Pythagorean fuzzy soft graphs with applications

Abstract

A Pythagorean fuzzy soft set, an extension of intuitionistic fuzzy soft set, plays an essential role to handle the vagueness in many real-life problems. We apply this concept to graph theory, and present certain new notions including, perfectly regular Pythagorean fuzzy soft graphs (PFSGs), perfectly edge-regular PFSGs and explore some of their properties. We formulate the notion of perfectly irregular PFSGs, perfectly edge-irregular PFSGs and open neighborhood degree sum $({\hat{O}}_{NDS})$ and closed neighborhood degree sum $({\hat{C}}_{NDS})$ of PFSGs. Finally, we discuss some decision-making problems of PFSGs.

Keywords

Pythagorean fuzzy soft graphs perfectly edge-regular perfectly edge-irregular

1 Introduction

Fuzzy set (FS) theory [40] is a mathematical model to handle the imprecise and incomplete data. For distinguishing the hesitancy more individually, FS was extended to intuitionistic fuzzy set (IFS) by Atanassov [10], which designates a membership value μ and a non-membership value ν to the objects, satisfying the condition μ + ν ≤ 1 and the hesitancy part π = 1 - μ - ν. In decision-making problems, when a scholar selects the membership value 0.6 and non-membership value 0.8 for some information, then IFS fails in this situation because 0.6 + 0.8 > 1, but (0.6) ² + (0.8) ² ≤ 1. To overcome this situation, the notion of Pythagorean fuzzy set (PFS) was introduced by Yager [36 –38] satisfying the condition μ² + ν² ≤ 1 . A PFS has more potential as compared to IFS in solving decision-making problems. The clue of Pythagorean fuzzy number (PFN) was established by Zhang and Xu [47]. Zhang [45] conferred the Pythagorean fuzzy weighted averaging operator. In decision-making problems, Garg [17, 18] considered the applications of PFSs.

Soft set (SS) theory was originated by Molodstov [23] for the parameterized point of view for uncertainty modeling and soft computing. For the hybrid models such as rough soft sets, soft rough sets and soft-rough fuzzy sets, Feng et al. [14, 15] joined the SSs with rough sets (RSs) and FSs. Ali et al. [9] renewed many new operations in SS theory. Som [35] popularized the idea of soft relation and fuzzy soft relation. Fuzzy soft sets (FSSs) were defined by Maji et al. [21]. Roy et al. [31] established many applications of FSSs. The idea of FSs and FSSs induced by SSs were handled by Ali [8]. Maji et al. [22] elaborated the theory of intuitionistic fuzzy soft set (IFSS). Peng et al. [28] introduced the idea of Pythagorean fuzzy soft set (PFSS). On the other hand, several decision making methods have been discussed in [41 , 50].

Graph theory plays a vital role for modeling complex problems. Fuzzy graph theory, generalized form of graph theory, is extensively used in numerous fields, including computer network, engineering science and pattern recognition. Kaufmann [19] proposed the concept of fuzzy graphs. Later, Rosenfeld [30] discussed several basic graph theoretical concepts, including paths, cycles, bridges and connectedness in fuzzy environment. Nagoor Gani and Radha [16] proposed the notion of regular fuzzy graph in 2008. Mordeson and Peng [24] introduced some operations on fuzzy graphs and studied their properties. Further, Nirmala and Vijaya [26] studied new operations on fuzzy graphs. Cary [11] proposed the concept of perfectly regular and perfectly edge-regular fuzzy graphs. Parvathi and Karunambigai [27] introduced the idea of intuitionistic fuzzy graphs (IFGs). Later, Akram and Davvaz [1] studied the IFGs. The edge regular intuitionistic fuzzy soft graphs were introduced by Shahzadi and Akram [33]. There are many decision-making problems, where IFGs are not applicable. So, Naz et al. [25] established the notion of Pythagorean fuzzy graphs (PFGs) to solve such complications. Some new operations of PFGs were studied by Akram et al. [3]. The theory of soft graphs (SGs) was studied by Thumbakara and George. Akram and Nawaz analyzed the ideas of SGs and vertex-induced SGs in broad spectrum. The idea of q-rung picture fuzzy graphs was studied by Akram and Habib [2]. Akram et al. [4] gave the idea of perfectly edge regular q-rung picture fuzzy graphs. Akram and Nawaz [5] introduced the concept of FSGs. The idea of IFSGs was given by Akram and Shahzadi [34]. The concept of novel intuitionistic fuzzy soft multiple-attribute decision-making methods was handled by Akram and Shahzadi [6]. The idea of hybrid soft computing models in graph theory was introduced by Akram and Zafar [7]. Yu et al. [39] introduced the novel consensus model for multi-attribute group decision making (MAGDM) problem based on multi-granular hesitant fuzzy linguistic term sets. The concept of managing multi-granular unbalanced hesitant fuzzy linguistic information in multi-attribute large-scale group decision making was handled by Zhang et al. [48]. The additive consistency analysis and improvement for hesitant fuzzy preference relations was studied by Zhang et al. [46]. For further terminologies which are not discussed in this paper, the readers are referred to [20 , 44]. In this paper, we present certain new notions including, perfectly regular PFSGs, perfectly edge-regular PFSGs and explore some of their properties. We formulate the notion of perfectly irregular PFSGs, perfectly edge-irregular PFSGs and open neighborhood degree sum $({\hat{O}}_{NDS})$ and closed neighborhood degree sum $({\hat{C}}_{NDS})$ of PFSGs. Finally, we discuss some decision-making problems of PFSGs.

The motivation of this article is described as follows:

1. PFSSs are more flexible to handle uncertainty where IFSSs fail and they are not only applicable to situations where the sum of membership and non-membership degree is equal to 1 but they satisfy μ² + ν² ≤ 1 corresponding to distinct attributes.

2. The main purpose of this article is to discuss the pairwise relationship between objects under Pythagorean fuzzy environment corresponding to different parameters.

2 Preliminaries

Definition 2.1. [36] Let V be a universe of discourse. A PFS P in V is given by P = {< c, μ_P (c) , ν_P (c) > |c ∈ V}

where μ_P (c) : V → [0, 1] denotes the degree of membership and ν_P (c) : V → [0, 1] denotes the degree of non-membership of the element c ∈ V to the set P, respectively, with the condition that 0 ≤ (μ_P (c)) ² + (ν_P (c)) ² ≤ 1. The degree of indeterminacy $π_{P} (c) = \sqrt{1 - (μ_{P} (c))^{2} - (ν_{P} (c))^{2}}$ . For convenience, Zhang and Xu [47] called (μ_P (c) , ν_P (c)) a PFN denoted by p = (μ_P, ν_P).

Definition 2.2. [28] Let V be a universe of discourse and W be the set of all parameters, O ⊆ W. P (V) denotes the set of all PFSs. $(\tilde{I}, O)$ is called an PFSS over V, where Pythagorean fuzzy approximation function is given by $\tilde{I} = (\tilde{I_{μ}}, \tilde{I_{ν}}) : O \to P (V)$ .

Throughout this paper, we will use the notations as defined in Table 1.

Table 1
Notations

Symbol Definition

$P_{G} = (\tilde{I}, \tilde{J}, O)$ Pythagorean fuzzy soft graph

o_i, i = 1, 2, ⋯ , m Parameters

$(\tilde{I}, O)$ Pythagorean fuzzy soft subsets

$(\tilde{J}, O)$ Pythagorean fuzzy soft relation

$\tilde{T} (o_{i}) = (\tilde{T_{μ}} (o_{i}), \tilde{T_{ν}} (o_{i})), \forall o_{i} \in O$ Pythagorean fuzzy subgraph

${\hat{O}}_{NDS}$ Open neighborhood degree sum

${\hat{C}}_{NDS}$ Closed neighborhood degree sum

Symbol	Definition
$P_{G} = (\tilde{I}, \tilde{J}, O)$	Pythagorean fuzzy soft graph
o_i, i = 1, 2, ⋯ , m	Parameters
$(\tilde{I}, O)$	Pythagorean fuzzy soft subsets
$(\tilde{J}, O)$	Pythagorean fuzzy soft relation
$\tilde{T} (o_{i}) = (\tilde{T_{μ}} (o_{i}), \tilde{T_{ν}} (o_{i})), \forall o_{i} \in O$	Pythagorean fuzzy subgraph
${\hat{O}}_{NDS}$	Open neighborhood degree sum
${\hat{C}}_{NDS}$	Closed neighborhood degree sum

3 Pythagorean fuzzy soft graphs

In this section, we introduce the idea of PFSGs, perfectly regular PFSGs, perfectly edge-regular PFSGs and discuss some of their properties.

Definition 3.1. A PFSG on a nonempty set V is a tuple $P_{G} = (\tilde{I}, \tilde{J}, O)$ such that

1. O is a nonempty set of parameters,

2. $(\tilde{I}, O)$ is a PFSS over V,

3. $(\tilde{J}, O)$ is a PFSS over E ⊆ V × V,

4. $(\tilde{I} (o_{i}), \tilde{J} (o_{i}))$ is a connected PF subgraph, ∀ o_i ∈ O, i = 1, 2, ⋯ , m, that is, $\tilde{J_{μ}} (o_{i}) (cd) \leq min {\tilde{I_{μ}} (o_{i}) (c), \tilde{I_{μ}} (o_{i}) (d)},$ $\tilde{J_{ν}} (o_{i}) (cd) \leq max {\tilde{I_{ν}} (o_{i}) (c), \tilde{I_{ν}} (o_{i}) (d)},$ such that $0 \leq {\tilde{J}}_{μ}^{2} (o_{i}) (cd) + {\tilde{J}}_{ν}^{2} (o_{i}) (cd) \leq 1, \forall o_{i} \in O, c, d \in V .$

The PF subgraph $(\tilde{I} (o_{i}), \tilde{J} (o_{i}))$ is denoted by $\tilde{T} (o_{i}) = (\tilde{T_{μ}} (o_{i}), \tilde{T_{ν}} (o_{i}))$ .

Definition 3.2. Let P_G be a PFSG on V. Then P_G is said to be regular PFSG if $\tilde{T} (o_{i})$ is a regular PFG for all o_i ∈ O. If $\tilde{T} (o_{i})$ is a regular PFG of degree $(r_{i}, r_{i}^{'})$ for all o_i ∈ O, then P_G is a regular PFSG.

Definition 3.3. Let P_G be a PFSG on V. Then P_G is said to be totally regular PFSG if $\tilde{T} (o_{i})$ is a totally regular PFG for all o_i ∈ O. If $\tilde{T} (o_{i})$ is a totally regular PFG of degree $(r_{i}, r_{i}^{'})$ for all o_i ∈ O, then P_G is a totally regular PFSG.

Definition 3.4. Let P_G be a PFSG on V. Then P_G is said to be perfectly regular PFSG if $\tilde{T} (o_{i})$ is a regular and totally regular PFG for all o_i ∈ O.

Example 3.1. Consider two nonempty sets V = {c₁, c₂, c₃} and E = {c₁c₂, c₂c₃, c₁c₃}. Let O = {o₁} and $(\tilde{I}, O)$ be a PFSS over V with its approximate function $\tilde{I} : O \to P (V)$ given by $\begin{matrix} \tilde{I} (o_{1}) & = & {(c_{1}, 0.5, 0.6), (c_{2}, 0.5, 0.6), (c_{3}, 0.5, 0.6)} . \end{matrix}$ Let $(\tilde{J}, O)$ be a PFSS over E with its approximate function $\tilde{J} : O \to P (E)$ given by $\begin{matrix} \tilde{J} (o_{1}) & = & {(c_{1} c_{2}, 0.5, 0.6), (c_{2} c_{3}, 0.5, 0.6), (c_{1} c_{3}, 0.5, \\ 0.6)} . \end{matrix}$ By routine calculations, it is easy to see that PFG $\tilde{T} (o_{1}) = (\tilde{I} (o_{1}), \tilde{J} (o_{1}))$ , is a regular and totally regular as shown in Fig. 1. Hence P_G is a perfectly regular PFSG.

Fig. 1

Perfectly regular PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

Proposition 3.1. For a perfectly regular PFSG $P_{G} = (\tilde{I}, \tilde{J}, O)$ , $\tilde{I}$ is a constant function.

Theorem 3.1. Let P_G be a PFSG. Then P_G is perfectly regular if and only if

1 . $\sum_{c \neq e} \tilde{J_{μ}} (o_{i}) (ce) = \sum_{d \neq e} \tilde{J_{μ}} (o_{i}) (de)$ and $\sum_{c \neq e} \tilde{J_{ν}} (o_{i}) (ce) = \sum_{d \neq e} \tilde{J_{ν}} (o_{i}) (de),$

2 . $\tilde{I_{μ}} (o_{i}) (c) = \tilde{I_{μ}} (o_{i}) (d), \tilde{I_{ν}} (o_{i}) (c) = \tilde{I_{ν}} (o_{i}) (d)$ , ∀ c, d ∈ V, o_i ∈ O.

Proof. First assume P_G is perfectly regular PFSG. So P_G is regular PFSG and from the definition of regular PFSG, we have deg_μ (c)=deg_μ (d) and deg_ν (c)=deg_ν (d), ∀ c, d ∈ V, o_i ∈ O. Then

$\sum_{c \neq e} \tilde{J_{μ}} (o_{i}) (ce) = \sum_{d \neq e} \tilde{J_{μ}} (o_{i}) (de)$

$\sum_{c \neq e} \tilde{J_{ν}} (o_{i}) (ce) = \sum_{d \neq e} \tilde{J_{ν}} (o_{i}) (de),$ ∀ c, d ∈ V, o_i ∈ O.

Thus (1) holds. By Proposition 3.1, (2) also holds.

Conversely, suppose that P_G is a PFSG such that it satisfies the conditions. From condition (1), we have

$\sum_{c \neq e} \tilde{J_{μ}} (o_{i}) (ce) = \sum_{d \neq e} \tilde{J_{μ}} (o_{i}) (de)$

$\sum_{c \neq e} \tilde{J_{ν}} (o_{i}) (ce) = \sum_{d \neq e} \tilde{J_{ν}} (o_{i}) (de),$ ∀ c, d ∈ V, o_i ∈ O.

⇒deg_μ (c) = deg_μ (d) = k_i and $\deg_{ν} (c) = \deg_{ν} (d) = k_{i}^{'}$ , ∀ c, d ∈ V, o_i ∈ O . This implies that $\tilde{T} (o_{i})$ is a regular PFG. Hence P_G is a regular PFSG.

Now from condition (2), $\tilde{I_{μ}} (o_{i}) (c) = \tilde{I_{μ}} (o_{i}) (d) = r_{i}, \tilde{I_{ν}} (o_{i}) (c) = \tilde{I_{ν}} (o_{i}) (d) = r_{i}^{'}$ , ∀ c, d ∈ V, o_i ∈ O.

Thus $\tilde{I}$ is a constant function. So, $\deg_{μ} [c] = \deg_{μ} (c) + \tilde{I} (o_{i}) (c) = k_{i} + r_{i}$ , $\deg_{μ} [d] = \deg_{μ} (d) + \tilde{I} (o_{i}) (d) = k_{i} + r_{i}$ , ∀ c, d ∈ V, o_i ∈ O . and $\deg_{ν} [c] = \deg_{ν} (c) + \tilde{I} (o_{i}) (c) = k_{i}^{'} + r_{i}^{'}$ , $\deg_{ν} [d] = \deg_{ν} (d) + \tilde{I} (o_{i}) (d) = k_{i}^{'} + r_{i}^{'}$ , ∀ c, d ∈ V, o_i ∈ O . Therefore, deg_μ [c] = deg_μ [d] = f_i(say) and $\deg_{ν} [c] = \deg_{ν} [d] = f_{i}^{'}$ (say), i.e, $\deg [c] = \deg [d] = (f_{i}, f_{i}^{'}), \forall c, d \in V, o_{i} \in O$ . So, $\tilde{T} (o_{i})$ is a $(f_{i}, f_{i}^{'})$ -totally regular PFG. Hence P_G is totally regular PFSG. This implies P_G is a perfectly regular PFSG.

□

Corollary 3.1. If P_G is a perfectly regular PFSG and $\tilde{I} = (\tilde{I_{μ}} (o_{i}) (c), \tilde{I_{ν}} (o_{i}) (c)) = (k_{i}, k_{i}^{'})$ , ∀ c ∈ V, o_i ∈ O is a constant function in $\tilde{T} (o_{i})$ , then $O (\tilde{T} (o_{i})) = | V | (k_{i}, k_{i}^{'})$ .

Theorem 3.2. Let $P_{G} = (\tilde{I}, \tilde{J}, O)$ be a perfectly regular PFSG. Then size of $\tilde{T} (o_{i})$ is $S (\tilde{T} (o_{i})) = \frac{| V |}{2} (f_{i}, f_{i}^{'})$ , where $(f_{i}, f_{i}^{'})$ is the degree of a vertex in $\tilde{T} (o_{i})$ , ∀ o_i ∈ O.

The converse of Theorem 3.2 need not be true as seen in the following example.

Example 3.2. Consider the PFSG P_G in Fig. 2. Here |V|=5 and deg (c₁) = deg₍c₂) = deg₍c₃) = deg (c₄) = deg (c₅) = (1, 1.4). Size of $\tilde{T} (o_{1})$ is $S (\tilde{T} (o_{1})) = (2.5, 3.5)$ . From Theorem 3.2, we have $S (\tilde{T} (o_{1})) = \frac{| V |}{2} (f_{i}, f_{i}^{'}) = \frac{| V |}{2} (f_{1}, f_{1}^{'}) = \frac{5}{2} (1, 1.4) = (2.5, 3.5)$ . Now deg [c₁] = (1.5, 2) and deg [c₂] = (1.6, 2), so deg [c₁] ≠ deg [c₂]. Therefore, P_G is not perfectly regular PFSG. Hence, the converse of Theorem 3.2 is not true.

Fig. 2

A connected PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

Definition 3.5. Let P_G be a PFSG on V. Then P_G is said to be edge-regular PFSG if $\tilde{T} (o_{i})$ is an edge-regular PFG for all o_i ∈ O. If $\tilde{T} (o_{i})$ is an edge-regular PFG of degree $(r_{i}, r_{i}^{'})$ for all o_i ∈ O, then P_G is an edge-regular PFSG.

Definition 3.6. Let P_G be a PFSG on V. Then P_G is said to be totally edge-regular PFSG if $\tilde{T} (o_{i})$ is a totally edge-regular PFG for all o_i ∈ O. If $\tilde{T} (o_{i})$ is a totally edge-regular PFG of degree $(r_{i}, r_{i}^{'})$ for all o_i ∈ O, then P_G is a totally edge-regular PFSG.

Definition 3.7. A perfectly edge-regular PFSG is a PFSG that is both edge-regular and totally edge-regular PFSG.

Example 3.3. Consider two nonempty sets V = {c₁, ₂, c₃, c₄} and E = {c₁c₂, c₂c₃, c₃c₄, c₁c₄}. Let O = {o₁, o₂} and $(\tilde{I}, O)$ be a PFSS over V with its approximate function $\tilde{I} : O \to P (V)$ given by $\begin{matrix} \tilde{I} (o_{1}) & = & {(c_{1}, 0.6, 0.6), (c_{2}, 0.6, 0.6), (c_{3}, 0.6, 0.6), \\ (c_{4}, 0.6, 0.6)}, \\ \tilde{I} (o_{2}) & = & {(c_{1}, 0.7, 0.6), (c_{2}, 0.7, 0.6), (c_{3}, 0.7, 0.6), \\ (c_{4}, 0.7, 0.6)} . \end{matrix}$ Let $(\tilde{J}, O)$ be a PFSS over E with its approximate function $\tilde{J} : O \to P (E)$ given by $\begin{matrix} \tilde{J} (o_{1}) & = & {(c_{1} c_{2}, 0.5, 0.5), (c_{2} c_{3}, 0.5, 0.5), (c_{3} c_{4}, 0.5, \\ 0.5), (c_{1} c_{4}, 0.5, 0.5)}, \\ \tilde{J} (o_{2}) & = & {(c_{1} c_{2}, 0.6, 0.6), (c_{2} c_{3}, 0.6, 0.6), (c_{3} c_{4}, 0.6, \\ 0.6), (c_{1} c_{4}, 0.6, 0.6), (c_{1} c_{3}, 0.6, 0.6), (c_{2} c_{4}, \\ 0.6, 0.6)} . \end{matrix}$ By routine calculations, it is easy to see that PFGs $\tilde{T} (o_{1}) = (\tilde{I} (o_{1}), \tilde{J} (o_{1})),$ $\tilde{T} (o_{2}) = (\tilde{I} (o_{2}), \tilde{J} (o_{2})),$ are edge-regular and totally edge-regular PFGs as shown in Fig. 3. Since deg (c₁c₂) = deg (c₂c₃) = deg (c₃c₄) = deg (c₁c₄) = (1.2, 1.2) and deg [c₁c₂] = deg [c₂c₃] = deg [c₃c₄] = deg [c₁c₄] = (1.7, 1.7) in $\tilde{T} (o_{1})$ and deg (c₁c₂) = deg (c₂c₃) = deg (c₃c₄) = deg (c₁c₄) = deg (c₁c₃) = deg (c₂c₄) = (2.4, 2.4) and deg [c₁c₂] = deg [c₂c₃] = deg [c₃c₄] = deg [c₁c₄] = deg [c₁c₃] = deg [c₂c₄] = (3, 3) in $\tilde{T} (o_{2})$ . Hence $P_{G} = (\tilde{T} (o_{1}), \tilde{T} (o_{2}))$ is a perfectly edge-regular PFSG.

Fig. 3

Perfectly edge-regular PFSG $P_{G} = {\tilde{T} (o_{1}), \tilde{T} (o_{2})}$ .

Proposition 3.2. For a perfectly edge-regular PFSG $P_{G} = (\tilde{I}, \tilde{J}, O)$ , $\tilde{J}$ is a constant function.

The converse of Proposition 3.2 need not be true as seen in the following example.

Example 3.4. From the PFSG P_G as shown in Fig. 4, we see that $\tilde{J} (o_{1}) (c_{1} c_{4}) = \tilde{J} (o_{1}) (c_{1} c_{3}) = \tilde{J} (o_{1}) (c_{1} c_{2}) = \tilde{J} (o_{1}) (c_{2} c_{3}) = \tilde{J} (o_{1}) (c_{3} c_{4}) = (0.5, 0.5) .$ So $\tilde{J}$ is a constant function. But deg₍c₁c₄) = (1.5, 1.5), deg₍c₁c₃) = (2, 2). But deg₍c₁c₄) = (1.5, 1.5) ≠ deg₍c₁c₃) = (2, 2) . This implies $\tilde{T} (o_{1})$ is not perfectly regular PFG. Hence P_G is not perfectly regular PFSG.

Fig. 4

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

Theorem 3.3. Let P_G be a PFSG. Then P_G is perfectly edge-regular PFSG if and only if

1 . $\sum_{c \neq e} \tilde{J_{μ}} (o_{i}) (ce) + \sum_{d \neq e} \tilde{J_{μ}} (o_{i}) (de) - 2 \tilde{J_{μ}} (o_{i}) (cd) = \sum_{x \neq z} \tilde{J_{μ}} (o_{i}) (xz) + \sum_{y \neq z} \tilde{J_{μ}} (o_{i}) (yz) - 2 \tilde{J_{μ}} (o_{i}) (xy),$

$\sum_{c \neq e} \tilde{J_{ν}} (o_{i}) (ce) + \sum_{d \neq e} \tilde{J_{ν}} (o_{i}) (de) - 2 \tilde{J_{ν}} (o_{i}) (cd) = \sum_{x \neq z} \tilde{J_{ν}} (o_{i}) (xz) + \sum_{y \neq z} \tilde{J_{ν}} (o_{i}) (yz) - 2 \tilde{J_{ν}} (o_{i}) (xy),$

2 . $\tilde{J_{μ}} (o_{i}) (cd) = \tilde{J_{μ}} (o_{i}) (xy), \tilde{J_{ν}} (o_{i}) (cd) = \tilde{J_{ν}} (o_{i}) (xy), \forall cd, xy \in E, o_{i} \in O$ .

Proof. Suppose P_G is perfectly edge-regular PFSG. So, P_G is edge-regular PFSG and from the definition of edge-regular PFSG, we have deg_μ (cd) = deg_μ (xy) and deg_ν (cd) = deg_ν (xy), ∀ cd, xy ∈ E. Then

$\sum_{c \neq e} \tilde{J_{μ}} (o_{i}) (ce) + \sum_{d \neq e} \tilde{J_{μ}} (o_{i}) (de) - 2 \tilde{J_{μ}} (o_{i}) (cd) = \sum_{x \neq z} \tilde{J_{μ}} (o_{i}) (xz) + \sum_{y \neq z} \tilde{J_{μ}} (o_{i}) (yz) - 2 \tilde{J_{μ}} (o_{i}) (xy),$

Thus condition (1) holds. By Proposition 3.2, condition (2) also holds.

Conversely, let P_G be a PFSG and conditions (1) and (2) hold. From condition (1), we have

deg_μ (cd) = deg_μ (xy) = k_i,

$\deg_{ν} (cd) = \deg_{ν} (xy) = k_{i}^{'}, \forall cd, xy \in E, o_{i} \in O .$ This implies $\tilde{T} (o_{i})$ is $(k_{i}, k_{i}^{'})$ edge-regular PFG. So, P_G is edge-regular PFSG.

Now from the condition (2), we have

$\tilde{J_{μ}} (o_{i}) (cd) = \tilde{J_{μ}} (o_{i}) (xy) = f_{i},$

$\tilde{J_{ν}} (o_{i}) (cd) = \tilde{J_{ν}} (o_{i}) (xy) = f_{i}^{'}, \forall cd, xy \in E, o_{i} \in O .$

Thus $\tilde{J}$ is a constant function. So,

$\deg_{μ} [cd] = \deg_{μ} (cd) + \tilde{J_{μ}} (o_{i}) (cd) = k_{i} + f_{i},$

$\deg_{μ} [xy] = \deg_{μ} (xy) + \tilde{J_{μ}} (o_{i}) (xy) = k_{i} + f_{i},$

$\deg_{ν} [cd] = \deg_{ν} (cd) + \tilde{J_{ν}} (o_{i}) (cd) = k_{i}^{'} + f_{i}^{'},$

$\deg_{ν} [xy] = \deg_{ν} (xy) + \tilde{J_{ν}} (o_{i}) (xy) = k_{i}^{'} + f_{i}^{'} .$

Therefore, deg_μ [cd] = deg_μ [xy] = t_i (say) and $\deg_{ν} [cd] = \deg_{ν} [xy] = t_{i}^{'} (say) .$ So $\tilde{T} (o_{i})$ is totally edge-regular PFG. Hence P_G is totally edge-regular PFSG. Therefore, P_G is a perfectly edge-regular PFSG.□

Corollary 3.2. Let P_G be a perfectly edge-regular PFSG. Then size of $\tilde{T} (o_{i})$ is $S (\tilde{T} (o_{i})) = | E | (k_{i}, k_{i}^{'})$ , where $(k_{i}, k_{i}^{'}) = (\tilde{J_{μ}} (o_{i}) (cd), \tilde{J_{ν}} (o_{i}) (cd))$ , ∀ cd ∈ E, o_i ∈ O.

The converse of Corollary 3.2 need not be true as seen in the following example.

Example 3.5. Consider the PFSG P_G as shown in Fig. 5. Here |E|=6 and size of $\tilde{T} (o_{1})$ is (3.6, 3.6). Now by Corollary 3.2, size of $\tilde{T} (o_{1})$ is $S (\tilde{T} (o_{1})) = | E | (k_{i}, k_{i}^{'}) = 6 (0.6, 0.6) = (3.6, 3.6)$ . But deg_μ (c₁c₂) = 1.2, deg_ν (c₁ c₂) = 1.2, deg_μ (c₁ c₄) =1.8, deg_ν (c₁c₄) =1.8. So deg (c₁c₂) ≠ deg (c₁c₄). Therefore, $\tilde{T} (o_{1})$ is not perfectly edge-regular PFG. Hence, P_G is not perfectly edge-regular PFSG. Thus the converse of Corollary 3.2 is not true.

Fig. 5

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

Theorem 3.4. Let P_G be a perfectly edge-regular PFSG and |V| = n. Then

$\sum_{c_{j} \in V} {\max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{μ} (P_{G}) \leq n,$

$- n \leq \sum_{c_{j} \in V} {\min \tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{ν} (P_{G}), 1 \leq j, k \leq n .$

Proof. Since |V| = n, so let V = {c₁, c₂, . . . , c_n}. $\tilde{I_{μ}} (o_{i}) (c_{j}) \leq 1$ and $\tilde{I_{ν}} (o_{i}) (c_{j}) \leq 1, 1 \leq j \leq n .$ So $\sum_{c_{j} \in V} \tilde{I_{μ}} (o_{i}) (c_{j}) \leq n$ and $\sum_{c_{j} \in V} \tilde{I_{ν}} (o_{i}) (c_{j}) \leq n$ , $- n \leq \sum_{c_{j} \in V} \tilde{I_{ν}} (o_{i}) (c_{j}), c_{j} \in V .$ This implies O_μ (P_G) ≤ n and -n ≤ O_ν (P_G) ⋯(1)

Now, $\tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) \leq \min {\tilde{I_{μ}} (o_{i}) (c_{j}), \tilde{I_{μ}} (o_{i}) (c_{k})}$ and $\tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) \leq \max {\tilde{I_{ν}} (o_{i}) (c_{j}), \tilde{I_{ν}} (o_{i}) (c_{k})}, 1 \leq j, k \leq n, j \neq k .$

$\Rightarrow \max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) \leq {\tilde{I_{μ}} (o_{i}) (c_{j}), \tilde{I_{μ}} (o_{i}) (c_{k})}$ , $\min \tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) \leq {\tilde{I_{ν}} (o_{i}) (c_{j}), \tilde{I_{ν}} (o_{i}) (c_{k})}, 1 \leq j, k \leq n .$

$\Rightarrow \sum_{c_{j} \in V} {\max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq \sum_{c_{j} \in V} \tilde{I_{μ}} (o_{i}) (c_{j})$ , $\sum_{c_{j} \in V} {\min \tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq \sum_{c_{j} \in V} \tilde{I_{ν}} (o_{i}) (c_{j}), 1 \leq j, k \leq n .$

$\Rightarrow \sum_{c_{j} \in V} {\max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{μ} (P_{G})$ and $\sum_{c_{j} \in V} {\min \tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{ν} (P_{G})$ ⋯ (2)

Combining (1) and (2), we have

$\sum_{c_{j} \in V} {\max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{μ} (P_{G}) \leq n$ and $- n \leq \sum_{c_{j} \in V} {\min \tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{ν} (P_{G}) .$ □

Example 3.6. For the PFSG P_G of Fig. 6, we have |V|=4, O (P_G) = (O_μ (P_G) , O_ν (P_G)) = (2.8, 2.7) , $\sum_{c_{j} \in V} {\max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) | j \neq k} = 0.6 + 0.6 + 0.6 + 0.6 = 2.4 .$ Now 2.4 ≤ 2.8 ≤ 4.

Fig. 6

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

$\Rightarrow \sum_{c_{j} \in V} {\max \tilde{J_{μ}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{μ} (P_{G}) \leq 4$ and $- 4 \leq \sum_{c_{j} \in V} {\min \tilde{J_{ν}} (o_{i}) (c_{j} c_{k}) | j \neq k} \leq O_{ν} (P_{G}), 1 \leq j, k \leq 4 .$ Thus P_G satisfies Theorem 3.4.

Theorem 3.5. If P_G is a regular PFSG and $\tilde{J}$ is a constant function, then P_G is perfectly edge-regular PFSG.

Proof. Let P_G be a regular PFSG and $\tilde{J} = (\tilde{J_{μ}} (o_{i}) (cd), \tilde{J_{μ}} (o_{i}) (cd)) = (k_{i}, k_{i}^{'}), \forall cd \in E$ be a constant function. Now, $\begin{matrix} \deg_{μ} (cd) & = & \deg_{μ} (c) + \deg_{μ} (d) - 2 \tilde{J_{μ}} (o_{i}) (cd) \\ = & f_{i} + f_{i} - 2 k_{i} \\ = & 2 (f_{i} - k_{i}), \\ \deg_{ν} (cd) & = & \deg_{ν} (c) + \deg_{ν} (d) - 2 \tilde{J_{ν}} (o_{i}) (cd) \\ = & f_{i}^{'} + f_{i}^{'} - 2 k_{i}^{'} \\ = & 2 (f_{i}^{'} - k_{i}^{'}), \forall cd \in E, o_{i} \in O . \end{matrix}$

This implies $\tilde{T} (o_{i})$ is an edge-regular PFG. Hence P_G is an edge-regular PFSG.

Now, $\deg_{μ} [cd] = \deg_{μ} (cd) + \tilde{J_{μ}} (o_{i}) (cd) = 2 (f_{i} - k_{i}) + k_{i} = 2 f_{i} - k_{i}$ and $\deg_{ν} [cd] = \deg_{ν} (cd) + \tilde{J_{ν}} (o_{i}) (cd) = 2 (f_{i}^{'} - k_{i}^{'}) + k_{i} = 2 f_{i}^{'} - k_{i}^{'}$ . Thus $\deg [cd] = (2 f_{i} - k_{i}, 2 f_{i}^{'} - k_{i}^{'}), \forall cd \in E, o_{i} \in O$ . Thus $\tilde{T} (o_{i})$ is totally edge-regular PFG. Therefore, P_G is totally edge-regular PFSG. Hence, P_G is perfectly edge-regular PFSG.□

If P_G is a totally regular PFSG and $\tilde{J}$ is a constant function, then P_G may not be perfectly edge-regular PFSG as seen in the following example.

Example 3.7. Consider the PFSG P_G, which is shown in Fig. 7. Here $\tilde{J} = (\tilde{J_{μ}} (o_{1}) (cd), \tilde{J_{ν}} (o_{1}) (cd)) = (0.2, 0.1), \forall cd \in E .$ deg [c₁] = deg [c₂] = deg [c₃] = deg [c₄] = (1, 1). So $\tilde{T} (o_{1})$ is totally regular PFG. Hence P_G is totally regular PFSG. Now deg (c₁c₂) = (0.6, 0.3) and deg (c₂c₄) = (0.8, 0.4) ≠ deg (c₁c₂) . Hence P_G is not perfectly edge-regular PFSG.

Fig. 7

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

Theorem 3.6. If P_G is perfectly regular and complete PFSG, then P_G is perfectly edge-regular PFSG.

Proof. Since P_G is perfectly regular PFSG, so by Proposition 3.1, we have

$\tilde{I_{μ}} (o_{i}) (c) = \tilde{I_{μ}} (o_{i}) (d),$

$\tilde{I_{ν}} (o_{i}) (c) = \tilde{I_{ν}} (o_{i}) (d), \forall c, d \in V, o_{i} \in O \dots (1) .$

As P_G is complete PFSG, so

$\tilde{J_{μ}} (o_{i}) (cd) = \min {\tilde{I_{μ}} (o_{i}) (c), \tilde{I_{μ}} (o_{i}) (d)},$

$\tilde{J_{ν}} (o_{i}) (cd) = \max {\tilde{I_{ν}} (o_{i}) (c), \tilde{I_{μ}} (o_{i}) (d)}, \forall cd \in E, o_{i} \in O \dots (2) .$

Combining (1) and (2), we say that $\tilde{J}$ is constant function. Since P_G is perfectly regular PFSG, so P_G is regular PFSG and $\tilde{J}$ is constant. Thus by Theorem 3.5, P_G is perfectly edge-regular PFSG.□

4 Perfectly irregular and perfectly edge-irregular PFSG

In this section, we discuss the perfectly irregular and perfectly edge-irregular PFSG.

Definition 4.1. A PFSG $P_{G} = (\tilde{I}, \tilde{J}, O)$ is said to be neighborly irregular PFSG if $\tilde{T} (o_{i})$ is neighborly irregular PFG ∀ o_i ∈ O, i.e, if the degree of every pair of adjacent vertices of $\tilde{T} (o_{i})$ are distinct, ∀ o_i ∈ O.

Definition 4.2. A PFSG $P_{G} = (\tilde{I}, \tilde{J}, O)$ is said to be totally neighborly irregular PFSG if $\tilde{T} (o_{i})$ is totally neighborly irregular PFG ∀ o_i ∈ O, i.e, if the total degree of every pair of adjacent vertices of $\tilde{T} (o_{i})$ are distinct, ∀ o_i ∈ O.

Example 4.1. Consider two nonempty sets V = {c₁, c₂, c₃, c₄} and E = {c₁c₂, c₂c₃, c₁c₄}. Let O = {o₁, o₂} and $(\tilde{I}, O)$ be a PFSS over V with its approximate function $\tilde{I} : O \to P (V)$ given by $\begin{matrix} \tilde{I} (o_{1}) & = & {(c_{1}, 0.4, 0.7), (c_{2}, 0.5, 0.6), (c_{3}, 0.6, 0.6), \\ (c_{4}, 0.7, 0.7)}, \\ \tilde{I} (o_{2}) & = & {(c_{1}, 0.5, 0.6), (c_{2}, 0.7, 0.4), (c_{3}, 0.6, 0.7), \\ (c_{4}, 0.6, 0.6)} . \end{matrix}$ Let $(\tilde{J}, O)$ be a PFSS over E with its approximate function $\tilde{J} : O \to P (E)$ given by $\begin{matrix} \tilde{J} (o_{1}) & = & {(c_{1} c_{2}, 0.4, 0.7), (c_{2} c_{3}, 0.5, 0.6), (c_{1} c_{4}, 0.4, \\ 0.7)}, \\ \tilde{J} (o_{2}) & = & {(c_{1} c_{2}, 0.5, 0.6), (c_{2} c_{3}, 0.6, 0.7), (c_{1} c_{4}, 0.5, \\ 0.6)} . \end{matrix}$ By routine calculations, it is easy to see that PFGs $\tilde{T} (o_{1}) = (\tilde{I} (o_{1}), \tilde{J} (o_{1})),$ $\tilde{T} (o_{2}) = (\tilde{I} (o_{2}), \tilde{J} (o_{2})),$ are neighborly irregular and totally neighborly irregular PFGs as shown in Fig. 8. Hence P_G is neighborly irregular and totally neighborly irregular PFSG.

Fig. 8

Neighborly irregular and totally neighborly irregular PFSG $P_{G} = {\tilde{T} (o_{1}), \tilde{T} (o_{2})}$ .

Definition 4.3. A PFSG P_G is said to be perfectly irregular if $\tilde{T} (o_{i})$ is perfectly irregular PFG for all o_i ∈ O, i.e,

1. The degrees of all vertices of $\tilde{T} (o_{i})$ are distinct.

2. The total degrees of all vertices of $\tilde{T} (o_{i})$ are distinct.

Theorem 4.1. If P_G is perfectly irregular PFSG, then P_G is necessarily neighborly irregular, totally neighborly irregular and highly irregular PFSG.

Proof. Let P_G be a perfectly irregular PFSG. So every vertex of $\tilde{T} (o_{i})$ has different degree. Then every two adjacent vertices of $\tilde{T} (o_{i})$ are of different degrees. Therefore, $\tilde{T} (o_{i})$ is neighborly irregular PFG. Hence P_G is neighborly irregular PFSG.

Since P_G is perfectly irregular PFSG, the total degrees of all the vertices of $\tilde{T} (o_{i})$ are distinct. Then every two adjacent vertices of $\tilde{T} (o_{i})$ are of different degrees. Therefore, $\tilde{T} (o_{i})$ is totally neighborly irregular PFG. Hence, P_G is totally neighborly irregular PFSG.

Since P_G is perfectly irregular PFSG, the degrees of all the vertices of $\tilde{T} (o_{i})$ are distinct. Thus the degrees of the adjacent vertices of every vertex of $\tilde{T} (o_{i})$ are distinct. Therefore, $\tilde{T} (o_{i})$ is highly irregular PFG. Hence P_G is highly irregular PFSG.□

Example 4.2. Consider a PFSG $P_{G} = (\tilde{I}, \tilde{J}, O)$ as shown in Fig. 9.

Fig. 9

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

deg (c₁) = (0.6, 0.6), deg (c₂) = (1.1, 1.1), deg (c₃) = (0.6, 0.6), deg (c₄) = (0.1, 0.1). Thus, P_G is neighborly irregular PFSG. Also, deg [c₁] = (1.2, 1.3), deg [c₂] = (1.7, 1.8), deg [c₃] = (1.2, 1.3), deg [c₄] = (0.7, 0.8) . Thus P_G is totally neighborly irregular PFSG. But deg (c₁) = deg (c₃), so P_G is not perfectly irregular PFSG.

Similarly, we can show that there exists a PFSG P_G, which is highly irregular PFSG but not perfectly irregular PFSG.

Theorem 4.2. (Sufficient Condition) The sufficient condition of a neighborly irregular and totally neighborly irregular PFSG to be perfectly irregular PFSG is that there exists an edge between every pair of vertices of $\tilde{T} (o_{i}), \forall o_{i} \in O$ .

Proof. Let P_G be a neighborly irregular and totally neighborly irregular PFSG and there exists an edge between every pair of vertices of $\tilde{T} (o_{i}), \forall o_{i} \in O$ . Since P_G is neighborly irregular PFSG, so deg (c) ≠ deg (d) for all adjacent vertices c, d ∈ V, o_i ∈ O ⋯(1)

But between every pair of vertices of there is an edge in $\tilde{T} (o_{i})$ . This means every pair of vertices of $\tilde{T} (o_{i})$ are adjacent, i.e, cd ∈ E, o_i ∈ O ⋯(2)

From (1) and (2), we have deg (c) ≠ deg (d) , ∀ c, d ∈ V, o_i ∈ O. Similarly, it can be proved that deg [c] ≠ deg [d] , ∀ c, d ∈ V, o_i ∈ O.

Therefore, the degree and total degree of all vertices of $\tilde{T} (o_{i})$ are distinct. Hence $\tilde{T} (o_{i})$ is perfectly irregular PFG, ∀ o_i ∈ O. So P_G is perfectly irregular PFSG.□

Corollary 4.1. For a perfectly irregular PFSG, $\tilde{I}$ need not be constant.

Theorem 4.3. If in a PFSG P_G, $\deg (c_{j}) \neq \deg (c_{k}), \tilde{I_{μ}} (o_{i}) (c_{j}) = \tilde{I_{μ}} (o_{i}) (c_{k})$ and $\tilde{I_{μ}} (o_{i}) (c_{j}) = \tilde{I_{μ}} (o_{i}) (c_{k}), j \neq k, \forall c_{j}, c_{k} \in V .$ Then P_G is perfectly irregular PFSG.

Proof. Let $\tilde{I_{μ}} (o_{i}) (c_{j}) = \tilde{I_{μ}} (o_{i}) (c_{k}) = f_{i}$ and $\tilde{I_{μ}} (o_{i}) (c_{j}) = \tilde{I_{μ}} (o_{i}) (c_{k}) = f_{i}^{'}, j \neq k, \forall c_{j}, c_{k} \in V .$ Now deg (c_j) ≠ deg (c_k).

⇒deg_μ (c_j) ≠ deg_μ (c_k) and deg_ν (c_j) ≠ deg_ν (c_k), ∀ c_j, c_k ∈ V .

⇒deg_μ (c_j) + f_i ≠ deg_μ (c_k) + f_i and deg_ν (c_j) + f_i ≠ deg_ν (c_k) + f_i, ∀ c_j, c_k ∈ V .

$\Rightarrow \deg_{μ} (c_{j}) + \tilde{I_{μ}} (o_{i}) (c_{j}) \neq \deg_{μ} (c_{k}) + \tilde{I_{μ}} (o_{i}) (c_{k})$ and $\deg_{ν} (c_{j}) + \tilde{I_{ν}} (o_{i}) (c_{j}) \neq \deg_{ν} (c_{k}) + \tilde{I_{ν}} (o_{i}) (c_{k})$ , ∀ c_j, c_k ∈ V .

⇒deg_μ [c_j] ≠ deg_μ [c_k] and deg_ν [c_j] ≠ deg_ν [c_k] , j ≠ k, ∀ c_j, c_k ∈ V .

⇒deg [c_j] ≠ deg [c_k] , j ≠ k, ∀ c_j, c_k ∈ V .

Hence $\tilde{T} (o_{i})$ is perfectly irregular PFSG. Therefore, P_G is perfectly irregular PFSG.□

Definition 4.4. A PFSG P_G is said to be perfectly edge-irregular PFSG if $\tilde{T} (o_{i})$ is perfectly edge-irregular PFG for all o_i ∈ O, i.e,

The degrees of all edges of $\tilde{T} (o_{i})$ are distinct.

The total degrees of all edges of $\tilde{T} (o_{i})$ are distinct.

Theorem 4.4. If P_G is perfectly edge-irregular PFSG, then P_G is necessarily neighborly edge-irregular, totally neighborly edge-irregular PFSG.

Proof. Let P_G be a perfectly edge-irregular PFSG. Then the degree of all edges of $\tilde{T} (o_{i})$ is distinct. Thus the degree of every pair of edges is distinct. Therefore, $\tilde{T} (o_{i})$ is neighborly edge-irregular PFG, ∀ o_i ∈ O. Hence P_G is neighborly edge-irregular PFSG. Similarly, P_G is totally neighborly edge-irregular PFSG.□

But the converse of Theorem 4.4 may not be true as seen in the following example.

Example 4.3. Consider a PFSG P_G as shown in Fig. 10.

Fig. 10

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

deg (c₁c₂) = (0.7, 0.7), deg (c₂c₃) = (1.2, 1.2), deg (c₃c₄) = (0.7, 0.7). So, $\tilde{T} (o_{1})$ is neighborly edge-irregular PFG. Hence P_G is neighborly edge-irregular PFSG.

deg [c₁c₂] = (1.3, 1.3), deg [c₂c₃] = (1.9, 1.9), deg [c₃c₄] = (1.3, 1.3). So, $\tilde{T} (o_{1})$ is totally neighborly edge-irregular PFG. Hence P_G is totally neighborly edge-irregular PFSG. But deg (c₁c₂) = deg (c₃c₄). So, P_G is not perfectly edge-irregular PFSG.

Corollary 4.2. For a perfectly edge-irregular PFSG, $\tilde{J}$ need not be a constant.

Example 4.4. For the PFSG P_G of Fig. 11. We have deg (c₁c₂) = (1.6, 1.7) ≠ deg (c₂c₃) = (1.1, 1.2) ≠ deg (c₁c₄) = (1.8, 1.8) ≠ deg (c₂c₄) = (2.4, 2.3) ≠ deg (c₄c₅) = (1, 1.1) . deg [c₁c₂] = (2.2, 2.3) ≠ deg [c₂c₃] = (1.7, 1.8) ≠ deg [c₁c₄] = (2.3, 2.3) ≠ deg [c₂c₄] = (2.9, 2.9) ≠ deg [c₄c₅] = (1.7, 1.7) . Hence $\tilde{T} (o_{1})$ is perfectly edge-irregular PFG. Therefore, P_G is perfectly edge-irregular PFSG.

Fig. 11

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

Corollary 4.3. For a PFSG P_G, which is both neighborly edge-irregular and totally neighborly edge-irregular PFSG, if there exits an edge between every pair of vertices, then PFSG P_G need not be perfectly edge-irregular PFSG.

{\hat{O}}_{NDS}

and

{\hat{C}}_{NDS}

in a PFSG

In this section, we discuss the idea of ${\hat{O}}_{NDS}$ and ${\hat{O}}_{NDS}$ of a vertex in a PFSG, ${\hat{O}}_{NDS}$ and ${\hat{O}}_{NDS}$ of an edge in a PFSG.

Definition 5.1. The $({\hat{O}}_{NDS})$ (open neighborhood degrees sum) of a vertex c₁ in a PFSG P_G is denoted by $({\hat{O}}_{NDS}) (c_{1})$ and is defined as $({\hat{O}}_{NDS}) (c_{1}) = (\deg_{μ} (c_{1}) + \deg_{ν} (c_{1})) .$ The $({\hat{C}}_{NDS})$ (closed neighborhood degree sum) of a vertex c₁ in a PFSG P_G is denoted by $({\hat{C}}_{NDS}) [c_{1}]$ and is defined as $({\hat{C}}_{NDS}) [c_{1}] = (\deg_{μ} [c_{1}] + \deg_{ν} [c_{1}]) .$

Theorem 5.1. For a perfectly regular PFSG, 1 . The $({\hat{O}}_{NDS})$ ’s of all nodes are same.

2 . The $({\hat{C}}_{NDS})$ ’s of all nodes are same.

Proof. Suppose P_G is a perfectly regular PFSG.

(1) Since P_G is perfectly regular PFSG, P_G must be $(f_{i}, f_{i}^{'})$ regular PFSG, ∀ o_i ∈ O.

Then deg_μ (c₁) = f_i and $\deg_{ν} (c_{1}) = f_{i}^{'}$ , ∀ c₁ ∈ V, o_i ∈ O.

Therefore, $({\hat{O}}_{NDS}) (c_{1}) = \deg_{μ} (c_{1}) + \deg_{ν} (c_{1}) = f_{i} + f_{i}^{'}$ =constant, ∀ c₁ ∈ V, o_i ∈ O.

Thus the $({\hat{O}}_{NDS})$ ’s of all nodes of P_G are same.

(2) Since P_G is perfectly regular PFSG, P_G must be $(r_{i}, r_{i}^{'})$ totally regular PFSG, ∀ o_i ∈ O.

Then deg_μ [c₁] = r_i and $\deg_{ν} [c_{1}] = r_{i}^{'}$ , ∀ c₁ ∈ V, o_i ∈ O.

Therefore, $({\hat{C}}_{NDS}) [c_{1}] = \deg_{μ} [c_{1}] + \deg_{ν} [c_{1}] = r_{i} + r_{i}^{'}$ =constant, ∀ c₁ ∈ V, o_i ∈ O.

Thus the $({\hat{C}}_{NDS})$ ’s of all nodes of P_G are same.□

The converse of Theorem 5.1 may not be true as seen in the following example.

Example 5.1. For a PFSG P_G of Fig. 12, we have deg (c₁) = deg (c₃) = (0.6, 0.6) ,

deg (c₂) = deg (c₄) = (0.7, 0.5) ,

deg [c₁] = deg [c₃] = (1.2, 1.3) ,

deg [c₂] = deg [c₄] = (1.4, 1.1) .

So,

$({\hat{O}}_{NDS}) (c_{1})$ =0.6+0.6=1.2= $({\hat{O}}_{NDS}) (c_{3}),$

$({\hat{O}}_{NDS}) (c_{2})$ =0.7+0.5=1.2= $({\hat{O}}_{NDS}) (c_{4})$ .

Thus $\begin{matrix} ({\hat{O}}_{NDS}) (c_{1}) & = & ({\hat{O}}_{NDS}) (c_{2}) \\ = & ({\hat{O}}_{NDS}) (c_{3}) \\ = & ({\hat{O}}_{NDS}) (c_{4}) . \end{matrix}$ Now, $\begin{matrix} ({\hat{C}}_{NDS}) (c_{1}) & = & 1.2 + 1.3 = 2.5 = ({\hat{C}}_{NDS}) (c_{3}), \end{matrix}$ $\begin{matrix} ({\hat{C}}_{NDS}) (c_{2}) & = & 1.4 + 1.1 = 2.5 = ({\hat{C}}_{NDS}) (c_{4}) . \end{matrix}$ Thus $({\hat{C}}_{NDS}) (c_{1}) = ({\hat{C}}_{NDS}) (c_{2}) = ({\hat{C}}_{NDS}) (c_{3}) = ({\hat{C}}_{NDS}) (c_{4}) .$ Hence, $({\hat{O}}_{NDS})$ ’s and $({\hat{C}}_{NDS})$ ’s of all the vertices of P_G are same.

But deg (c₁) ≠ deg (c₂). So, P_G is not regular PFSG. Hence P_G is not perfectly regular PFSG.

Definition 5.2. The $({\hat{O}}_{NDS})$ (open neighborhood degree sum) of an edge cd in a PFSG P_G is denoted by $({\hat{O}}_{NDS}) (cd)$ and is defined as $({\hat{O}}_{NDS}) (cd) = (\deg_{μ} (cd) + \deg_{ν} (cd)) .$ The $({\hat{C}}_{NDS})$ (closed neighborhood degree sum) of an edge cd in a PFSG P_G is denoted by $({\hat{C}}_{NDS}) [cd]$ and is defined as $({\hat{C}}_{NDS}) [cd] = (\deg_{μ} [cd] + \deg_{ν} [cd]) .$

Theorem 5.2. For a perfectly edge-regular PFSG, 1 . The $({\hat{O}}_{NDS})$ ’s of all edges are same.

2 . The $({\hat{C}}_{NDS})$ ’s of all edges are same.

Proof. Suppose P_G is a perfectly edge-regular PFSG.

(1) Since P_G is perfectly edge-regular PFSG, P_G must be $(k_{i}, k_{i}^{'})$ edge-regular PFSG, ∀ o_i ∈ O.

Then deg_μ (cd) = k_i and $\deg_{ν} (cd) = k_{i}^{'}$ , ∀ cd ∈ E, o_i ∈ O. Therefore, $({\hat{O}}_{NDS}) (cd) = \deg_{μ} (cd) + \deg_{ν} (cd) = k_{i} + k_{i}^{'}$ =constant, ∀ cd ∈ E, o_i ∈ O. Thus the $({\hat{O}}_{NDS})$ ’s of all edges of P_G are same.

(2) Since P_G is perfectly edge-regular PFSG, P_G must be $(h_{i}, h_{i}^{'})$ totally edge-regular PFSG, ∀ o_i ∈ O. Then deg_μ [cd] = h_i and $\deg_{ν} [cd] = h_{i}^{'}$ , ∀ cd ∈ E, o_i ∈ O. Therefore, $({\hat{C}}_{NDS}) [cd] = \deg_{μ} [cd] + \deg_{ν} [cd] = h_{i} + h_{i}^{'}$ =constant, ∀ cd ∈ E, o_i ∈ O. Thus the $({\hat{C}}_{NDS})$ ’s of all edges of P_G are same.□

The converse of Theorem 5.2 may not be true as seen in the following example.

Example 5.2. For the PFSG P_G of Fig. 12, we have $({\hat{O}}_{NDS}) (c_{1} c_{3}) = 0 + 0 = 0 = ({\hat{O}}_{NDS}) (c_{2} c_{4})$ , $({\hat{C}}_{NDS}) [c_{1} c_{3}] = 0.6 + 0.6 = 1.2, ({\hat{C}}_{NDS}) [c_{2} c_{4}] = 0.7 + 0.5 = 1.2 .$ So, $({\hat{C}}_{NDS}) [c_{1} c_{3}] = ({\hat{C}}_{NDS}) [c_{2} c_{4}]$ . Hence $({\hat{O}}_{NDS})$ ’s and $({\hat{C}}_{NDS})$ ’s of all the edges of P_G are same. But deg [c₁c₃] ≠ deg [c₂c₄]. Thus P_G is not totally edge-regular PFSG. Hence P_G is not perfectly edge-irregular PFSG.

Fig. 12

PFSG $P_{G} = {\tilde{T} (o_{1})}$ .

6 Applications

In decision-making problems, PFSGs have numerous applications and used to handle with uncertainties from our daily life problems. In this section, we utilize the idea of PFSGs in a decision-making problems.

6.1 Selection of a teacher for the improvement of students:

Education is a way of instilling knowledge, skills, and values into people by various modes. It is the way by which an individual understands and gains knowledge and uses it for the betterment of him and the society. Schools and colleges are the medium of providing education to their students. Education helps in enhancing the mind of person as well it creates an individual perspective on a particular subject. Education plays a crucial role in the socio-economic development of a country. It also improves the state of mind, thoughts and ideas of a person.

An educated person has the ability to differentiate between right and wrong or good and evil. It is the foremost responsibility of a society to educate its citizens. A person becomes perfect with education as he is not only gaining something from it, but also contributing to the growth of a nation. In educational sector, teacher plays a main role. If he or she is good in communicate the knowledge to students, then it is easy for students to gain education and improve themselves.

This application, which we develop, aims to select a suitable teacher for the improvement of education system in mathematics department. Therefore, in the selection of teacher some attributes such as (i) patience teaching skills, (ii) strong communication and collaboration skill with students and parents, and (iii) subject matter expertise are under considerable. Here, we suppose that, we have four teachers in mathematics department. Let V = {t₁, t₂, t₃, t₄} be the set of four teachers to be consider as the universal set and O = {o₁, o₂, o₃} be the set of parameters that particularize the teacher, the parameters o₁, o₂ and o₃ symbolize the patience teaching skills, strong communication and collaboration skill with students and parents, and subject matter expertise, respectively. Consider the PFSS $(\tilde{I}, O)$ over V which defines the “efficiency of teacher” corresponding to the given parameters that we want to select. $(\tilde{J}, O)$ is a PFSS over E = {t₁t₂, t₁t₃, t₁t₄, t₂t₃, t₂t₄, t₃t₄} defines degree of membership and degree of non-membership of the connection between two teachers corresponding to the selected attributes o₁, o₂ and o₃. The PFGs $\tilde{T} (o_{1})$ , $\tilde{T} (o_{2})$ and $\tilde{T} (o_{3})$ of PFSGs $P_{G} = {\tilde{T} (o_{1}), \tilde{T} (o_{2}), \tilde{T} (o_{3})}$ corresponding to the parameters “patience teaching skills”, “strong communication and collaboration skill with students and parents”, and “subject matter expertise”, respectively are shown in Fig. 13.

Fig. 13

PFSG $P_{G} = {\tilde{T} (o_{1}), \tilde{T} (o_{2}), \tilde{T} (o_{3})}$ .

Now we calculate the $({\hat{O}}_{NDS})$ and $({\hat{C}}_{NDS})$ of each vertex corresponding to given attribute.

In $\tilde{T} (o_{1})$ , $(\deg_{μ} (t_{1}), \deg_{ν} (t_{1})) = (1.7, 2.2),$ $(\deg_{μ} (t_{2}), \deg_{ν} (t_{2})) = (1.6, 2.4),$ $(\deg_{μ} (t_{3}), \deg_{ν} (t_{3})) = (1.2, 2.4),$ $(\deg_{μ} (t_{4}), \deg_{ν} (t_{4})) = (1.7, 2.2) .$ Similarly, $(\deg_{μ} [t_{1}], \deg_{ν} [t_{1}]) = (2.5, 2.8),$ $(\deg_{μ} [t_{2}], \deg_{ν} [t_{2}]) = (2.2, 3.2),$ $(\deg_{μ} [t_{3}], \deg_{ν} [t_{3}]) = (1.6, 3.2),$ $(\deg_{μ} [t_{4}], \deg_{ν} [t_{4}]) = (2.4, 2.6) .$ Now, $({\hat{O}}_{NDS})_{1} (t_{1}) = \deg_{μ} (t_{1}) + \deg_{ν} (t_{1}) = 3.9,$ $({\hat{O}}_{NDS})_{1} (t_{2}) = \deg_{μ} (t_{2}) + \deg_{ν} (t_{2}) = 4,$ $({\hat{O}}_{NDS})_{1} (t_{3}) = \deg_{μ} (t_{3}) + \deg_{ν} (t_{3}) = 3.6,$ $({\hat{O}}_{NDS})_{1} (t_{4}) = \deg_{μ} (t_{4}) + \deg_{ν} (t_{4}) = 3.9 .$ Similarly, $({\hat{C}}_{NDS}) [t_{1}] = \deg_{μ} [t_{1}] + \deg_{ν} [t_{1}] = 5.3,$ $({\hat{C}}_{NDS})_{1} [t_{2}] = \deg_{μ} [t_{2}] + \deg_{ν} [t_{2}] = 5.4,$ $({\hat{C}}_{NDS})_{1} [t_{3}] = \deg_{μ} [t_{3}] + \deg_{ν} [t_{3}] = 4.8,$ $({\hat{C}}_{NDS})_{1} [t_{4}] = \deg_{μ} [t_{4}] + \deg_{ν} [t_{4}] = 5 .$ In $\tilde{T} (o_{2})$ ,

$(\deg_{μ} (t_{1}), \deg_{ν} (t_{1})) = (1.7, 2),$ $(\deg_{μ} (t_{2}), \deg_{ν} (t_{2})) = (1.4, 2),$ $(\deg_{μ} (t_{3}), \deg_{ν} (t_{3})) = (1.5, 2),$ $(\deg_{μ} (t_{4}), \deg_{ν} (t_{4})) = (1.7, 2) .$ Similarly, $(\deg_{μ} [t_{1}], \deg_{ν} [t_{1}]) = (2.3, 2.6),$ $(\deg_{μ} [t_{2}], \deg_{ν} [t_{2}]) = (1.9, 2.6),$ $(\deg_{μ} [t_{3}], \deg_{ν} [t_{3}]) = (2.1, 2.7),$ $(\deg_{μ} [t_{4}], \deg_{ν} [t_{4}]) = (2.4, 2.7) .$ Now, $({\hat{O}}_{NDS})_{2} (t_{1}) = \deg_{μ} (t_{1}) + \deg_{ν} (t_{1}) = 3.7,$ $({\hat{O}}_{NDS})_{2} (t_{2}) = \deg_{μ} (t_{2}) + \deg_{ν} (t_{2}) = 3.4,$ $({\hat{O}}_{NDS})_{2} (t_{3}) = \deg_{μ} (t_{3}) + \deg_{ν} (t_{3}) = 3.5,$ $({\hat{O}}_{NDS})_{2} (t_{4}) = \deg_{μ} (t_{4}) + \deg_{ν} (t_{4}) = 3.7 .$ Similarly, $({\hat{C}}_{NDS})_{2} [t_{1}] = \deg_{μ} [t_{1}] + \deg_{ν} [t_{1}] = 4.9,$ $({\hat{C}}_{NDS})_{2} [t_{2}] = \deg_{μ} [t_{2}] + \deg_{ν} [t_{2}] = 4.5,$ $({\hat{C}}_{NDS})_{2} [t_{3}] = \deg_{μ} [t_{3}] + \deg_{ν} [t_{3}] = 4.8,$ $({\hat{C}}_{NDS})_{2} [t_{4}] = \deg_{μ} [t_{4}] + \deg_{ν} [t_{4}] = 5.1 .$ In $\tilde{T} (o_{3})$ , $(\deg_{μ} (t_{1}), \deg_{ν} (t_{1})) = (1.8, 2),$ $(\deg_{μ} (t_{2}), \deg_{ν} (t_{2})) = (2, 2),$ $(\deg_{μ} (t_{3}), \deg_{ν} (t_{3})) = (1.8, 2),$ $(\deg_{μ} (t_{4}), \deg_{ν} (t_{4})) = (1.9, 2.1) .$

Similarly, $(\deg_{μ} [t_{1}], \deg_{ν} [t_{1}]) = (2.6, 2.5),$ $(\deg_{μ} [t_{2}], \deg_{ν} [t_{2}]) = (2.6, 2.8),$ $(\deg_{μ} [t_{3}], \deg_{ν} [t_{3}]) = (2.7, 2.4),$ $(\deg_{μ} [t_{4}], \deg_{ν} [t_{4}]) = (2.5, 2.9) .$

Now, $({\hat{O}}_{NDS})_{3} (t_{1}) = \deg_{μ} (t_{1}) + \deg_{ν} (t_{1}) = 3.8,$ $({\hat{O}}_{NDS})_{3} (t_{2}) = \deg_{μ} (t_{2}) + \deg_{ν} (t_{2}) = 4,$ $({\hat{O}}_{NDS})_{3} (t_{3}) = \deg_{μ} (t_{3}) + \deg_{ν} (t_{3}) = 3.8,$ $({\hat{O}}_{NDS})_{3} (t_{4}) = \deg_{μ} (t_{4}) + \deg_{ν} (t_{4}) = 4 .$

Similarly, $({\hat{C}}_{NDS})_{3} [t_{1}] = \deg_{μ} [t_{1}] + \deg_{ν} [t_{1}] = 5.1,$ $({\hat{C}}_{NDS})_{3} [t_{2}] = \deg_{μ} [t_{2}] + \deg_{ν} [t_{2}] = 5.4,$ $({\hat{C}}_{NDS})_{3} [t_{3}] = \deg_{μ} [t_{3}] + \deg_{ν} [t_{3}] = 5.1,$ $({\hat{C}}_{NDS})_{3} [t_{4}] = \deg_{μ} [t_{4}] + \deg_{ν} [t_{4}] = 5.4 .$

The decision is τ if $τ = (max {min ({\hat{O}}_{NDS})_{q}}, max {min ({\hat{C}}_{NDS})_{q}})$ $τ = (max {3.7, 3.4, 3.5, 3.7}, max {4.9, 4.5, 4.8, 5})$ $τ = (3.7, 5) .$ As $max ({\hat{O}}_{NDS}) (t_{i}) = 3.7 = max ({\hat{O}}_{NDS}) (t_{1}) = max ({\hat{O}}_{NDS}) (t_{4})$ , but $max ({\hat{C}}_{NDS}) (t_{1}) = 4.9, max ({\hat{C}}_{NDS}) (t_{4}) = 5 .$ Therefore, teacher t₄ is the most suitable teacher for the improvements of students.

6.2 Selection of Appropriate Cell Phone:

We establish an algorithm for most suitable selection of an object in a multiple criteria decision-making problem.

Algorithm:

1. Input the set of attributes o₁, o₂, ⋯ , o_m.

2. Input the PFSSs $(\tilde{I}, O)$ and $(\tilde{J}, O)$ .

3. Input the PFGs $\tilde{T} (o_{1}), \tilde{T} (o_{2}), \dots, \tilde{T} (o_{m}) .$

4. Compute the accuracy function of PFGs $\tilde{T} (o_{1}), \tilde{T} (o_{2}), \dots, \tilde{T} (o_{m})$ using formula $S_{jk} = (μ_{k})^{2} + (ν_{k})^{2}$ (1)

5. Compute the choice values of C_q = ∑_kS_jk for all j = 1, 2, ⋯ , n and q = 1, 2, ⋯ , l.

6. The decision is S_j if S_j = max {min C_q} .

7. If j has more than one value then any one of S_j may be chosen.

Probably connecting with the world has become equally important today as we take food for livelihood. One of the basic necessities of our life now include connectivity. The information of the world is now a click away or at the pores of our fingers. Cell phone is answer to the problem. We stay connected with a palm-top device known as cell phone. Now the question arises what phone, which type and which category. These questions are equally vital as the need of having cell phone. Only having a cell phone is not sufficient but the choice of an appropriate cell phone is actual quest of technology now-a-days.

This application, which we develop, aims to select a suitable cell phone for communication and connectivity. The performance of cell phone is badly affected by the wrong selection. The features of mobile phones are the set of capabilities, services and applications that they offer to their users. Therefore, in the selection of cell phone some features such as roaming feature, GPS navigation feature and memo recording are under considerable. Mr. Y should be an expert or at least familiar with the cell phone features, to select a suitable and best cell phone among the parameters, i.e, roaming feature (is allow us to communicate world widely without any hurdles like voice calls, text messages etc), GPS navigation feature and memo recording. Let V = {c₁, c₂, c₃, c₄, c₅} be the set of five cell phones to be consider as the universal set and O = {o₁, o₂, o₃} be the set of parameters that particularize the cell phone, the parameters o₁, o₂ and o₃ stands for roaming feature, GPS navigation feature and memo recording, respectively. Consider the PFSS $(\tilde{I}, O)$ over V which defines the “attractiveness of cell phone” corresponding to the given parameters that Mr. Y wants to select. $(\tilde{J}, O)$ is a PFSS over E = {c₁c₂, c₁c₃, c₁c₄, c₁c₅, c₂c₃, c₂c₄, c₂c₅, c₃c₄, c₃c₅, c₄c₅} defines degree of membership and degree of nonmembership of comparison between two cell phones corresponding to the selected attributes o₁, o₂ and o₃. The PFGs $\tilde{T} (o_{1})$ , $\tilde{T} (o_{2})$ and $\tilde{T} (o_{3})$ of PFSGs $P_{G} = {\tilde{T} (o_{1}), \tilde{T} (o_{2}), \tilde{T} (o_{3})}$ corresponding to the parameters “roaming feature”, “GPS navigation feature”, and “memo recording”, respectively are shown in Fig. 14.

Fig. 14

PFSG P_G= ${\tilde{T} (o_{1})$ , $\tilde{T} (o_{2})$ , $\tilde{T} (o_{3})}$ .

Tabular representation of accuracy values of $\tilde{T} (o_{1})$ , $\tilde{T} (o_{2})$ and $\tilde{T} (o_{3})$ with accuracy function S_jk = (μ_k) ² + (ν_k) ² and choice value for each cell phone j = 1, 2, 3, 4, 5 is given in Tables 2, 3, 4 for each parameter, respectively.

Table 2

Tabular representation of accuracy function values and choice values of $\tilde{T} (o_{1})$

Cell phones	c ₁	c ₂	c ₃	c ₄	c ₅	C _q
c ₁	0	0.97	0.8	0	0.89	2.66
c ₂	0.97	0	0.97	0.41	0	2.35
c ₃	0.8	0.97	0	0.41	0.41	2.59
c ₄	0	0.41	0.41	0	0.5	1.32
c ₅	0.89	0	0.41	0.5	0	1.8

Table 3

Tabular representation of accuracy function values and choice values of $\tilde{T} (o_{2})$

Cell phones	c ₁	c ₂	c ₃	c ₄	c ₅	C _q
c ₁	0	1	0.89	0.89	0	2.78
c ₂	1	0	0.89	0.89	1	3.78
c ₃	0.89	0.89	0	0.61	0	2.39
c ₄	0.89	0.89	0.61	0	0.61	3
c ₅	0	1	0	0.61	0	1.61

Table 4

Tabular representation of accuracy function values and choice values of $\tilde{T} (o_{3})$

Cell phones	c ₁	c ₂	c ₃	c ₄	c ₅	C _q
c ₁	0	1	0.89	1	1	3.89
c ₂	1	0	0.74	0.85	1	2.59
c ₃	0.89	0.74	0	0.5	0.5	2.63
c ₄	1	0.85	0.5	0	0.61	2.96
c ₅	1	0	0.5	0.61	0	2.11

The decision is S_j if S_j = max {min C_q} = max {2.66, 2.35, 2.39, 1.32, 1.61} =2.66. Clearly, the maximum score value is 2.66, scored by c₁. Mr. Y will purchase the cell phone c₁.

7 Conclusions

A Pythagorean fuzzy soft graph, an extension of intuitionistic fuzzy soft graph, is a powerful tool to handle the pairwise relationship between objects corresponding to different parameters and relax the condition of IFSGs. In this research article, we have depicted the idea of perfectly regular PFSGs, perfectly irregular PFSGs and discussed their different results. We have introduced the idea of open neighborhood degree sum of PFSGs, closed neighborhood degree sum of PFSGs and used these in decision-making problems. We have applied the concept of PFSGs in daily life problems, including selection of a teacher and selection of a cell phone and got appropriate results by using score function. We plan to extend our study to (i) Pythagorean fuzzy soft hypergraphs; (ii) Rough Pythagorean fuzzy soft graphs.

Conflict of Interest: The authors declare that they have no conflict of interest.

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