipolar f-morphism on bipolar fuzzy graphs and regular bipolar fuzzy graphs. We study the action of bipolar f-morphism on bipolar fuzzy graphs and derive some elegant results on weak and co-weak isomorphism. Also, we define d₂-degree and total d₂-degree of a vertex in bipolar fuzzy graphs and study (2, k)-regularity and totally (2, k)-regularity of bipolar fuzzy graphs. The concept of f-morphism on bipolar fuzzy graphs can be applied in various areas of engineering, computer science:database theory, expert systems, neural networks, artificial intelligence, signalprocessing, pattern recognition, robotics, computer networks, and medical diagnosis.

2 Preliminaries

By a graph G^* = (V, E), we mean a non-trivial, finite connected and undirected graph without loops or multiple edges. A fuzzy graph G = (σ, μ) is a pair of functions σ : V → [0, 1] and μ : V × V → [0, 1] with μ (u, v) ≤ σ (u) ∧ σ (v), for all u, v ∈ V, where V is a finite non-empty set and ∧ denote minimum.

Definition 2.1. [24] Let X be a non-empty set. A bipolar fuzzy set B in X is an object having the form B = {(x, μ^P (x) , μ^N (x)) | x ∈ X}, where μ^P : X → [0, 1] and μ^N : X → [-1, 0] are mapping.

Definition 2.2. [24] Let X be a non-empty set. Then we call a mapping A = (μ_{A
^P}, μ_{A
^N}) : X × X → [-1, 0] × [0, 1] a bipolar fuzzy relation on X such that μ_{A
^P} (x, y) ∈ [0, 1] and μ_{A
^N} (x, y) ∈ [-1, 0].

Definition 2.3. [24] Let A = (μ_{A
^P}, μ_{A
^N}) and B = (μ_{B
^P}, μ_{B
^N}) be bipolar fuzzy sets on a set X. If A = (μ_{A
^P}, μ_{A
^N}) is a bipolar fuzzy relation on a set X, then A = (μ_{A
^P}, μ_{A
^N}) is called a bipolar fuzzy relation on B = (μ_{B
^P}, μ_{B
^N}) if μ_{A
^P} (x, y) ≤ min(μ_{B
^P} (x) , μ_{B
^P} (y)) and μ_{A
^N} (x, y) ≥ max(μ_{B
^N} (x) , μ_{B
^N} (y)), for all x, y ∈ X. A bipolar fuzzy relation A on X is called symmetric if μ_{A
^P} (x, y) = μ_{A
^P} (y, x) and μ_{A
^N} (x, y) = μ_{A
^N} (y, x), for all x, y ∈ X.

For a given set V, we define an equivalence relation ∼ on V × V - {(x, x) : x ∈ V} as follows: (x₁, y₁) ∼ (x₂, y₂) ⇔ either (x₁, y₁) = (x₂, y₂) or x₁ = y₂ and y₁ = x₂. The quotient set obtained in this way is denoted by ${\tilde{V}}^{2}$ , and the equivalent class that contains the element (x, y) is denoted as xy or yx.

Definition 2.4. [22] A bipolar fuzzy graph of a graph G^* = (V, E) is a pair G = (A, B), where A = [μ_{A
^P}, μ_{A
^N}] is a bipolar fuzzy set in V and B = [μ_{B
^P}, μ_{B
^N}] is a bipolar fuzzy set in ${\tilde{V}}^{2}$ such that μ_{B
^P} (xy) ≤ min {μ_{A
^P} (x) , μ_{A
^P} (y)} for all $xy \in {\tilde{V}}^{2}$ , μ_{B
^N} (xy) ≥ max {μ_{A
^N} (x) , μ_{A
^N} (y)} for all $xy \in {\tilde{V}}^{2}$ and μ_{B
^P} (xy) = μ_{B
^N} (xy) =0 for all $xy \in {\tilde{V}}^{2} - E$ .

A bipolar fuzzy graph G = (A, B) of a graph G^∗ = (V, E) is called strong if μ_{B
^P} (xy) = min(μ_{A
^P} (x) , μ_{A
^P} (y)), μ_{B
^N} (xy) = max(μ_{A
^N} (x) , μ_{A
^N} (y)) for all xy ∈ E.

Definition 2.5. The bipolar fuzzy graph G is said to be regular if ∑_{v_j,v_i≠v_j}μ_{B
^P} (v_iv_j) and ∑_{v_j,v_i≠v_j}μ_{B
^N} (v_iv_j) are constant, for all v_i ∈ V. Moreover, it is called strong regular if

μ_B
^P (v_iv_j) = min {μ_A
^P (v_i) , μ_A
^P (v_j)} and μ_B
^N (v_iv_j) = max {μ_A
^N (v_i) , μ_A
^N (v_j)}.

∑_{v_j,v_i≠v_j}μ_B
^P (v_iv_j) and ∑_{v_j,v_i≠v_j}μ_B
^N (v_iv_j)are constant.

The complement [22] of a strong bipolar fuzzy graph G = (A, B) of a graph G^∗ = (V, E) is a bipolar fuzzy graph $\bar{G} = (\bar{A}, \bar{B})$ of $\bar{G^{*}} = (V, V \times V)$ , where $\bar{A} = A = [μ_{A^{P}}, μ_{A^{N}}]$ and $\bar{B} = [\bar{μ_{B^{P}}}, \bar{μ_{B^{N}}}]$ is defined by

$\bar{μ_{B^{P}}} (xy) = min (μ_{A^{P}} (x), μ_{A^{P}} (y)) - μ_{B^{P}} (xy)$ for all $x, y \in V, xy \in {\tilde{V}}^{2}$ ,

$\bar{μ_{B^{N}}} (xy) = max (μ_{A^{N}} (x), μ_{A^{N}} (y)) - μ_{B^{N}} (xy)$ for all $x, y \in V, xy \in {\tilde{V}}^{2}$ .

Definition 2.6. Let G₁ and G₂ be two bipolar fuzzy graphs.

A homomorphism h : G₁ → G₂ is a mapping h : V₁ → V₂ which satisfies the followingconditions:

$μ_{A_{1}^{P}} (x_{1}) \leq μ_{A_{2}^{P}} (h (x_{1}))$ and $μ_{A_{1}^{N}} (x_{1}) \geq μ_{A_{2}^{N}} (h (x_{1}))$ ,

$μ_{B_{1}^{P}} (x_{1} y_{1}) \leq μ_{B_{2}^{P}} (h (x_{1}) h (y_{1}))$ and $μ_{B_{1}^{N}} (x_{1} y_{1})$ $\geq μ_{B_{2}^{N}} (h (x_{1}) h (y_{1}))$ , for all x₁ ∈ V₁ and $x_{1} y_{1} \in {\tilde{V_{1}}}^{2}$ .

An isomorphism h : G₁ → G₂ is a bijective mapping h : V₁ → V₂ which satisfies the following conditions:

$μ_{A_{1}^{P}} (x_{1}) = μ_{A_{2}^{P}} (h (x_{1}))$ and $μ_{A_{1}^{N}} (x_{1}) = μ_{A_{2}^{N}}$ (h (x₁)),

$μ_{B_{1}^{P}} (x_{1} y_{1}) = μ_{B_{2}^{P}} (h (x_{1}) h (y_{1}))$ and $μ_{B_{1}^{N}} (x_{1} y_{1})$ $= μ_{B_{2}^{N}} (h (x_{1}) h (y_{1}))$ , for all x₁ ∈ V₁ and $x_{1} y_{1} \in {\tilde{V_{1}}}^{2}$ .

A weak isomorphism h : G₁ → G₂ is a bijective mapping h : V₁ → V₂ which satisfies the following conditions:

h is homomorphism,

$μ_{A_{1}^{P}} (x_{1}) = μ_{A_{2}^{P}} (h (x_{1}))$ and $μ_{A_{1}^{N}} (x_{1}) = μ_{A_{2}^{N}}$ (h (x₁)), for all x₁ ∈ V₁ and $x_{1} y_{1} \in {\tilde{V_{1}}}^{2}$ .

A co-weak isomorphism h : G₁ → G₂ is a bijective mapping h : V₁ → V₂ which satisfies:

h is homomorphism,

$μ_{B_{1}^{P}} (x_{1} y_{1}) = μ_{B_{2}^{P}} (h (x_{1}) h (y_{1}))$ and $μ_{B_{1}^{N}} (x_{1} y_{1})$ $= μ_{B_{2}^{N}} (h (x_{1}) h (y_{1}))$ , for all x₁, y₁ ∈ V₁ and $x_{1} y_{1} \in {\tilde{V_{1}}}^{2}$ .

Let G = (A, B) be a bipolar fuzzy graph. The open degree of a vertex u is defined as deg (u) = (d_P (u) , d_N (u)), where d_P (u) = ∑_{substacku≠vv∈V}μ_{B
^P} (uv) and d_N (u) = ∑_{substacku≠vv∈V}μ_{B
^N} (uv). The order of G is defined and denoted as

O (G) = (\sum_{u \in V} μ_{A^{P}} (u), \sum_{u \in V} μ_{A^{N}} (u))

and the size of G is S (G) = (S_P (G) , S_N (G)) = (∑_{substacku≠vv,u∈V}μ_{B
^P} (uv) , ∑_{substacku≠vv,u∈V}μ_{B
^N} (uv)).

3 Regularity on isomorphic bipolar fuzzy graphs

One of the most widely studied classes of fuzzy graphs are regular fuzzy graphs. They show up in many contexts. For example, r-regular fuzzy graphs with connectivity and edge-connectivity equal to r play a key role in designing reliable communication networks. In this section, we introduce the notion of bipolar f-morphism on bipolar fuzzy graphs and regular bipolar fuzzy graphs. We derive some elegant results on weak and co-weak isomorphism. Also, we define d₂-degree andtotal d₂-degree of a vertex in bipolar fuzzy graphs and study (2, k)-regularity and totally (2, k)-regularity of bipolar fuzzy graphs.

Definition 3.1. Let G = (A, B) be a bipolar fuzzy graph. The d₂-degree of a vertex u in G is $d_{2} (u) = (d_{2}^{P} (u), d_{2}^{N} (u))$ , where $d_{2}^{P} (u) = \sum μ_{B^{P}}^{2} (u, v)$ and $d_{2}^{N} (u) = \sum μ_{B^{N}}^{2} (u, v)$ such that $\begin{matrix} μ_{B^{P}}^{2} (u, v) & = & Sup {μ_{B^{P}} (u, u_{1}) \land μ_{B^{P}} (u_{1}, v)} and \\ μ_{B^{N}}^{2} (u, v) & = & \inf {μ_{B^{N}} (u, u_{1}) \lor μ_{B^{N}} (u_{1}, v)} . \end{matrix}$ The minimum d₂-degree of G is δ₂ (G) = ∧ {d₂ (v) : v ∈ V}. The maximum d₂-degree of G is △₂ (G) = ∨ {d₂ (v) : v ∈ V}.

Example 3.2. Consider a bipolar fuzzy graph G such that V = {u, v, w, x, y} and

E = {uv, vw, wx, xy, yu} (See. Fig. 1).

By routine computation we have $\begin{matrix} d_{2}^{P} (u) = {0.2 \land 0.3} + {0.1 \land 0.2} = 0.3, \\ d_{2}^{N} (u) = {- 0.2 \lor - 0.4} + {- 0.2 \lor - 0.3} = - 0.4, \\ d_{2}^{P} (v) = {0.2 \land 0.1} + {0.3 \land 0.3} = 0.4, \\ d_{2}^{N} (v) = {- 0.2 \lor - 0.2} + {- 0.4 \lor - 0.2} = - 0.4, \\ d_{2}^{P} (w) = {0.3 \land 0.2} + {0.3 \land 0.2} = 0.4, \\ d_{2}^{N} (w) = {- 0.2 \lor - 0.3} + {- 0.4 \lor - 0.2} = - 0.4 . \end{matrix}$ Hence, d₂ (u) = (0.3, - 0.4) , d₂ (v) = (0.4, - 0.4) ,

d₂ (w) = (0.4, - 0.4).

Definition 3.3. Let G = (A, B) be a bipolar fuzzy graph. If d₂ (v) = k, for all v ∈ V then G is said to be (2, k)-regular bipolar fuzzy graph.

Example 3.4. Consider a bipolar fuzzy graph G such that V = {u, v, w, x} and E = {uv, vw, wx}.

Here, d₂ (u) = d₂ (v) = d₂ (w) = d₂ (x) = (0.2, - 0.2). This graph is a (2, (0.2, - 0.2))-regular bipolar fuzzy graph (See. Fig. 2).

Definition 3.5. Let G = (A, B) be a bipolar fuzzy graph. The total d₂-degree of a vertex u ∈ V is defined as ${td}_{2} (u) = ({td}_{2}^{P} (u), {td}_{2}^{N} (u))$ , where ${td}_{2}^{P} (u) = \sum μ_{B^{P}}^{2} (u, v) + μ_{A^{P}} (u)$ and ${td}_{2}^{N} (u) = \sum μ_{B^{N}}^{2} (u, v) + μ_{A^{N}} (u) .$

Note 3.6. If each vertex of G has the same total d₂-degree k, then G is said to be totally (2, k)-regular bipolar fuzzy graph.

Example 3.7. Consider a bipolar fuzzy graph G such that V = {u, v, w, x, y} and E = {uv, vw, wx, xy, yu} (See. Fig. 3).

It is easy to show that d₂ (u) = (0.6, - 0.4), d₂ (v) = (0.4, - 0.6), d₂ (w) = (0.5, - 0.4), d₂ (x) = (0.6, - 0.4), d₂ (y) = (0.5, - 0.4) and td₂ (u) = td₂ (v) = td₂ (w) = td₂ (x) = td₂ (y) = (1, - 0.9). Each vertex has same total d₂-degree (1, - 0.9). Therefore, G is totally (2, (1, - 0.9))-regular bipolar fuzzy graph. Note that G is not (2, k)-regular bipolar fuzzy graph.

Theorem 3.8. Let G = (A, B) be a bipolar fuzzy graph. Then, μ_{A
^P} (u) = c and μ_{A
^N} (u) = c, for all u ∈ V if and only if the following conditions are equivalent.

(i) G is a (2, k)-regular bipolar fuzzy graph.

(ii) G is a totally (2, k + c)-regular bipolar fuzzy graph.

Proof. Suppose that μ_{A
^P} (u) = c and μ_{A
^N} (u) = c, for all u ∈ V.

(i) ⇒ (ii) From (i) we have d₂ (u) = k, for all u ∈ V. Hence td₂ (u) = k + c, for all u ∈ V. Therefore, G is a totally (2, k + c)-regular bipolar fuzzy graph.

(ii) ⇒ (i) Suppose G is a totally (2, k + c)-regular bipolar fuzzy graph. Then we have

\begin{matrix} {td}_{2} (u) & = & k + c, for all u \in V, \\ d_{2} (u) + c & = & k + c, for all u \in V . \end{matrix}

So, d₂ (u) = k, for all u ∈ V. Hence, G is a (2, k)-regular bipolar fuzzy graph. Therefore, (i) and (ii) are equivalent. Conversely assume that (i) and (ii) are equivalent. Let G be both totally (2, k + c)-regular and (2, k)-regular bipolar fuzzy graph. We have

\begin{matrix} {td}_{2}^{P} (u) = k + c and d_{2}^{P} (u) = k, for all u \in V, \\ d_{2}^{P} (u) + μ_{A^{P}} (u) = k + c and d_{2}^{P} (u) = k, \end{matrix}

for all u ∈ V. So, μ_{A
^P} (u) = c, for all u ∈ V.

Similarly we can show that μ_{A
^N} (u) = c, for all u ∈ V.

□

Definition 3.9. Let G₁ and G₂ be two bipolar fuzzy graphs. Then bijective function f : V₁ → V₂ is called a bipolar morphism or bipolar f-morphism if there exists positive numbers k₁ and k₂ such that

$(i) μ_{A_{2}^{P}} (f (u)) = k_{1} μ_{A_{1}^{P}} (u)$ and $μ_{A_{2}^{N}} (f (u)) = k_{1} μ_{A_{1}^{N}} (u)$ , for all u ∈ V₁.

$(ii) μ_{B_{2}^{P}} (f (u) f (v)) = k_{2} μ_{B_{1}^{P}} (uv)$ and $μ_{B_{2}^{N}} (f (u) f (v)) = k_{2} μ_{B_{1}^{N}} (uv)$ , for all $uv \in {\tilde{V_{1}}}^{2}$ .

In such a case, f will be called a (k₁, k₂) bipolar f-morphism from G₁ to G₂. If k₁ = k₂ = k, we call f is a bipolar k-morphism. When k = 1 we obtain usual bipolar morphism.

Note 3.10. Let G₁ = (A₁, B₁), G₂ = (A₂, B₂) and G₃ = (A₃, B₃) be three bipolar fuzzy graphs on (V₁, E₁), (V₂, E₂) and (V₃, E₃), respectively. $μ_{A_{1}^{P}}$ , $μ_{A_{2}^{P}}$ and $μ_{A_{3}^{P}}$ denote membership function of the vertices in G₁, G₂, G₃ respectively; $μ_{A_{1}^{N}}$ , $μ_{A_{2}^{N}}$ , $μ_{A_{3}^{N}}$ denote non-membership functions of the vertices in G₁, G₂, G₃ respectively; $μ_{B_{1}^{P}}$ , $μ_{B_{2}^{P}}$ , $μ_{B_{3}^{P}}$ denote membership functions of the edges in G₁, G₂, G₃ respectively; $μ_{B_{1}^{N}}$ , $μ_{B_{2}^{N}}$ , $μ_{B_{3}^{N}}$ denote non-membership functions of the edges in G₁, G₂, G₃ respectively.

Theorem 3.11. The relation f-morphism is an equivalence relation in the collection of bipolar fuzzy graphs.

Proof. Consider the collection of bipolar fuzzy graphs. Define the relation G₁ ≈ G₂ if there exists a (k₁, k₂) f-morphism from G₁ to G₂ where both k₁ and k₂ are non-zero. Consider the identity morphism from G₁ to G₁. It is one to one morphism from G₁ to G₁ and hence ≈ is reflexive.

Let G₁ ≈ G₂. Then there exists a (k₁, k₂) morphism from G₁ to G₂ for some non-zero k₁ and k₂. Therefore $μ_{A_{2}^{P}} (f (u)) = k_{1} μ_{A_{1}^{P}} (u)$ , $μ_{A_{2}^{N}} (f (u)) = k_{1} μ_{A_{1}^{N}} (u)$ , for all u ∈ V₁ and $μ_{B_{2}^{P}} (f (u) f (v)) = k_{2} μ_{B_{1}^{P}} (uv)$ and $μ_{B_{2}^{N}} (f (u) f (v)) = k_{2} μ_{B_{1}^{N}} (uv)$ , for all $uv \in {\tilde{V_{1}}}^{2}$ . Consider f^-1 : G₂ → G₁. Let x, y ∈ V₂. Since f^-1 is bijective, x = f (u) and y = f (v), for some u, v ∈ V₂. Now, $\begin{matrix} μ_{A_{1}^{P}} (f^{- 1} (x)) = μ_{A_{1}^{P}} (f^{- 1} (f (u))) = μ_{A_{1}^{P}} (u) \\ = \frac{1}{k_{1}} μ_{A_{2}^{P}} (f (u)) = \frac{1}{k_{1}} μ_{A_{2}^{P}} (x) . \\ μ_{A_{1}^{N}} (f^{- 1} (x)) = μ_{A_{1}^{N}} (f^{- 1} (f (u))) = μ_{A_{1}^{N}} (u) \\ = \frac{1}{k_{1}} μ_{A_{2}^{N}} (f (u)) = \frac{1}{k_{1}} μ_{A_{2}^{N}} (x) . \\ μ_{B_{1}^{P}} (f^{- 1} (x) f^{- 1} (y)) = μ_{B_{1}^{P}} (f^{- 1} (f (u)) f^{- 1} (f (v))) \\ = μ_{B_{1}^{P}} (uv) = \frac{1}{k_{2}} μ_{B_{2}^{P}} (f (u) f (v)) = \frac{1}{k_{2}} μ_{B_{2}^{P}} (xy), \\ μ_{B_{1}^{N}} (f^{- 1} (x) f^{- 1} (y)) = μ_{B_{1}^{N}} (f^{- 1} (f (u)) f^{- 1} (f (v))) \\ = μ_{B_{1}^{N}} (uv) = \frac{1}{k_{2}} μ_{B_{2}^{N}} (f (u) f (v)) = \frac{1}{k_{2}} μ_{B_{2}^{N}} (xy) . \end{matrix}$ Thus there exists $(\frac{1}{k_{1}}, \frac{1}{k_{2}})$ morphism from G₂ to G₁. Therefore G₂ ≈ G₁ and hence ≈ is symmetric.

Let G₁ ≈ G₂ and G₂ ≈ G₃. Then there exists a (k₁, k₂) morphism from G₁ to G₂ say f for some non-zero k₁ and k₂ and there exists a (k₃, k₄) morphism from G₂ to G₃ say g for some non-zero k₃ and k₄. Hence, $μ_{A_{3}^{P}} (g (x)) = k_{3} μ_{A_{2}^{P}} (x)$ and $μ_{A_{3}^{N}} (g (x)) = k_{3} μ_{A_{2}^{N}} (x)$ for all x ∈ V₂ and $μ_{B_{3}^{P}} (g (x) g (y)) = k_{4} μ_{B_{2}^{P}} (xy)$ and $μ_{B_{3}^{N}} (g (x) g (y)) = k_{4} μ_{B_{2}^{N}} (xy)$ , for all $xy \in {\tilde{V_{1}}}^{2}$ .

Let h = gof : G₁ → G₃. Now $\begin{matrix} μ_{A_{3}^{P}} (h (u)) = μ_{A_{3}^{P}} (gof (u)) = μ_{A_{3}^{P}} (g (f (u))) \\ = k_{3} μ_{A_{3}^{P}} (f (u)) = k_{3} k_{1} μ_{A_{1}^{P}} (u) \\ μ_{A_{3}^{N}} (h (u)) = μ_{A_{3}^{N}} (gof (u)) = μ_{A_{3}^{N}} (g (f (u))) \\ = k_{3} μ_{A_{3}^{N}} (f (u)) = k_{3} k_{1} μ_{A_{1}^{N}} (u) \\ μ_{B_{3}^{P}} (h (u) h (v)) = μ_{B_{3}^{P}} (gof (u) gof (v)) \\ = μ_{B_{3}^{P}} (g (f (u)) g (f (v))) = k_{4} μ_{B_{2}^{P}} (f (u) f (v)) \\ = k_{4} k_{2} μ_{B_{1}^{P}} (uv) \\ μ_{B_{3}^{N}} (h (u) h (v)) = μ_{B_{3}^{N}} (gof (u) gof (v)) \\ = μ_{B_{3}^{N}} (g (f (u)) g (f (v))) = k_{4} μ_{B_{2}^{N}} (f (u) f (v)) \\ = k_{4} k_{2} μ_{B_{1}^{N}} (uv) . \end{matrix}$ Thus, there exists (k₃k₁, k₄k₂) morphism h from G₁ to G₃. Therefore, G₁ ≈ G₃ and hence ≈ is transitive. So, the relation f-morphism is an equivalence relation in the collection of bipolar fuzzy graphs. □

Theorem 3.12. Let G₁ and G₂ be two bipolar fuzzy graphs such that G₁ is (k₁, k₂) bipolar fuzzy morphism to G₂ for some non-zero k₁ and k₂. This image of strong edge in G₁ is strong edge in G₂ if and only if k₁ = k₂.

Proof. Let (u, v) be strong edge in G₁ such that f (u), f (v) is also strong edge in G₂. Now as G₁ ≈ G₂ we have $\begin{matrix} k_{2} μ_{B_{1}^{P}} (uv) = μ_{B_{2}^{P}} (f (u) f (v)) \\ = μ_{A_{2}^{P}} (f (u)) \land μ_{A_{2}^{P}} (f (v)) \\ = k_{1} μ_{A_{1}^{P}} (u) \land k_{1} μ_{A_{1}^{P}} (v) = k_{1} μ_{B_{1}^{P}} (uv), \\ for all u \in V_{1} . \end{matrix}$ Hence, $k_{2} μ_{B_{1}^{P}} (uv) = k_{1} μ_{B_{1}^{P}} (uv), for all u \in V_{1}$ (3.1) $\begin{matrix} k_{2} μ_{B_{1}^{N}} (uv) = μ_{B_{2}^{N}} (f (u) f (v)) \\ = μ_{A_{2}^{N}} (f (u)) \lor μ_{A_{2}^{N}} (f (v)) \\ = k_{1} μ_{A_{1}^{N}} (u) \lor k_{1} μ_{A_{1}^{N}} (v) = k_{1} μ_{B_{1}^{N}} (uv), \\ for all u \in V_{1} . \end{matrix}$ Hence, $k_{2} μ_{B_{1}^{N}} (uv) = k_{1} μ_{B_{1}^{N}} (uv), for all u \in V_{1}$ (3.2) Equations (3.1) and (3.2) holds if and only if k₁ = k₂.

□

Corollary 3.13. Let G₁ and G₂ be two bipolar fuzzy graphs and G₁ be a (k₁, k₂) bipolar fuzzy morphism to G₂. If G₁ is strong, then G₂ is strong if and only if k₁ = k₂.

Theorem 3.14. If bipolar fuzzy graph G₁ is co-weak isomorphic to G₂ and G₁ is regular, then G₂ is regular, too.

Proof. As bipolar fuzzy graph G₁ is co-weak isomorphic to G₂, then there exists a co-weak isomorphism h : G₁ → G₂ which is bijective that satisfies $μ_{A_{1}^{P}} (u) \leq μ_{A_{2}^{P}} (h (u))$ and $μ_{A_{1}^{N}} (u) \geq μ_{A_{2}^{N}} (h (u))$ , $μ_{B_{1}^{P}} (uv) = μ_{B_{2}^{P}} (h (u) h (v))$ and $μ_{B_{1}^{N}} (uv) = μ_{B_{2}^{N}}$ (h (u) h (v)), for all u, v ∈ V₁. As G₁ is regular, for u ∈ V we have $\sum_{u \neq v u, v \in V_{1}} μ_{B_{1}^{P}} (uv) = constant$ and

$\sum_{u \neq vu, v \in V_{1}} μ_{B_{1}^{N}} (uv) = constant$ .

Now, $\begin{matrix} \sum_{h (u) \neq h (v)} μ_{B_{2}^{P}} (h (u) h (v)) = \sum_{u \neq v, v \in V_{1}} μ_{B_{1}^{P}} (uv) \\ = constant and \\ \sum_{h (u) \neq h (v)} μ_{B_{2}^{N}} (h (u) h (v)) = \sum_{u \neq v, v \in V_{1}} μ_{B_{1}^{N}} (uv) \\ = constant . \end{matrix}$ Therefore, G₂ is regular. □

Remark 3.15. If bipolar fuzzy graph G₁ is co-weak isomorphic to G₂ and G₁ is strong, then G₂ need not be strong.

Theorem 3.16. Let G₁ and G₂ be two bipolar fuzzy graphs. If G₁ is weak isomorphic to G₂ and G₁ is strong, then G₂ is strong, too.

Proof. As bipolar fuzzy graph G₁ is weak isomorphic with G₂, then there exists a weak isomorphism h : G₁ → G₂ which is bijective that satisfies $μ_{A_{1}^{P}} (u) = μ_{A_{2}^{P}} (h (u))$ , $μ_{A_{1}^{N}} (u) = μ_{A_{2}^{N}} (h (u))$ , $μ_{B_{1}^{P}} (uv) \leq μ_{B_{2}^{P}} (h (u) h (v))$ and $μ_{B_{1}^{N}} (uv) \geq μ_{B_{2}^{N}} (h (u) h (v))$ . As G₁ is strong, $μ_{B_{1}^{P}} (uv) = min (μ_{A_{1}^{P}} (u), μ_{A_{1}^{P}} (v))$ and $μ_{B_{1}^{N}} (uv) = max (μ_{A_{1}^{N}} (u), μ_{A_{1}^{N}} (v))$ . Now we have $\begin{matrix} μ_{B_{2}^{P}} (h (u) h (v)) \geq μ_{B_{1}^{P}} (uv) \\ = min (μ_{A_{1}^{P}} (u), μ_{A_{1}^{P}} (v)) \\ = min (μ_{A_{2}^{P}} (h (u)), μ_{A_{2}^{P}} (h (v))) . \end{matrix}$ By the definition,

$μ_{B_{2}^{P}} (h (u) h (v)) \leq min (μ_{A_{2}^{P}} (h (u)), μ_{A_{2}^{P}} (h (v)))$ .

Therefore, $μ_{B_{2}^{P}} (h (u) h (v)) = min (μ_{A_{2}^{P}} (h (u)),$

$μ_{A_{2}^{P}} (h (v)))$ . Similarly, $\begin{matrix} μ_{B_{2}^{N}} (h (u) h (v)) \leq μ_{B_{1}^{N}} (uv) \\ = max (μ_{A_{1}^{N}} (u), μ_{A_{1}^{N}} (v)) \\ = max (μ_{A_{2}^{N}} (h (u)), μ_{A_{2}^{N}} (h (v))) . \end{matrix}$ By the definition, $max (μ_{A_{2}^{N}} (h (u)), μ_{A_{2}^{N}} (h (v))) \leq μ_{B_{2}^{N}} (h (u) h (v))$ . Therefore, $μ_{B_{2}^{N}} (h (u) h (v)) = max (μ_{A_{2}^{N}} (h (u)), μ_{A_{2}^{N}} (h (v)))$ . Hence, G₂ is strong. □

Corollary 3.17. Let G₁ and G₂ be two bipolar fuzzy graphs. If G₁ is weak isomorphic to G₂ and G₁ is regular, then G₂ need not be regular.

Theorem 3.18. If the bipolar fuzzy graph G₁ is co-weak isomorphic with a strong regular bipolar fuzzy graph G₂, then G₁ is strong regular bipolar fuzzy graph.

Proof. As bipolar fuzzy graph G₁ is co-weak isomorphic with a bipolar fuzzy graph G₂, then there exists a co-weak isomorphism h : G₁ → G₂ which is bijective that satisfies $μ_{A_{1}^{P}} (u) \leq μ_{A_{2}^{P}} (h (u))$ , $μ_{A_{1}^{N}} (u) \geq μ_{A_{2}^{N}} (h (u)),$ $μ_{B_{1}^{P}} (uv) = μ_{B_{2}^{P}} (h (u) h (v))$ and $μ_{B_{1}^{N}} (uv) = μ_{B_{2}^{N}} (h (u) h (v))$ , for all u, v ∈ V₁. Now we have $\begin{matrix} μ_{B_{1}^{P}} (uv) = μ_{B_{2}^{P}} (h (u) h (v)) \\ = min (μ_{A_{2}^{P}} (h (u)), μ_{A_{2}^{P}} (h (v))) \\ \geq min (μ_{A_{1}^{P}} (u), μ_{A_{1}^{P}} (v)) \end{matrix}$ $\begin{matrix} μ_{B_{1}^{N}} (uv) = μ_{B_{2}^{N}} (h (u) h (v)) \\ = max (μ_{A_{2}^{N}} (h (u)), μ_{A_{2}^{N}} (h (v))) \\ \leq max (μ_{A_{1}^{N}} (u), μ_{A_{1}^{N}} (v)) . \end{matrix}$ But by definition, $μ_{B_{1}^{P}} (uv) \leq min (μ_{A_{1}^{P}} (u), μ_{A_{1}^{P}} (v))$ and $μ_{B_{1}^{N}} (uv) \geq max (μ_{A_{1}^{N}} (u), μ_{A_{1}^{N}} (v))$ .

So, $μ_{B_{1}^{P}} (uv) = min (μ_{A_{1}^{P}} (u), μ_{A_{1}^{P}} (v))$ and $μ_{B_{1}^{N}} (uv)$

$= max (μ_{A_{1}^{N}} (u), μ_{A_{1}^{N}} (v))$ . Therefore, G₁ is strong. Also, for u ∈ V₁, $\sum_{u \neq v v \in V_{1}} μ_{B_{1}^{P}} (uv) = \sum μ_{B_{2}^{P}} (h (u) h (v)) = constant$ as G₂ is regular and $\sum_{u \neq v} μ_{B_{1}^{N}} (uv) = \sum μ_{B_{2}^{N}} (h (u) h (v)) = constant$ as G₂ is regular. Therefore, G₁ is regular. □

Theorem 3.19. Let G₁ and G₂ be two isomorphic bipolar fuzzy graphs, then G₁ is strong regular if and only if G₂ is strong regular.

Proof. As a bipolar fuzzy graph G₁ is isomorphic with bipolar fuzzy graph G₂, then there exists an isomorphism h : G₁ → G₂ which is bijective and satisfies $μ_{A_{1}^{P}} (u) = μ_{A_{2}^{P}} (h (u))$ and $μ_{A_{1}^{N}} (u) = μ_{A_{2}^{N}} (h (u))$ , for all u ∈ V₁ and $μ_{B_{1}^{P}} (uv) = μ_{B_{2}^{P}} (h (u) h (v))$ and $μ_{B_{1}^{N}} (uv) = μ_{B_{2}^{N}} (h (u) h (v))$ , for all $uv \in {\tilde{V_{1}}}^{2}$ .

Now, G₁ is strong if and only if $μ_{B_{1}^{P}} (uv) = min (μ_{A_{1}^{P}}$

$(u), μ_{A_{1}^{P}} (v)) and μ_{B_{1}^{N}} (uv) = max (μ_{A_{1}^{N}} (u), μ_{A_{1}^{N}} (v))$ if and only if $μ_{B_{2}^{P}} (h (u) h (v)) = min (μ_{A_{2}^{P}} (h (u)), μ_{A_{2}^{P}}$

(h (v))) and $μ_{B_{2}^{N}} (h (u) h (v)) = max (μ_{A_{2}^{N}} (h (u)), μ_{A_{2}^{N}}$

(h (v))) if and only if G₂ is strong. G₁ is regular if and only if $\sum_{u \neq v, v \in V_{1}} μ_{B_{1}^{P}} (uv) = constant$ , for all u ∈ V₁ and $\sum_{u \neq v, v \in V_{1}} μ_{B_{1}^{N}} (uv) = constant$ , for all u ∈ V₁ if and only if $\sum_{h (u) \neq h (v), h (v) \in V_{2}} μ_{B_{2}^{P}} (h (u) h (v)) = constant$ and $\sum_{h (u) \neq h (v), h (v) \in V_{2}} μ_{B_{2}^{N}} (h (u) h (v)) = constant$ for all h (v) ∈ V₂ if and only if G₂ is regular.

□

Theorem 3.20. A strong bipolar fuzzy graph G is strong regular if and only if its complement strong bipolar fuzzy graph $\bar{G}$ is strong regular.

Proof. According to the definition of complement of a strong bipolar fuzzy graph we have $μ_{A^{P}} = {\bar{μ}}_{A^{P}}$ , $μ_{A^{N}} = {\bar{μ}}_{A^{N}}$ , ${\bar{μ}}_{B^{P}} (uv) = μ_{A^{P}} (u) \land μ_{A^{P}} (v) - μ_{B^{P}} (uv)$ and ${\bar{μ}}_{B^{N}} (uv) = μ_{A^{N}} (u) \lor μ_{A^{N}} (v) - μ_{B^{N}} (uv)$ , for all $u, v \in V, uv \in {\tilde{V}}^{2}$ . G is strong regular if and only if μ_{B
^P} (uv) = min(μ_{A
^P} (u) , μ_{A
^P} (v)) and μ_{B
^N} (uv) = max(μ_{A
^N} (u) , μ_{A
^N} (v)) if and only if ${\bar{μ}}_{B^{P}} (uv) = μ_{A^{P}} (u) \land μ_{A^{P}} (v) - μ_{B^{P}} (uv) = μ_{B^{P}} (uv) - μ_{B^{P}} (uv) = 0$ and ${\bar{μ}}_{B^{N}} (uv) = μ_{A^{N}} (u) \lor μ_{A^{N}} (v) - μ_{B^{N}} (uv) = μ_{B^{N}} (uv) - μ_{B^{N}} (uv) = 0$ . if and only if $\sum {\bar{μ}}_{B^{P}} (uv) = 0$ and $\sum {\bar{μ}}_{B^{N}} (uv) = 0$ if and only if $\bar{G}$ is strong regular. □

Definition 3.21. Let G = (A, B) be a connected bipolar fuzzy graph. Then G is said to be a highly irregular bipolar fuzzy graph if every vertex of G is adjacent to vertices with distinct degrees.

Example 3.22. Consider a bipolar fuzzy graph G such that V = {u₁, u₂, u₃, u₄} and E = {u₁u₂, u₂u₃, u₃u₄,

u₄u₁} (See. Fig. 4).

By routine computations, we have $\begin{matrix} \deg (u_{1}) = (0.3, - 0.5), \deg (u_{2}) = (0.2, - 0.7), \\ \deg (u_{3}) = (0.4, - 0.8), \deg (u_{4}) = (0.5, - 0.6) . \end{matrix}$ We see that every vertex of G is adjacent to vertices with distinct degrees. So, G is highly irregular bipolar fuzzy graph.

Theorem 3.23. For any two isomorphism bipolar fuzzy graphs, their order and size are same.

Proof. If h from G₁ to G₂ is an isomorphism between the bipolar fuzzy graphs G₁ and G₂ with the underlying sets V₁ and V₂, respectively then,

$μ_{A_{1}^{P}} (u_{1}) = μ_{A_{2}^{P}} (h (u_{1})), μ_{A_{1}^{N}} (u_{1}) = μ_{A_{2}^{N}} (h (u_{1})),$

for all u₁ ∈ V₁, $\begin{matrix} μ_{B_{1}^{P}} (u_{1} v_{1}) = μ_{B_{2}^{P}} (h (u_{1}) h (v_{1})), \\ μ_{B_{1}^{N}} (u_{1} v_{1}) = μ_{B_{2}^{N}} (h (u_{1}) h (v_{1})), \\ for all u_{1} \in V_{1} and u_{1} v_{1} \in {\tilde{V_{1}}}^{2} . \end{matrix}$ Hence, we have $\begin{matrix} O (G_{1}) = (\sum_{u_{1} \in V_{1}} μ_{A_{1}^{P}} (u_{1}), \sum_{u_{1} \in V_{1}} μ_{A_{1}^{N}} (u_{1})) \\ = (\sum μ_{A_{2}^{P}} (h (u_{1})), \sum μ_{A_{2}^{N}} (h (u_{1}))) \\ = (\sum_{u_{2} \in V_{2}} μ_{A_{2}^{P}} (u_{2}), \sum_{u_{2} \in V_{2}} μ_{A_{2}^{N}} (u_{2})) = O (G_{2}) . \\ S (G_{1}) = (\sum μ_{B_{1}^{P}} (u_{1} v_{1}), \sum μ_{B_{1}^{N}} (u_{1} v_{1})) \\ = (\sum_{u_{1}, v_{1} \in V_{1}} μ_{B_{2}^{P}} (h (u_{1}) h (v_{1})), \\ \sum_{u_{1}, v_{1} \in V_{1}} μ_{B_{2}^{N}} (h (u_{1}) h (v_{1}))) \\ = (\sum μ_{B_{2}^{P}} (u_{2} v_{2}), \sum μ_{B_{2}^{N}} (u_{2} v_{2})) = S (G_{2}) . \end{matrix}$ □

Theorem 3.24. If G₁ and G₂ are isomorphic bipolar fuzzy graphs then, the degrees of the corresponding vertices u and h (u) are preserved.

Proof. If h : G₁ → G₂ is an isomorphism between the bipolar fuzzy graphs G₁ and G₂ with the underlying sets V₁ and V₂ respectively then, $μ_{B_{1}^{P}} (u_{1} v_{1}) = μ_{B_{2}^{P}} (h (u_{1}) h (v_{1}))$ and $μ_{B_{1}^{N}} (u_{1} v_{1}) = μ_{B_{2}^{N}} (h (u_{1}) h (v_{1}))$ , for all u₁ ∈ V₁ and $u_{1} v_{1} \in {\tilde{V_{1}}}^{2}$ . Therefore, $\begin{matrix} d_{P} (u_{1}) = \sum_{u_{1}, v_{1} \in V_{1}} μ_{B_{1}^{P}} (u_{1} v_{1}) \\ = \sum_{u_{1}, v_{1} \in V_{1}} μ_{B_{2}^{P}} (h (u_{1}) h (v_{1})) = d_{P} (h (u_{1})) \end{matrix}$

$\begin{matrix} d_{N} (u_{1}) = \sum_{u_{1}, v_{1} \in V_{1}} μ_{B_{1}^{N}} (u_{1} v_{1}) \\ = \sum_{u_{1}, v_{1} \in V_{1}} μ_{B_{2}^{N}} (h (u_{1}) h (v_{1})) = d_{N} (h (u_{1})), \end{matrix}$ for all u₁ ∈ V₁. That is, the degrees of the corresponding vertices of G₁ and G₂ are the same. □

4 Conclusion

Graph theory is becoming increasingly significant as it is applied to other areas of mathematics, science, and technology. In computer science, graphs are used to represent networks of communication, data organization, computational devices, and the flow of computation. The bipolar fuzzy sets constitute a generalization of Zadeh’s fuzzy set theory. The bipolar fuzzy models give more precision, flexibility, and compatibility to the system as compared to the classical and fuzzy models. In this paper, we introduced the notion of bipolar f-morphism on bipolar fuzzy graphs and regular bipolar fuzzy graphs. We studied the action of bipolar f-morphism on bipolar fuzzy graphs and derive some elegant results on weak and co-weak isomorphism. Also, we defined d₂-degree and total d₂-degree of a vertex in bipolar fuzzy graphs and studied (2, k)-regularity and totally (2, k)-regularity of bipolar fuzzy graphs.

Acknowledgments

The authors are thankful to all the reviewers, Associate editor, Editor-in-Chief of the journal for their important suggestions to improve the presentation of the paper.

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