A study on regular picture fuzzy graph with applications in communication networks

Abstract

Picture fuzzy graph (PFG) is an extended version of intuitionistic fuzzy graph (IFG) to model the uncertain real world problems, in which IFG may fail to model those problems properly. PFG is more precise, flexible and compatible than IFG to deal the real-life scenarios which consists of information these types: yes, abstain, no and refusal. The main focus of our study is to present the concept of isomorphic PFG, regular PFG (RPFG) and picture fuzzy multigraph. In this paper, we present the notation of RPFG. Many different types of RPFGs such as regular strong PFG, regular complete PFG, complete bipartite PFG and regular complement PFG are introduced. We also describe the concepts of d_n and td_n-degree of a vertex in a RPFG. Based on those two types of degrees, we classify the regularity of PFG into 3 type’s namely, d_n- RPFG, td_n-RPFG and n- highly irregular PFG. Several theorems of those RPFG are presented here. We define the busy vertex and free vertex in a RPFG. We present the notations of μ-complement, homomorphism, isomorphism, weak isomorphism and co weak isomorphism of RPFG. Some significant theorems on isomorphism and μ- complement of RPFG are derived here. We also introduce the notation of picture fuzzy multigraph. We present a mathematical model of communication network and transportation network by using picture fuzzy multigraph and real time data are collected so that the transportation network/communication network can work efficiently.

Keywords

Regular fuzzy graph picture fuzzy set intuitionistic fuzzy graph

1 Introduction

The concept of fuzzy set was presented by Zadeh [57] in 1965. It has been very successfully utilized to model several real life problems, which are generally uncertain. It is an extension of classical set, where each element of the fuzzy set has a membership degree. Since the classical set is based on two membership degrees: "false" (0) and "true"(1). It may not be tackled to handle the uncertainties of the real life problem properly. However, the fuzzy set can allow its elements to have the membership grade within the interval [0,1] for better result, rather than taking only two values: 0 or 1. In a fuzzy set, the membership grade of any object is not equivalent to the probability. The membership grade describes the belongings value of the object to that fuzzy set and it is only one specific value between 0 and 1. However, this type of membership degree is unable to properly cover the uncertainties/vagueness due to insufficient information of the real world scenarios.

In [4], Atanassov described the idea of intuitionistic fuzzy set (IFS) to handle the uncertainties in the real life problems considering an extra membership grade which defined as hesitation margin. IFS is nothing but a generalized version of type 1 fuzzy set (T1FS). The T1FS considers one and only membership grade value of each and every element in the T1FS and the addition of non membership grade and membership grade must be one, while IFS has two independent degree of membership. One is membership grade and another is non membership grade where the main necessary criteria is that sum of two membership grade need to be not bigger than 1. IFS is more efficient comparing type 1 fuzzy set to handle the uncertainty/vagueness due to the hesitation margin. The IFS uses frequently in real life problems, where human perception are involved. Human perception is in generally inaccurate and it is not totally reliable. IFS is mainly used in social network [55], image processing [7], optimization problem [20], machine learning [56], artificial intelligence [43], market future prediction [18] etc.

The idea of degree of neutrality cannot be accepted in IFS theory. However, we need to consider the neutrality degree in many real world problems. For example, in a democratic voting system [14], 20000 people will participate in the voting process. They generally give several types of choice: refusal, no, yes and abstain. The office of election commission will issue 20000 ballot papers and each person can consider almost one ballot paper for selecting his/her choice. The result of the voting process is provided in 4 different groups based on ballot namely "voting for the candidate" (10000), "abstain in vote" (5000), "voting against candidate" (2000) and "refusal of vote" (3000). The person with "abstain in vote" indicates that the ballot paper is a blank paper rejecting both "voting for the candidate" and "voting against candidate" but still considers the vote. The "refusal of vote" means either invalid ballot paper or bypassing the vote. We cannot model this types of real life scenarios by IFS. In this regards, [8], Cuong and Kreinovich presented the idea of Picture fuzzy set (PFS). It is an extension of IFS and type 1 fuzzy set. PFS consists of three membership grades: positive, negative and neutral. The refusal membership grade is calculated based on those three membership values. PFS can deal the complexity and uncertainty of the human perception in practice.

Several research works [9 , 44–53] have been done on the PFS. In [12], the author presented many theorems of PFS and they have presented a ranking method of PFG based on distance measurement. In [29], the Phong el at. introduced many types relationship of PFS. In [11], the authors have presented the complements, disjunctions, implications and conjunctions of PFSs. In [28], the authors have presented a method for PFS and applied it in an optimization problem.

Graph theory has emerged as a mathematical tool to model many real world applications such as telecommunication, financial system analysis, operation research, supply chain management, traffic planning, computer wireless network, traveling sell man problem and scheduling problem. However, the real world problems are generally uncertain and it is always challenging to model those problems using graph. Rosenfeld [37] presented the graph theory in fuzzy environment. This type of graph is called fuzzy graph. It is very useful to model the uncertain decision making problems and it provides more flexible and compatible environment to deal with real life problem than crisp graph theory. Atanassov [42] described the concepts of relationship between two IFSs. Based on this relationship, they have introduced the intuitionistic fuzzy graph (IFG) and presented many proprieties and theorems in [42]. Parvathi et al. [25 –27] presented some different products between two IFGs. Rashmanlou et al [35] presented several type of products (lexicographic, direct, and strong) between two IFGs. They also presented the idea of Cartesian production, join, composition and union of 2 IFGs. For details study on IFG, please read the papers [1 , 38].

For any node or arc in a classical graph, there are two possibilities: one is either in the graph or it is not the graph. Therefore, classical graph cannot model uncertain optimization problem. Since the real world problems are generally very uncertain and it is difficult to model the uncertain real life problems using classical graph. Fuzzy set [57] is an extended version of classical set, where the objects have varying membership degree. A fuzzy set gives its objects to have different membership degrees between 0 and 1. The membership degree is not same as probability; rather it describes membership in vaguely defined sets. The concept of fuzziness in graph theory has been described by Kaufmann [19] using the fuzzy relation. Rosenfeld [37] introduced some concept like bridges, cycles, paths, trees and connectedness of fuzzy graph and presented many concepts of fuzzy graph. Gani and Radha [16] have presented the notation of regular fuzzy graph. Many other researchers, such as Samanta and Pal [41], Rashmanlou and Pal [32], Rashmanlou et al. [33, 36], Paramik [30], Nandhini [24], Ghorai and Pal [17] and Borzooei [5] have done a lot of work in the domain of fuzzy graph and several applications in real life. Samanta and Pal [41] and Rashmanlou and Pal [6] presented the concept of irregular and regular fuzzy graph. They have also described some applications of those graphs.

The degree of nodes provides an important way to describe the relationship among the nodes in a network/graph, and we can also use the node degree to analyze the network. In [15], the authors presented the definition of totally irregularity, total degree and irregularity of a vertex in a type-1 network. Maheswari and Sekar [22] presented the idea of d₂- node of a fuzzy network. They introduced some properties and theorems on d₂- node degree in a fuzzy network. Darabian et al. [13] have described some definitions of regular vague graph such as d_m regular, m- highly irregular, td_m regular, and m- highly total regular. Several applications (e.g., wireless networks, fullerene molecules, and road networks) of regular vague graph were presented in this research. PFG is very precise, effective and flexible to deal the uncertainties in decision making problem.

Fuzzy graph emerges as a mathematical model of many decision making problems. A number of extensions of fuzzy graph [2 , 40] has been proposed to handle the uncertainty of the complex decision making problem. Human perception cannot be confined to yes (true) or no (false). However, our opinion may be yes, no, abstain, and refusal. Although, the theoretical concept of fuzzy graph is still not sufficient to deal such kind of real life scenarios in a broad manner. PFS [52, 54] can be used to deal this type of uncertainties. Based on picture fuzzy relation, PFG would be very prominent research direction in the domain of fuzzy graph theory. This motivated us to work on PFG. The novelty of this paper is described as follows:

To the extent of our knowledge, no researcher has worked on the regular PFG still now. In this study, we present the first time the notation of regular PFG, star PFG, regular strong PFG, and complete bipartite PFG.

We present two different types of degrees, (total d_m and d_m) of a node in a PFG. We have also presented the idea of busy node and free node in a regular PFG here.

Based on the degree value of the vertex, we classify three different types of regular PFG such as td_m-regular, m- highly irregular PFG and d_m-regular. Some properties and theorems of those regular PFG are also described in this manuscript.

The basic idea of isomorphic and complement of a PFG are described in this manuscript. We also describe the notation of h-morphism and μ-complement a PFG. We provide several theorems of complement and isomorphism of PFG here.

We also describe an applications of PFG in communication networks.

The rest of the paper is as follows: Section 2 gives the preliminary. Section 3 presents new notions of regular PFG. Section 4 studies the regularity on complement and isomorphic PFG. Section 5 defines Complete bipartite PFG. Section 6 shows an application of Picture Fuzzy Multigraph in Communication Networks. Lastly, Section 7 shows the conclusions.

2 Preliminaries

We will describe about PFG, picture fuzzy path, picture fuzzy adjacent node, picture fuzzy isolated node, strength of a picture fuzzy path, strong PFG, complement PFG and complete PFG in this section.

Definition 2.1. A PFS [8] P on an universe set X is an element of the form P = {p, μ_P (x) , η_P (x) , ν_P (x) |x ∈ X}. Here, μ_P (x) ∈ [0, 1] is the positive membership grade of x in P, η_P (x) ∈ [0, 1] is the neutral membership grade x in P, and ν_P (x) ∈ [0, 1] is the negative membership grade x in P. The PFS P must satisfy μ_P (x) + η_P (x) + ν_P (x) ≤1. The refusal membership grade equals to (1 - (μ_P (x) + η_P (x) + ν_P (x))).

Definition 2.2. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a PFG [3, 58] where $\tilde{X}$ and $\tilde{Y}$ are considered two PFSs on V and E which fulfill the following equations.

$μ_{\tilde{Y}} (i, j) \leq min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j))$

$η_{\tilde{Y}} (i, j) \geq max (η_{\tilde{X}} (i), η_{\tilde{X}} (j))$

$ν_{\tilde{Y}} (i, j) \geq max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j))$

Here, j and k are 2 nodes of $\tilde{G}$ and (i, j) ∈ E is an arc in the PFG G.

Definition 2.3. The two nodes i and j of a PFG $\tilde{G}$ are called picture fuzzy adjacent nodes if and only if

$(μ_{\tilde{X}} (i, j), η_{\tilde{Y}} (i, j), ν_{\tilde{Y}} (i, j)) > 0$

The picture fuzzy nodes i and j are said to be neighbor node.

Definition 2.4. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ is a PFG [3, 58] and P is a picture fuzzy path of PFG G. The picture fuzzy path P is a set of some different picture fuzzy nodes, j₀, j₁, j₂, . . . , j_n such that

(μ_M (j_m-1, j_m) , η_M (j_m-1, j_m) , ν_M (j_m-1, j_m)) > 0

The length of picture fuzzy path P is l. A single picture fuzzy node, i.e., j_m is also considered as a picture fuzzy path and the length of this path is (0, 0, 0). The picture fuzzy path is is a picture fuzzy cycle if j₀ = j_l and l ≥ 3.

Definition 2.5. A picture fuzzy node j_m ∈ V is said to be isolated picture fuzzy node if there is no incident picture fuzzy edge to the node j_m.

Definition 2.6. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ is a PFG. If the PFG G have a picture fuzzy path P with length picture fuzzy length l between vertex j and k such as P = {j = m₁, (m₁, m₂) , m₂, . . . , m_l-1 (m_l-1, m_l) , a_l = k)} then $μ_{\tilde{Y}}^{l} (i, j)$ , $η_{\tilde{Y}}^{l} (i, j)$ and $ν_{\tilde{Y}}^{l} (i, j)$ are described as follows.

$μ_{\tilde{Y}}^{l} (i, j) =$ $sup (μ_{\tilde{Y}} (j, m_{1}) \land μ_{\tilde{Y}} (m_{1}, m_{2}) \land . . μ_{\tilde{Y}} (m_{k - 1}, m_{k}))$

$η_{\tilde{Y}}^{l} (i, j) =$ $inf (η_{\tilde{Y}} (j, m_{1}) \lor η_{\tilde{Y}} (m_{1}, m_{2}) \lor . . η_{\tilde{Y}} (m_{k - 1}, m_{k}))$

$ν_{\tilde{Y}}^{l} (i, j) =$ $inf (ν_{\tilde{Y}} (a, m_{1}) \lor ν_{\tilde{Y}} (m_{1}, m_{2}) \lor . . ν_{\tilde{Y}} (m_{k - 1}, m_{k}))$

Definition 2.7. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a PFG. We can find the strength of a picture fuzzy path from the node j and k using $(μ_{\tilde{Y}}^{\infty} (i, j), η_{\tilde{Y}}^{\infty} (i, j), ν_{\tilde{Y}}^{\infty} (i, j))$ as follows.

$μ_{\tilde{Y}}^{\infty} (i, j) = sup {μ_{\tilde{Y}}^{k} (i, j) | l = 1, 2, . . .}$

$η_{\tilde{Y}}^{\infty} (i, j) = inf {η_{\tilde{Y}}^{k} (i, j) | l = 1, 2, . . .}$

$ν_{\tilde{Y}}^{\infty} (i, j) = inf {ν_{B}^{k} (i, j) | l = 1, 2, . . .}$

Definition 2.8. The PFG is a connected PFG if there is no isolated picture fuzzy node is in that PFG.

Definition 2.9. The picture fuzzy degree of a node is the picture fuzzy addition of the degree value of true memberships, picture fuzzy addition of the degree value of indeterminacy memberships and picture fuzzy addition of the degree value of false memberships of each and every the incident edges to the vertex x. The picture fuzzy degree of picture fuzzy vertex x is described by d (x) = (d_μ (x) , d_η (x) , d_ν (x)) where $\begin{matrix} d_{μ} (x) = \sum_{x \in V x \neq k} μ_{\tilde{Y}} (x, k), \\ d_{η} (x) = \sum_{x \in V x \neq k} η_{\tilde{Y}} (x, k), \\ d_{ν} (i) = \sum_{x \in V i \neq j} ν_{\tilde{Y}} (x, k) \end{matrix}$ Here, d_μ (x), d_η (x) and d_ν (x) are used to denote the degree value of true membership, value of indeterminacy membership degree and degree value of false membership severally of the node x.

Definition 2.10. The picture fuzzy order of a PFG $\tilde{G}$ is represented by $O (\tilde{G}) = (O_{μ} (\tilde{G}), O_{η} (\tilde{G}), O_{ν} (\tilde{G}))$ where $\begin{matrix} O_{μ} (\tilde{G}) = \sum_{j \in V} μ_{\tilde{X}} (i), \\ O_{η} (\tilde{G}) = \sum_{j \in V} η_{\tilde{X}} (i), \\ O_{ν} (\tilde{G}) = \sum_{j \in V} ν_{\tilde{X}} (i) \end{matrix}$ Here, $O_{μ} (\tilde{G})$ is the order value of true membership degree, $O_{η} (\tilde{G})$ is the order value of indeterminacy membership degree and $O_{ν} (\tilde{G})$ is the order value of false membership degree.

Definition 2.11. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ is PFG. The picture fuzzy size of PFG $\tilde{G}$ is represented as $S (\tilde{G}) = (S_{μ} (\tilde{G}), S_{η} (\tilde{G}), S_{ν} (\tilde{G}))$ where $\begin{matrix} S_{μ} (\tilde{G}) = \sum_{j, i \in V j \neq i} μ_{\tilde{Y}} (j, i), \\ S_{η} (\tilde{G}) = \sum_{j, i \in V j \neq i} η_{\tilde{Y}} (j, i), \\ S_{ν} (\tilde{G}) = \sum_{j, i \in V j \neq i} ν_{\tilde{Y}} (j, i) \end{matrix}$ Here, $S_{μ} (\tilde{G})$ , $S_{η} (\tilde{G})$ and $S_{ν} (\tilde{G})$ are respectively the size of the membership grade of the true, indeterminacy and false of $\tilde{G}$ .

Definition 2.12. The PFG $\tilde{G}$ is a strong PFG if

$μ_{\tilde{Y}} (i, j) = min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j))$

$η_{\tilde{Y}} (i, j) = max (η_{\tilde{X}} (i), η_{\tilde{X}} (j))$

$ν_{\tilde{Y}} (i, j) = max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j))$ , ∀ (i, j) ∈ E

Definition 2.13. Let $G = (\tilde{X}, \tilde{Y})$ is a PFG. The PFG $\tilde{G}$ is a complete PFG [3, 58] if

$μ_{\tilde{Y}} (i, j) = min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j))$

$η_{\tilde{Y}} (i, j) = max (η_{\tilde{X}} (i), η_{\tilde{X}} (j))$

$ν_{\tilde{Y}} (i, j) = max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j))$ , ∀i, j ∈ V

Definition 2.14. Let $G = (\tilde{X}, \tilde{Y})$ is a PFG. The $G^{c} = ({\tilde{X}}^{c}, {\tilde{Y}}^{c})$ is a complement PFG [3, 58] if ${\tilde{X}}^{c} = \tilde{X}$ and ${\tilde{Y}}^{c}$ is calculated as follows.

$μ_{\tilde{Y}}^{c} (i, j) = min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j)) - μ_{\tilde{X}} (i, j)$

$η_{\tilde{Y}}^{c} (i, j) = max (η_{\tilde{X}} (i), η_{\tilde{X}} (j)) - η_{\tilde{Y}} (i, j)$

$ν_{\tilde{Y}}^{c} (i, j) = max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j)) - ν_{\tilde{Y}} (i, j)$ , ∀a, b ∈ V

Here, $μ_{\tilde{Y}}^{c} (i, j)$ , $η_{\tilde{Y}}^{c} (i, j)$ and $ν_{\tilde{Y}}^{c} (i, j)$ are represented the true membership, intermediate membership and false membership grade for the edge (i, j) of the PFG of G^c.

3 Regular, d_m-regular and td_m-regular PFG

In this section, first we define regular PFG, regular strong PFG, d_m-degree and td_m-degree of nodes in a PFG. Then, we propose the notions of d_m and td_m-regular PFGs and prove the necessary and sufficient conditions which under this conditions the d_m-regular with td_m-regular PFG is equivalent.

Definition 3.1. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a PFG. $\tilde{G}$ is a regular PFG if $\begin{matrix} \sum_{i \neq j} μ_{\tilde{X}} (i, j) = Constant, \\ \sum_{i \neq j} η_{\tilde{Y}} (i, j) = Constant, \\ \sum_{i \neq j} ν_{\tilde{Y}} (i, j) = Constant \end{matrix}$

Definition 3.2. The PFG $\tilde{G}$ is a complete regular PFG if and only satisfies the following three conditions for each and every vertices i, j ∈ V in $\tilde{G}$ .

$\begin{matrix} μ_{\tilde{X}} (i, j) = min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j)) and \\ \sum_{i \neq j} μ_{\tilde{Y}} (i, j) = Constant \\ η_{\tilde{Y}} (i, j) = max (η_{\tilde{X}} (i), η_{\tilde{X}} (j)) and \\ \sum_{i \neq j} η_{\tilde{Y}} (i, j) = Constant \\ ν_{\tilde{Y}} (i, j) = max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j)) and \\ \sum_{i \neq j} ν_{\tilde{Y}} (i, j) = Constant \end{matrix}$

Definition 3.3. The PFG $\tilde{G}$ is a strong regular PFG if and only satisfies the following three conditions for each and every arc i, j ∈ E in $\tilde{G}$ .

Definition 3.4. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a PFG. ${\tilde{G}}^{c} = ({\tilde{X}}^{c}, {\tilde{Y}}^{c})$ is said to be complement regular PFG if and only satisfies the following three conditions for each and every arc i, j ∈ E in $\tilde{G}$ .

$\begin{matrix} μ_{\tilde{Y}}^{c} (i, j) = min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j)) - μ_{\tilde{Y}} (i, j) and \\ \sum_{i \neq j} μ_{\tilde{Y}}^{c} (i, j) = Constant \\ η_{\tilde{Y}}^{c} (i, j) = max (η_{\tilde{X}} (i), η_{\tilde{X}} (j)) - η_{\tilde{Y}} (i, j) and \\ \sum_{i \neq j} η_{\tilde{Y}}^{c} (i, j) = Constant \\ ν_{\tilde{Y}}^{c} (i, j) = max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j)) - ν_{\tilde{Y}} (i, j) and \\ \sum_{i \neq j} ν_{\tilde{Y}}^{c} (i, j) = Constant \end{matrix}$

Definition 3.5. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ represents a PFG and the d_n degree of a particular vertex i in $\tilde{G}$ is symbolized as d_n (j) where $\begin{matrix} d_{n} (i) = & (\sum_{i \neq j \in V} μ_{Y}^{n} (i, j), \sum_{i \neq j \in V} η_{Y}^{n} (i, j), \\ \sum_{i \neq j \in V} ν_{Y}^{n} (i, j)) \end{matrix}$ The path {j, j₁, j₂, . . . , j_n-1, k} is used to represent the shortest path between the two vertices i and j and n is the path length of this shortest path.

Example 3.1. We have considered an Example of a PFG G, presented in Fig. 1.

Fig. 1

A PFG G.

Then, the d₂ degree of the nodes in G are computed as follows.

d₂ (a) = (0.1 + 0.1, 0.1 + 0.2, 0.1 + 0.1)

=(0.2, 0.3, 0.3)

d₂ (b) = (0.1 + 0.1, 0.1 + 0.1, 0.1 + 0.2)

=(0.2, 0.2, 0.3)

d₂ (c) = (0.1 + 0.1, 0.1 + 0.2, 0.1 + 0.1)

=(0.2, 0.3, 0.2)

Definition 3.6. The total d_n-degree of the vertex i is represented by td_n-degree and the td_n-degree is described as follows. $\begin{matrix} {td}_{n} (i) = (\sum_{i \neq j \in V} μ_{\tilde{Y}}^{n} (i, j) + μ_{\tilde{X}} (i)), \\ (\sum_{i \neq j \in V} η_{\tilde{Y}}^{n} (i, j) + η_{\tilde{X}} (i)), \\ (\sum_{i \neq j \in V} μ_{\tilde{Y}}^{n} (i, j) + ν_{\tilde{X}} (i)) \end{matrix}$

Example 3.2. Let us consider an example of a PFG, shown in Fig. 1. Then the td₂-degree of the nodes in G as follows.

td₂ (a) = ((0.1 + 0.1) + 0.1) , ((0.1 + 0.2) + 0.1) ,

((0.1 + 0.2) + 0.1) = (0.3, 0.4, 0.3)

td₂ (b) = ((0.1 + 0.1) + 0.1) , ((0.1 + 0.1) + 0.1) ,

((0.1 + 0.2) + 0.1) = (0.3, 0.3, 0.4)

td₂ (a) = ((0.1 + 0.1) + 0.1) , ((0.1 + 0.2) + 0.1) ,

((0.1 + 0.1) + 0.1) = (0.3, 0.4, 0.3)

Definition 3.7. The PFG $\tilde{G}$ is called (n, (y₁, b₂, b₃))- regular PFG or d_n regular PFG if and only if for each and every vertices j ∈ V of $\tilde{G}$ , d_n (j) = (y₁, b₂, b₃).

Definition 3.8. The PFG $\tilde{G}$ is called (n, (c₁, c₂, c₃))- total regular PFG or td_n regular PFG if and only if for each and every vertices j ∈ V of $\tilde{G}$ , d_n (j) = (c₁, c₂, ₃).

Definition 3.9. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a regular PFG and j ∈ V is a specific vertex of G. The picture fuzzy value of node j is represented by D (j) = (D_μ (i) , D_η (i) , D_ν (i)), where $\begin{matrix} D_{μ} (i) = \sum min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j_{i})), \\ D_{η} (i) = \sum max (η_{\tilde{X}} (i), η_{\tilde{X}} (j_{i})) \\ D_{ν} (i) = \sum max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j_{i})) \end{matrix}$

Here, j_i is used to represent the neighboring vertex of j of regular PFG $\tilde{G}$ .

Definition 3.10. The busy value of a RPFG $\tilde{G}$ is a value of the sum of each and every busy values of all the vertices in graph G.

Definition 3.11. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a regular PFG and j ∈ V is a specific vertex of G. The node j is said to be a picture busy vertex if $μ_{X} (j) \leq d_{μ} (i), η_{X} (j) \leq d_{η} (i), ν_{X} (j) \geq d_{ν} (i)$

The vertex j is said to be free vertex if and only if vertex j is not a busy vertex.

Definition 3.12. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a regular PFG and {i, j} ∈ E is a specific arc of G. The arc {i, j} is said to be a strong picture fuzzy arc if and only if $μ_{\tilde{Y}} (ij) \geq μ_{\tilde{Y}}^{\infty} (ij), η_{\tilde{Y}} (ij) \leq η_{\tilde{Y}}^{\infty} (ij), ν_{\tilde{Y}} (ij) \leq ν_{\tilde{Y}}^{\infty} (ij)$

Definition 3.13. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a regular PFG and {i, j} ∈ E is a specific arc of G. The arc {i, j} is said to be a effective picture fuzzy arc if and only if $\begin{matrix} μ_{\tilde{Y}} (i, j) = min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j)), \\ η_{\tilde{Y}} (i, j) = max (η_{\tilde{X}} (i), η_{\tilde{X}} (j)), \\ ν_{\tilde{Y}} (i, j) = max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j)) \end{matrix}$

4 Regularity on complement and isomorphic PFG

Definition 4.1. Let ${\tilde{G}}_{1} = ({\tilde{X}}_{1}, {\tilde{Y}}_{1})$ and ${\tilde{G}}_{2} = ({\tilde{X}}_{2}, {\tilde{Y}}_{2})$ are two RPFGs on G₁ = (V₁, E₁) and G₂ = (V₂, E₂).

Let h : V₁ → V₂ be a bijective picture fuzzy function. The picture fuzzy function h is said to be picture fuzzy morphism or picture fuzzy h morphism if satisfies the following two conditions:

∀x₁ ∈ V₁, x₁y₁ ∈ E₁ and the k₁ and k₂ are two positive numbers.

$μ_{{\tilde{X}}_{2}} (h (x_{1})) = k_{1} μ_{{\tilde{X}}_{1}} (x_{1})$

$η_{{\tilde{X}}_{2}} (h (x_{1})) = k_{1} η_{{\tilde{X}}_{1}} (x_{1}),$

$ν_{{\tilde{X}}_{2}} (h (x_{1})) = k_{1} ν_{{\tilde{X}}_{1}} (x_{1})$

$μ_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = k_{2} μ_{{\tilde{Y}}_{1}} (x_{1} y_{1})$

$η_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = k_{2} η_{{\tilde{Y}}_{1}} (x_{1} y_{1})$

$ν_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = k_{2} ν_{{\tilde{Y}}_{1}} (x_{1} y_{1})$

This function h can be named as a (k₁, k₂) picture fuzzy h morphism between ${\tilde{G}}_{1}$ to ${\tilde{G}}_{1}$ .

Let h : V₁ → V₂ be a bijective picture fuzzy function. The picture fuzzy function h is said to be a picture fuzzy isomorphism between a RPFG ${\tilde{G}}_{1}$ and an other RPFG ${\tilde{G}}_{2}$ if satisfies the following two conditions:

∀x₁ ∈ V₁, x₁y₁ ∈ E₁

$μ_{{\tilde{X}}_{2}} (h (x_{1})) = μ_{{\tilde{X}}_{1}} (x_{1}),$

$η_{{\tilde{X}}_{2}} (h (x_{1})) = η_{{\tilde{X}}_{1}} (x_{1}),$

$ν_{{\tilde{X}}_{2}} (h (x_{1})) =$

$ν_{{\tilde{X}}_{1}} (x_{1})$

$μ_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = μ_{{\tilde{Y}}_{1}} (x_{1} y_{1})$ ,

$η_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = η_{{\tilde{Y}}_{1}} (x_{1} y_{1}),$

$ν_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = ν_{{\tilde{Y}}_{1}} (x_{1} y_{1})$

The picture fuzzy function h is said to be weak isomorphism a RPFG ${\tilde{G}}_{1}$ and an other RPFG ${\tilde{G}}_{2}$ if satisfies the following two conditions:

∀x₁ ∈ V₁

h is picture fuzzy morphism or picture fuzzy h morphism.

$μ_{{\tilde{X}}_{2}} (h (x_{1})) = μ_{{\tilde{X}}_{1}} (x_{1}),$

$η_{{\tilde{X}}_{2}} (h (x_{1})) = η_{{\tilde{X}}_{1}} (x_{1}),$

$ν_{{\tilde{X}}_{2}} (h (x_{1})) = ν_{{\tilde{X}}_{1}} (x_{1})$ .

The picture fuzzy function h is said to be co weak isomorphism between a RPFG ${\tilde{G}}_{1}$ and an other RPFG ${\tilde{G}}_{2}$ if satisfies the following two conditions:

∀x₁y₁ ∈ E₁

h is picture fuzzy morphism or picture fuzzy h morphism.

$μ_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = μ_{{\tilde{Y}}_{1}} (x_{1} y_{1})$ ,

$η_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = η_{{\tilde{Y}}_{1}} (x_{1} y_{1}),$

$ν_{{\tilde{Y}}_{2}} (h (x_{1}) h (y_{1})) = ν_{{\tilde{Y}}_{1}} (x_{1} y_{1})$

Theorem 4.1. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ are two RPFGs. The RPFG ${\tilde{G}}_{1}$ is (k₁, k₂) picture fuzzy morphism to RPFG ${\tilde{G}}_{2}$ . The strong arc (x₁y₁) in ${\tilde{G}}_{1}$ is also strong arc in ${\tilde{G}}_{2}$ if the values of k₁ and k₂ are equal.

Proof. The (x₁y₁) is a strong arc in ${\tilde{G}}_{1}$ such that h (x₁), h (₁y) is also strong arc in ${\tilde{G}}_{2}$ . Now,

$k_{2} μ_{\tilde{Y_{1}}} (x_{1} y_{1}) = μ_{\tilde{Y_{1}}} (h (x_{1}) h (y_{1}))$

$= μ_{\tilde{X_{2}}} (h (x_{1}) \land h (y_{1})) = k_{1} μ_{\tilde{X_{1}}} (x_{1}) \land k_{1} μ_{\tilde{X_{1}}} (y_{1})$

$= k_{1} μ_{\tilde{Y_{1}}} (x_{1} y_{1}) \forall x_{1} \in V_{1}$ .

Hence, $k_{2} μ_{\tilde{Y_{1}}} (x_{1} y_{1}) = k_{1} μ_{\tilde{Y_{1}}} (x_{1} y_{1})$ , ∀x₁ ∈ V₁.

$k_{2} η_{\tilde{Y_{1}}} (x_{1} y_{1}) = η_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1}))$

$= η_{\tilde{X_{2}}} (h (x_{1}) \lor h (y_{1})) = k_{1} η_{\tilde{X_{1}}} (x_{1}) \lor k_{1} η_{\tilde{X_{1}}} (y_{1})$ = $k_{1} η_{\tilde{Y_{1}}} (x_{1} y_{1})$ , ∀x₁ ∈ V₁.

Hence $k_{2} η_{\tilde{Y_{1}}} (x_{1} y_{1}) = k_{1} η_{\tilde{Y_{1}}} (x_{1} y_{1})$ , ∀x₁ ∈ V₁.

$k_{2} ν_{\tilde{Y_{1}}} (x_{1} y_{1}) = ν_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1}))$ = $ν_{\tilde{X_{2}}} (h (x_{1}) \lor h (y_{1})) = k_{1} ν_{\tilde{X_{1}}} (x_{1}) \lor k_{1} η_{\tilde{X_{1}}} (y_{1})$ = $k_{1} ν_{\tilde{Y_{1}}} (x_{1} y_{1}) \forall x_{1} \in V_{1}$ .

Hence $k_{2} ν_{\tilde{Y_{1}}} (x_{1} y_{1}) = k_{1} ν_{\tilde{Y_{1}}} (x_{1} y_{1})$ , ∀x₁ ∈ V₁. The strong arc (x₁y₁) in ${\tilde{G}}_{1}$ is also strong arc in ${\tilde{G}}_{2}$ if the values of k₁ and k₂ are equal.□

Theorem 4.2. Let ${\tilde{G}}_{1}$ is a RPFG. If ${\tilde{G}}_{1}$ is co weak picture fuzzy isomorphic to ${\tilde{G}}_{2}$ then ${\tilde{G}}_{1}$ is also regular.

Proof. If ${\tilde{G}}_{1}$ is co weak picture fuzzy isomorphic to ${\tilde{G}}_{2}$ then it satisfies $μ_{\tilde{X_{1}}} (x_{1}) \leq μ_{\tilde{X_{2}}} (h (x_{1}))$ , $η_{\tilde{X_{1}}} (x_{1}) \geq η_{\tilde{X_{2}}} (h (x_{1}))$ , $ν_{\tilde{X_{1}}} (x_{1}) \geq ν_{\tilde{X_{2}}} (h (x_{1}))$ , $μ_{\tilde{Y_{1}}} (x_{1} y_{1}) = μ_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1}))$ , $η_{\tilde{Y_{1}}} (x_{1} y_{1}) = η_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1}))$ , and $ν_{\tilde{Y_{1}}} (x_{1} y_{1}) = ν_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1}))$ ∀ x₁, y₁ ∈ V₁. ${\tilde{G}}_{1}$ is a RPFG, for u ∈ v, $\sum_{x_{1} \neq y_{1}, y_{1} \in V_{1}} μ_{\tilde{Y_{1}}} (x_{1} y_{1}) = constant$ , $\sum_{x_{1} \neq y_{1}, y_{1} \in V_{1}} η_{\tilde{Y_{1}}} (x_{1} y_{1}) = constant$ and $\sum_{x_{1} \neq y_{1}, y_{1} \in V_{1}} ν_{\tilde{Y_{1}}} (x_{1} y_{1}) = constant$ . $\begin{matrix} \sum_{h (x_{1}) \neq h (y_{1})} μ_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})) = \\ \sum_{x_{1} \neq y_{1}, y_{1} \in V_{1}} μ_{\tilde{Y_{1}}} (uv) = constant . \\ \sum_{h (x_{1}) \neq h (y_{1})} η_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})) = \\ \sum_{x_{1} \neq y_{1}, y_{1} \in V_{1}} η_{\tilde{Y_{1}}} (uv) = constant . \\ \sum_{h (x_{1}) \neq h (y_{1})} ν_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})) = \\ \sum_{x_{1} \neq y_{1}, y_{1} \in V_{1}} ν_{\tilde{Y_{1}}} (uv) = constant . \end{matrix}$ It proves that ${\tilde{G}}_{2}$ is a RPFG.□

Definition 4.2. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a PFG. $\tilde{G}$ G is called a highly irregular PFG (HIPFG) if every node of $\tilde{G}$ is connected with other nodes with different picture fuzzy degrees.

Theorem 4.3. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two isomorphic HIPFGs. Then the order and size of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ are always equal.

Proof. If ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two isomorphic HIPFGs then

∀x₁ ∈ V₁, x₁y₁ ∈ E₁

$μ_{\tilde{X_{1}}} (x_{1}) = μ_{\tilde{X_{2}}} (h (x_{1})),$

$η_{\tilde{X_{1}}} (x_{1}) = η_{\tilde{X_{2}}} (h (x_{1})),$

$ν_{\tilde{X_{1}}} (x_{1}) = ν_{\tilde{X_{2}}} (h (x_{1}))$

$μ_{\tilde{Y_{1}}} (x_{1} y_{1}) = μ_{B_{2}} (h (x_{1}) h (y_{1})), η_{\tilde{Y_{1}}} (x_{1} y_{1}) = η_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})), ν_{\tilde{Y_{1}}} (x_{1} y_{1}) = ν_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1}))$

So, we have

$\begin{matrix} S ({\tilde{G}}_{1}) & = \sum_{x_{1} y_{1} \in E_{1}} μ_{\tilde{Y_{1}}} (x_{1} y_{1}), \sum_{x_{1} y_{1} \in E_{1}} η_{\tilde{Y_{1}}} (x_{1} y_{1}), \\ \sum_{x_{1} y_{1} \in E_{1}} ν_{\tilde{Y_{1}}} (x_{1} y_{1}) \\ = \sum_{x_{1}, y_{1} \in V_{1}} μ_{\tilde{Y_{2}}} (h (x_{1}), h (y_{1})), \\ \sum_{x_{1}, y_{1} \in V_{1}} η_{\tilde{Y_{2}}} (h (x_{1}), h (y_{1})), \\ \sum_{x_{1} y_{1} \in V_{1}} ν_{\tilde{Y_{2}}} (h (x_{1}), h (y_{1})) \\ = \sum_{x_{2} y_{2} \in E_{2}} μ_{\tilde{Y_{2}}} (x_{2} y_{2}), \sum_{x_{2} y_{2} \in E_{2}} η_{\tilde{Y_{2}}} (x_{2} y_{2}), \\ \sum_{x_{2} y_{2} \in E_{2}} ν_{\tilde{Y_{2}}} (x_{2} y_{2}) = S ({\tilde{G}}_{2}) \end{matrix}$ $\begin{matrix} O ({\tilde{G}}_{1}) & = \sum_{x_{1} \in V_{1}} μ_{\tilde{X_{1}}} (x_{1}), \sum_{x_{1} \in V_{1}} η_{\tilde{X_{1}}} (x_{1}), \\ \sum_{x_{1} \in V_{1}} ν_{\tilde{X_{1}}} (x_{1}) \\ = \sum_{x_{1} \in V_{1}} μ_{\tilde{X_{2}}} (h (x_{1})), \sum_{x_{1} \in V_{1}} η_{\tilde{X_{2}}} (h (x_{1})), \\ \sum_{x_{1} \in V_{1}} ν_{\tilde{X_{2}}} (h (x_{1})) \\ = \sum_{x_{2} \in V_{2}} μ_{\tilde{X_{2}}} (x_{2}), \sum_{x_{2} \in V_{2}} η_{\tilde{X_{2}}} (x_{2}), \\ \sum_{x_{2} \in V_{2}} ν_{\tilde{X_{2}}} (x_{2}) = O ({\tilde{G}}_{2}) \end{matrix}$

□

Theorem 4.4. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ be two isomorphic HIPFGs. Then the picture fuzzy degrees of vertices a and h (a) of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ are always same.

Proof. The ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ are two isomorphic HIPFGs. Then it satisfies the following conditions.

$μ_{\tilde{Y_{1}}} (x_{1} y_{1}) = μ_{\tilde{Y_{1}}} (h (x_{1}) h (y_{1}))$ ,

$η_{\tilde{Y_{1}}} (x_{1} y_{1}) = η_{\tilde{Y_{1}}} (h (x_{1}) h (y_{1}))$ and

$ν_{\tilde{Y_{1}}} (x_{1} y_{1}) = ν_{\tilde{Y_{1}}} (h (x_{1}) h (y_{1}))$ for all x₁, y₁ ∈ V₁.

Therefore, $\begin{matrix} d_{μ} (x_{1}) & = \sum_{x_{1}, y_{1} \in V_{1}} μ_{\tilde{Y_{1}}} (x_{1} y_{1}) \\ = \sum_{x_{1}, y_{1} \in V_{1}} μ_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})) = d_{μ} (h (x_{1})) \\ d_{η} (x_{1}) & = \sum_{x_{1}, y_{1} \in V_{1}} η_{\tilde{Y_{1}}} (x_{1} y_{1}) \\ = \sum_{x_{1}, y_{1} \in V_{1}} η_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})) = d_{η} (h (x_{1})) \\ d_{ν} (x_{1}) & = \sum_{x_{1}, y_{1} \in V_{1}} ν_{\tilde{Y_{1}}} (x_{1} y_{1}) \\ = \sum_{x_{1}, y_{1} \in V_{1}} ν_{\tilde{Y_{2}}} (h (x_{1}) h (y_{1})) = d_{n} u (h (x_{1})) \end{matrix}$ for all x₁ ∈ V₁. So, the degree of two vertices x₁ and h (x₁) of ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ are always same.□

Definition 4.3. Let $\tilde{G}$ be a connected PFG. G can be described as

Self co weak picture fuzzy complementary isomorphic graph if $\tilde{G}$ is co weak isomorphic with ${\tilde{G}}^{c}$ .

Self complementary PFG if ${\tilde{G}}^{c} ≅ \tilde{G}$ .

Theorem 4.5. Let $\tilde{G}$ be a self complementary co weak HIPFG. Then $\begin{matrix} \sum_{i \neq j} μ_{\tilde{Y}} (i, j) & \leq \frac{1}{2} \sum_{i \neq j} min (μ_{\tilde{X}} (i), μ_{\tilde{X}} (j))) \\ \sum_{i \neq j} η_{\tilde{Y}} (i, j) & \geq \frac{1}{2} \sum_{i \neq j} max (η_{\tilde{X}} (i), η_{\tilde{X}} (j))) \\ \sum_{i \neq j} ν_{\tilde{Y}} (i, j) & \geq \frac{1}{2} \sum_{i \neq j} max (ν_{\tilde{X}} (i), ν_{\tilde{X}} (j))) \end{matrix}$

Proof. Let G = (A, B) be a self complementary co weak HIPFG. If there is a weak isomorphism $h : \tilde{G} \to {\tilde{G}}^{c}$ such that ∀x₁, y₁ ∈ V then $μ_{\tilde{X}} (x_{1}) = μ_{\tilde{X}}^{c} (h (x_{1})) = μ_{\tilde{X}} (h (x_{1}))$ , $η_{\tilde{X}} (x_{1}) = η_{\tilde{X}}^{c} (h (x_{1})) = η_{\tilde{X}} (h (x_{1})), and ν_{\tilde{X}} (x_{1}) = ν_{\tilde{X}}^{c} (h (x_{1})) = ν_{\tilde{X}} (h (x_{1})), μ_{\tilde{Y}} (x_{1} y_{1}) \leq {\bar{μ}}_{\tilde{Y}} (h (x_{1}) h (y_{1}))$ , $η_{\tilde{Y}} (x_{1} y_{1}) \geq {\bar{η}}_{\tilde{Y}} (h (x_{1}) h (y_{1}))$ and $ν_{\tilde{Y}} (x_{1} y_{1}) \geq {\bar{ν}}_{\tilde{Y}} (h (x_{1}) h (y_{1}))$

For complement PFG, $\begin{matrix} μ_{\tilde{Y}} (x_{1} y_{1}) \leq μ_{\tilde{Y}}^{c} (h (x_{1}) h (y_{1})) = \\ min (μ_{\tilde{X}} (h (x_{1})), μ_{\tilde{X}} (h (y_{1}))) - μ_{\tilde{Y}} (h (x_{1}) h (y_{1})) \\ η_{\tilde{Y}} (x_{1} y_{1}) \geq η_{\tilde{Y}}^{c} (h (x_{1}) h (y_{1})) = \\ η_{\tilde{Y}} (h (x_{1}) h (y_{1})) - max (η_{\tilde{X}} (h (x_{1})), η_{\tilde{X}} (h (y_{1}))) \\ ν_{\tilde{Y}} (x_{1} y_{1}) \geq ν_{\tilde{Y}}^{c} (h (x_{1}) h (y_{1})) = \\ ν_{\tilde{Y}} (h (x_{1}) h (y_{1})) - max (ν_{\tilde{X}} (h (x_{1})), ν_{\tilde{X}} (h (y_{1}))) \\ η_{\tilde{Y}} (x_{1} y_{1}) + η_{\tilde{Y}} (h (x_{1}) h (y_{1})) \\ \leq max (η_{\tilde{X}} (h (x_{1})), η_{\tilde{X}} (h (y_{1}))) \end{matrix}$

$\begin{matrix} ν_{\tilde{Y}} (x_{1} y_{1}) + ν_{\tilde{Y}} (h (x_{1}) h (y_{1})) \\ \leq max (ν_{\tilde{X}} (h (x_{1})), ν_{\tilde{X}} (h (y_{1}))) \end{matrix}$ So,

$\sum_{x_{1} \neq y_{1}} μ_{\tilde{Y}} (x_{1} y_{1}) + \sum_{x_{1} \neq y_{1}} μ_{\tilde{Y}} (h (x_{1}) h (y_{1})) \leq \sum_{x_{1} \neq y_{1}} min (μ_{\tilde{X}} (h (x_{1})), μ_{\tilde{X}} (h (y_{1}))) and \sum_{x_{1} \neq y_{1}} η_{\tilde{Y}} (x_{1} y_{1}) + \sum_{x_{1} \neq y_{1}} η_{\tilde{Y}} (h (x_{1}) h (y_{1})) \geq \sum_{x_{1} \neq y_{1}} min (η_{\tilde{X}} (h (x_{1})), η_{\tilde{X}} (h (y_{1})))$ (4.1)

Hence, $2 \sum_{x_{1} \neq y_{1}} μ_{\tilde{Y}} (x_{1} y_{1}) \leq \sum_{x_{1} \neq y_{1}} min (μ_{\tilde{X}} (x_{1}), μ_{\tilde{X}} (y_{1}))$ and $2 \sum_{x_{1} \neq y_{1}} η_{\tilde{Y}} (x_{1} y_{1}) \geq \sum_{x_{1} \neq y_{1}} max (η_{\tilde{X}} (x_{1}), η_{\tilde{X}} (y_{1}))$

Now we have $\begin{matrix} \sum_{x_{1} \neq y_{1}} μ_{\tilde{Y}} (x_{1} y_{1}) \\ \leq \frac{1}{2} \sum_{x_{1} \neq y_{1}} min (μ_{\tilde{X}} (x_{1}), μ_{\tilde{X}} (y_{1})) \\ \sum_{x_{1} \neq y_{1}} η_{\tilde{Y}} (x_{1} y_{1}) \\ \geq \frac{1}{2} \sum_{x_{1} \neq y_{1}} max (η_{\tilde{X}} (x_{1}), η_{\tilde{X}} (y_{1})) \end{matrix}$

□

Definition 4.4. Let $\tilde{G} = (\tilde{X}, \tilde{Y})$ be a PFG and $G^{μ} = ({\tilde{X}}^{μ}, {\tilde{Y}}^{μ})$ be the μ complement of $\tilde{G}$ where X = X^μ and ${\tilde{Y}}^{μ} = (μ_{\tilde{Y}}^{μ}, η_{\tilde{Y}}^{μ}, ν_{\tilde{Y}}^{μ})$ where

$μ_{\tilde{Y}}^{μ} (i, j) = {\begin{matrix} min (μ_{\tilde{X}} (j), μ_{\tilde{X}} (k)) - μ_{\tilde{X}} (i, j) if μ_{\tilde{Y}} (i, j) > 0 \\ 0 if μ_{\tilde{Y}} (i, j) = 0 \end{matrix}$

$η_{\tilde{Y}}^{μ} (i, j) = {\begin{matrix} η_{\tilde{Y}} (i, j) - max (η_{\tilde{X}} (j), η_{\tilde{X}} (k)) if η_{\tilde{Y}} (i, j) > 0 \\ 0 if η_{\tilde{Y}} (i, j) = 0 \end{matrix}$

$ν_{\tilde{Y}} (i, j) = {\begin{matrix} ν_{\tilde{Y}} (i, j) - max (ν_{\tilde{X}} (j), ν_{\tilde{X}} (k)) if ν_{\tilde{Y}} (i, j) > 0 \\ 0 if ν_{\tilde{Y}} (i, j) = 0 \end{matrix}$

Theorem 4.6. Let $\tilde{G}$ be a highly irregular PFG and ${\tilde{G}}^{μ}$ be μ-complement of $\tilde{G}$ . The ${\tilde{G}}^{μ}$ is not always highly irregular PFG.

Proof. For all nodes of $\tilde{G}$ , the neighbor nodes with different picture fuzzy degrees or the non neighbor nodes with different picture fuzzy degrees may exist to be neighbor nodes with equal picture fuzzy degrees. This scenarios does not match the concept of highly irregular PFG.□

Theorem 4.7. Let ${\tilde{G}}_{1}$ and ${\tilde{G}}_{2}$ are two HIPFGs. If there exists an isomorphism relation from ${\tilde{G}}_{1}$ to ${\tilde{G}}_{2}$ ,then there exist also an isomorphism relation between $G_{1}^{μ}$ and $G_{2}^{μ}$ .

5 Complete bipartite PFGs

Definition 5.1. Le $\tilde{G} = (\tilde{X}, \tilde{Y})$ be PFG. $\tilde{G}$ is defined as a complete bipartite PFG if the node set V is divided into 2 sets N₁ and N₂, such that all x₁, x₂ ∈ N₁ or N₂

$(μ_{\tilde{Y}} (x_{1}, x_{2}), η_{\tilde{Y}} (x_{1}, x_{2}), ν_{\tilde{Y}} (x_{1}, x_{2})) = (0, 1)$ and

∀ x₁ ∈ V₁ and x₂ ∈ V₂

$(μ_{\tilde{Y}} (x_{1}, x_{2}), η_{\tilde{Y}} (x_{1}, x_{2}), ν_{\tilde{Y}} (x_{1}, x_{2}))$ =

$min (μ_{\tilde{X}} (x_{1}), μ_{\tilde{X}} (x_{2})),$

$max (η_{\tilde{X}} (x_{1}), η_{\tilde{X}} (x_{2}))$

$max (ν_{\tilde{X}} (x_{1}), ν_{\tilde{X}} (x_{2}))$

An example of complete bipartite PFG is shown in Fig. 2.

Fig. 2

An example of a complete bipartite PFG.

6 Application of picture fuzzy multigraph

Graph theory has several practical applications in many branches of technology, in specially in Civil Engineering, Electrical Engineering, transportation systems, Computer Science and Engineering, Social Science, Operation Research, Economics, Management Science, Medical Science, etc. However, many communication networks can not be efficiently represented by simple graph. For e.g., in an Adhoc network, there may exist several multiple path from one adjacent node to another adjacent nodes. Two adjacent routers in an Adhoc network can share multiple direct links (instead of single connection) which helps to minimize the bandwidth of communication link as compared to the type of single link. We need to use multigraph network to model this type of network. A graph is said to be multigraph if any two nodes are joined by multiple arcs/edges. Multigraph is generally two types: undirected and directed. If the arc (a, b) and the arc (b, a) are identical then the G is an undirected multigraph otherwise G is a directed multigraph. We have shown a directed multigraph in Fig. 3.

Fig. 3

A multigraph G.

For any real life communication network, it is very hard to determine complete topology of a communication network at a specific time because it may possible that some of its communication links may be temporarily not working properly due to several reasons, e.g., internal damage, external attack, blockage in links, etc. So, the arc lengths of a multigraphs are not real numbers but picture fuzzy numbers. Here, we present the idea of ’picture fuzzy multigraph’ model. It is very flexible and effective appropriate network model because it can incorporate the actual real time data of the communication network system to facilitate the expert to find the optimal solutions/results.

Definition 6.1. A fuzzy graph is a picture fuzzy multigraph if and only the arc lengths of fuzzy graph are picture fuzzy numbers and multiple arc may link the same pair of nodes.

We have shown picture fuzzy multigraph in Fig. 4. In the Fig. 4, we have modeled a transportation network using a picture fuzzy multigraph. Here, we have the picture fuzzy number to describe the traveling cost of each arc. Picture fuzzy number is an extended form of simple fuzzy numbers involving three independently guessed membership grades: positive, neutral and negative.

Fig. 4

A Picture fuzzy multigraph.

6.1 Neighbor point

For a specific vertex x, the point y will be pointed as a neighbor point of x if vertex y has at least one edge from vertex x to vertex y. For e.g., we take an Adhoc Network which consists of more than one paths within two neighbor points. In some real life scenarios, single or multiple paths may be temporary not working properly. So, it is not possible to send a packets properly from a sender point x to its neighbor point y at that time. This type of information is very significant for a communication network service if the sender point gets this information in advance. Here, we take the a directed picture fuzzy multigraph $\tilde{G}$ where the arc length are picture fuzzy number.

Fig. 5

A Picture fuzzy multigraph with some damaged links temporarily.

We can collect the real time information from the communication network which the multigraphs not static, more useful, and hence more efficient to the users. Every node of the multigraph carries an information vector corresponding to each of its neighbour nodes.

6.2 Link picture fuzzy status vector (LPFSV)

Every adjacent vertex x, link picture fuzzy status Vector (LPFSV) I_xy = (j₁, j₂, j₃, . . . , j_n) of x, where any time of j_r can take any of values between 0 and 1. The communication line j_r =0 if it is nonworking and j_r =1 if the communication line is working.

6.3 Temporarily stopped link

In a specific time, the value of j_r may be 0. It means that communication line is nonworking then communication line is a temporarily stopped link from x. The j_r value is used to represent the communication line status of the line. In real life scenarios, we may be unable to find the complete picture fuzzy multigraph because due to existence of non-working status for few line. There exists of few Temporarily stopped Link and we have to consider a sub picture fuzzy multigraph.

6.4 Temporarily stopped neighbor & reachable neighbor

Let x be a neighbor vertex of a vertex y. If the value of I_xy be null vector in a specific time. The vertex x is said to be a temporarily stopped neighbor of vertex y.

6.5 Communicable picture fuzzy vertex

If the value of I_x is not null vector for a specific vertex x and there exists minimum one vertex of I_x is not null at that time, then the vertex x is said to communicable picture fuzzy vertex. The value of I_x will be null if there exists no communicable picture fuzzy vertex.

7 Conclusions

In this manuscript, we present the concept of regularity in PFG. We define different types of degree such as d_m and td_m degree of a vertex in a regular PFG. We introduce several different type of regular PFGs such as regular strong, regular, complete bipartite, td_m-regular and d_m-regular PFG. The definition of complement and isomorphism of a PFG are described in this manuscript. We also define the highly irregular PFG and μ complement PFG. Finally, we present a model to represent the communication network using picture fuzzy multigraph. In future, we will try to define the intersection graphs, interval graphs, hyper graphs, planar graphs in picture fuzzy environment. In this manuscript, we have presented one simple numerical example of picture fuzzy multigraph to represent one small communication network. We have considered the small sized example to describe the utility of our proposed model. Communication network is based on millions of devices and big data paradigm. Therefore, as future work, we need to model a practical communication network using the picture fuzzy multigraph. Furthermore, we try to find some algorithms to find some measurements of any communication network.

Footnotes

Acknowledgments

This work was supported by the 13th Five Years Programs for Natural Science Research of the Education Department of Jilin Province (No. JJKH20181164KJ), and the Natural Science Foundation of Changchun Normal University (No. 2016-010).

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A study on regular picture fuzzy graph with applications in communication networks

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Regular, d m -regular and td m -regular PFG

6.3 Temporarily stopped link

6.4 Temporarily stopped neighbor & reachable neighbor

6.5 Communicable picture fuzzy vertex

7 Conclusions

Footnotes

Acknowledgments

References

3 Regular, d_m-regular and td_m-regular PFG