Certain types of m -polar interval-valued fuzzy graph

Abbreviations

FS

Fuzzy set

1 Introduction

Observing the vast application of graph theory which is a mathematical structure defined as an ordered pair G = (V, E) together with a set of vertices V and a set of edges E. A fuzzy model is needed to describe a fuzzy graph when there is a vagueness either in vertices or in edges or in both. Recently, applications of graph theory are mostly used in the fields of computer networks, electrical network, coding theory, operational research, architecture, data mining etc. In 1736 graph theoretical concept was first modeled by Swiss mathematician Euler from the concept of Konigsberg bridge problem.

For solving various decision making problems in real life fuzzy set, a non-empty set V together with a fuzzy set and a fuzzy relation have been applied. Ten years after Zadeh’s concept on fuzzy set in 1965, Kauffman [1] proposed the basic idea of fuzzy graph. Later Rosenfeld [2] developed this concept which have been successfully applied in information theory, modern science, engineering science, manufacturing, medical diagnosis, etc. In fact, he also detailed definition of several fuzzy concepts such as fuzzy vertex, edges, cycles, paths, connectedness, and so on. In 1994, Zhang [3, 4] gave the extension of the concept of fuzzy set and the idea of bipolar fuzzy sets. Mordeson and Peng [5–7] incorporated the idea of complement fuzzy graph and demonstrated a few properties and operations on it. Later this concept was described by Sunitha and Kumar [8, 9].

In 2006, Nagoorgani and Chandrasekaran [10] studied on the degree of a node in fuzzy graph. Later on Shannon and Atanassov [11] elaborated the conceptualization of intuitionistic fuzzy relation and intuitionistic fuzzy graphs and described various properties. In 2008, Nagoorgani and Radha [12] introduced regular fuzzy graph which was later used to find density. In 2009, Hongmei and Lianhua [13] made a deeper study on IVFG, in 2011, Akram and Dudek [14] and in 2018 Rashmanlou et al. [15] studied some operations on it. Hawary [16] defined complete fuzzy graph and studied some new operations on it. Nagoorgani and Malarvizhu [17, 18] detailed isomorphic properties on fuzzy graphs and also gave the definition of the self-complementary fuzzy graphs. Then, Akram [19] presented the notations of bipolar fuzzy graphs and further researched on it [20, 52]. Later, the extension of bipolar fuzzy set and the idea of m-polar fuzzy sets (m-PFS) were introduced by Chen et al. [21] in 2014. Later on Rashmanlou briefly described categorical properties on it [22]. Samanta and Pal investigated on fuzzy tolerance graph [23], fuzzy threshold graph [24], fuzzy k-competition graphs [25], p-competition fuzzy graphs and also fuzzy planar graphs [26]. Some properties related to isomorphism and complement on IVFG were studied by Talebi and Rashmanlou [27]. Later, Ghorai and Pal [28, 29] described various properties on m-PFGs. They examined isomorphic properties on m-PFG. Ghorai and Pal [30] described the density of m-polar fuzzy graphs and some operations on it. Raut and Pal [31] studied on fuzzy permutation graph and its complements in 2018. Various extensions of fuzzy graphs [32] have been incorporated to handle the vagueness of the real world problem [33]. In this uncertain environment to solve various decision making problems [34], fuzzy graphs have used several models. Different types of research on generalized fuzzy graphs were discussed in [36–42]. A through study on shortest path problem has made by Nayeem and Pal [51]. Some remarkable research studies regarding fuzzy graph are given in Table 1.

The main contribution of this study are as follows:

Concept of m-PIVFGs and complement of m-PIVFGs are introduced with examples.

The rest of the article is arranged as follows: Section 1 describes historical backgrounds of FSs, FGs, m-PFGs and IVFGs. Our aim and motivation are also written in this section. In section 2, some preliminary concepts regarding FGs, m-PFGs, IVFGs are described. Section 3 discussed about m-PIVFG and complement of m-PIVFG. In section 4, open and closed neighbourhood degree of a vertex of m-PIVFG are defined. Balanced m-PIVFG, regular m-PIVFG and also density of an m-PIVFG are also investigated and supported with examples. Some theorems related to these definitions are discussed with their proofs. In section 5, we define the irregularity of an m-PIVFGs with examples and some theorems with their proofs. Section 6 demonstrated the definition of busy nodes, free nodes with the help of examples and some basic theorems related to these. Section 7 is based on a summary of this article along with future works.

In this part of this paper we briefly study some basic terminologies of FS, FGs, m-PFS and m-PFGs which are used in this paper. The basic definition of IFGs are also demonstrated.

Fuzzy sets, extension of the classical notion of the set were proposed by Zadeh [43] in 1965. A fuzzy set is a set of elements having degrees of membership. A fuzzy set A = (S, m) is a pair of set S and a membership function m : S → [0, 1]. Throughout this article, we use the notation G^* as a classical graph and G as a graph defined in a fuzzy environment.

Definition 2.1. ([21]) An m-PFS (or a [0, 1]  ^m -set) on a set X is a function A : X → [0, 1]  ^m . The collection of all m-PFS on X is addressed as m (X).

Definition 2.2. [44] Let A be an m-PFS on X. An m-polar fuzzy relation on A is an m-PFS B of X × X such that B (x, y) ≤ min {A (x) , A (y)} ∀x, y ∈ X i.e. for each i = 1, 2, . . . , m and ∀x, y ∈ X p _i  ∘ B (x, y) ≤ min {p _i  ∘ A (x) , p _i  ∘ A (y)}, where p _i means membership value of ith pole.

Definition 2.3. [44] An m-PFG of a crisp graph G^* = (V, E) is a pair G = (A, B) where A : V → [0, 1]  ^m is an m-PFS in V and B : V × V → [0, 1]  ^m is an m-PFS in V × V such that for each i = 1, 2, . . . , m; p _i  ∘ B (xy) ≤ min {p _i  ∘ A (x) , p _i  ∘ A (y)} ∀xy ∈ V × V and B (xy) =0 ∀xy ∈ (V × V) - E, where 0 = (0, 0, . . . , 0) is the smallest element in [0, 1]  ^m . A and B are known as the m-polar fuzzy vertex set and edge set of G, respectively.

Definition 2.4. [45] An IVFS A on V is defined as A = { ( x , [ μ A l ( x ) , μ A u ( x ) ] ) : x ∈ V } , where μ A l ( x ) and μ A u ( x ) are fuzzy subsets on V such that μ A l ( x ) ≤ μ A u ( x ) , ∀x ∈ V.

Based on this set a graph called IVFG is defined.

Definition 2.5. [13] By an IVFG of a crisp graph G^* = (V, E) we mean G = (A, B), where A = [ μ A l ( x ) , μ A u ( x ) ] is an IVFS on V and B = [ μ B l ( xy ) , μ B u ( xy ) ] is an IVFS on edge set E, such that μ B l ( xy ) ≤ min { μ A l ( x ) , μ A l ( y ) } ,   μ B u ( xy ) ≤ min { μ A u ( x ) , μ A u ( y ) }   ∀xy ∈ E.

Definition 2.6. [46] An m-PIVFG of a graph G^* = (V, E) is a pair G = (V, A, B) consists of a non-empty set V together with pair of interval-valued function A : V → [0, 1]  ^m is an m-PFS in V and B : V × V → [0, 1]  ^m and μ : V × V → [0, 1]  ^m for each i = 1, 2, . . . , m; p i ∘ μ A ( x ) = [ p i ∘ μ A l ( x ) , p i ∘ μ A u ( x ) ] , 0 ≤ μ A l ( x ) ≤ μ A u ( x ) ≤ 1 and p i ∘ μ B ( xy ) = [ p i ∘ μ B l ( xy ) , p i ∘ μ B u ( xy ) ] , 0 ≤ μ B l ( xy ) ≤ μ B u ( xy ) ≤ 1 and for every i, the interval number of a vertex x and of an edge xy in G satisfying p i ∘ μ B l ( xy ) ≤ p i ∘ min { μ A l ( x ) , μ A l ( y ) } , p i ∘ μ B u ( xy ) ≤ p i ∘ min { μ A u ( x ) , μ A u ( y ) } , ∀x, y ∈ V.

Definition 2.7. [46] Let G = (V, A, B) of G^* = (V, E) be an m-PIVFG. The complement G ¯ of G is an m-PIVFG G ¯ = ( A ¯ , B ¯ ) , where p i ∘ μ B ¯ ( xy ) = [ p i ∘ μ B ¯ l ( xy ) , p i ∘ μ B ¯ u ( xy ) ] , p i ∘ μ B ¯ l ( xy ) = min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } - p i ∘ μ B l ( xy ) , and p i ∘ μ B ¯ u ( xy ) = min { p i ∘ μ A u ( x ) , p i ∘ μ A u ( y ) } - p i ∘ μ B u ( xy ) for every i and x, y ∈ V.

Lots of works have been done on this graph [47–50]. In the following section balanced m-PIVFG, regularity m-IVFG are defined with suitable examples. Some theorems related to regularity of m-IVFG are also discussed.

3 Balanced m-PIVFG

Herein, the concept of neighbourhood of m-PIVFG are introduced and demonstrated with examples. Also, in this part we described balanced and regularity of m-PIVFG with appropriate examples and some theorems related to these are also explained with their proofs.

Definition 3.1. Let G = (V, A, B) be an m-PIVFG of a crisp graph G^* = (V, E). The open-neighbourhood degree of a vertex x in G is defined by deg ( x ) = ( [ p 1 ∘ deg μ l ( x ) , p 1 ∘ deg μ u ( x ) ] , [ p 2 ∘ deg μ l ( x ) , p 2 ∘ deg μ u ( x ) ] , . . . , [ p i ∘ deg μ l ( x ) , p i ∘ deg μ u ( x ) ] , . . . , [ p m ∘ deg μ l ( x ) , p m ∘ deg μ u ( x ) ] ) , for i = 1, 2, . . . , m where p i ∘ deg μ l ( x ) and p i ∘ deg μ u ( x ) is defined by p i ∘ deg μ l ( x ) = ∑ y ∈ N ( x ) p i ∘ μ B l ( xy ) and p i ∘ deg μ u ( x ) = ∑ y ∈ N ( x ) p i ∘ μ B u ( xy ) , for every i. Moreover, p i ∘ μ B l ( xy ) > 0 , p i ∘ μ B u ( xy ) > 0 , for xy ∈ E and p i ∘ μ B l ( xy ) = 0 , p i ∘ μ B u ( xy ) = 0 , for xy ∉ E, x, y ∈ A.

Definition 3.2. Let G = (V, A, B) be an m-PIVFG of a crisp graph G^* = (V, E). If all the vertices have a similar open-neighborhood degree, then G is is known as a regular m-PIVFG.

Example 3.1. Let G = (V, A, B) be an m-PIVFG (Fig. 1) where, A = ( a < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.6 , 0.7 ] > , b < [ 0.2 , 0.3 ] , [ 0.4 , 0.5 ] , [ 0.6 , 0.8 ] > , c < [ 0.1 , 0.4 ] , [ 0.3 , 0.6 ] , [ 0.6 , 0.7 ] > , d < [ 0.1 , 0.5 ] , [ 0.3 , 0.8 ] , [ 0.5 , 0.6 ] > , e < [ 0.1 , 0.2 ] , [ 0.5 , 0.6 ] , [ 0.6 , 0.9 ] > , f < [ 0.1 , 0.3 ] , [ 0.3 , 0.5 ] , [ 0.6 , 0.8 ] > ) . B = ( ab < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.5 , 0.8 ] > , bc < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.5 , 0.8 ] > , cd < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.5 , 0.8 ] > , de < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.5 , 0.6 ] > , ef < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.5 , 0.8 ] > , af < [ 0.1 , 0.2 ] , [ 0.3 , 0.5 ] , [ 0.5 , 0.8 ] > ) .

OPEN IN VIEWER

Fig. 1

An open neighborhood degree of a 3-PIVFG G.

Here , p 1 ∘ deg μ l ( a ) = ∑ y ∈ N ( a ) p 1 ∘ μ B l ( ay ) = 0.1 + 0.1 = 0.2 , p 2 ∘ deg μ l ( a ) = ∑ y ∈ N ( a ) p 2 ∘ μ B l ( ay ) = 0.3 + 0.3 = 0.6 , p 3 ∘ deg μ l ( a ) = ∑ y ∈ N ( a ) p 3 ∘ μ B l ( ay ) = 0.5 + 0.5 = 1.0 . Similarly , p 1 ∘ deg μ u ( a ) = 0.4 , p 2 ∘ deg μ u ( a ) = 1.0 , p 3 ∘ deg μ u ( a ) = 1.2 .

Thus, deg (a) = ([0.2, 0.4] , [0.6, 1.0] , [1.0, 1.2]),

deg (b) = ([0.2, 0.4] , [0.6, 1.0] , [1.0, 1.2]),

deg (c) = ([0.2, 0.4] , [0.6, 1.0] , [1.0, 1.2]).

Definition 3.3. Let us consider an m-PIVFG G = (V, A, B) of a crisp graph G^* = (V, E). Then the closed-neighbourhood degree of a vertex x in the m-PIVFG G is defined by deg [ x ] = ( [ p 1 ∘ deg μ l [ x ] , p 1 ∘ deg μ u [ x ] ] , [ p 2 ∘ deg μ l [ x ] , p 2 ∘ deg μ u [ x ] ] , . . . , [ p i ∘ deg μ l [ x ] , p i ∘ deg μ u [ x ] ] , . . . , [ p m ∘ deg μ l [ x ] , p m ∘ deg μ u [ x ] ] ) , where, p i ∘ deg μ l [ x ] = p i ∘ deg μ l ( x ) + p i ∘ μ l ( x ) , p i ∘ deg μ u [ x ] = p i ∘ deg μ u ( x ) + p i ∘ μ u ( x ) , for every i.

Definition 3.4. Let G = (V, A, B) be an m-PIVFG of a crisp graph G^* = (V, E). If all the vertices of G have a similar closed neighborhood degree for every pole, then G is known as a totally regular m-PIVFG.

Example 3.2. Let us consider a graph G such that vertex set V = {a, b, c, d} and edge set E = {ab, bc, cd, ad}(Fig. 2). Let A be an m-PIVFS on V and B be an m-PIVFS on E defined by A = ( a < [ 0.2 , 0.4 ] , [ 0.3 , 0.5 ] , [ 0.4 , 0.7 ] > , b < [ 0.2 , 0.4 ] , [ 0.3 , 0.5 ] , [ 0.4 , 0.7 ] > , c < [ 0.2 , 0.4 ] , [ 0.3 , 0.5 ] , [ 0.4 , 0.7 ] > , d < [ 0.2 , 0.4 ] , [ 0.3 , 0.5 ] , [ 0.4 , 0.7 ] > ) . B = ( ab < [ 0.1 , 0.3 ] , [ 0.2 , 0.5 ] , [ 0.3 , 0.7 ] > , bc < [ 0.1 , 0.3 ] , [ 0.2 , 0.5 ] , [ 0.3 , 0.7 ] > , cd < [ 0.1 , 0.3 ] , [ 0.2 , 0.5 ] , [ 0.3 , 0.7 ] > , da < [ 0.1 , 0.3 ] , [ 0.2 , 0.5 ] , [ 0.3 , 0.7 ] > ) .

By calculation , deg ( a ) = ( [ 0.2 , 0.6 ] , [ 0.4 , 1.0 ] , [ 0.6 , 1.4 ] ) , deg [ a ] = ( [ 0.4 , 1.0 ] , [ 0.7 , 1.5 ] , [ 1.0 , 2.1 ] ) , deg ( b ) = ( [ 0.2 , 0.6 ] , [ 0.4 , 1.0 ] , [ 0.6 , 1.4 ] ) , deg [ b ] = ( [ 0.4 , 1.0 ] , [ 0.7 , 1.5 ] , [ 1.0 , 2.1 ] ) ,

deg ( c ) = ( [ 0.2 , 0.6 ] , [ 0.4 , 1.0 ] , [ 0.6 , 1.4 ] ) , deg [ c ] = ( [ 0.4 , 1.0 ] , [ 0.7 , 1.5 ] , [ 1.0 , 2.1 ] ) , deg ( d ) = ( [ 0.2 , 0.6 ] , [ 0.4 , 1.0 ] , [ 0.6 , 1.4 ] ) , deg [ d ] = ( [ 0.4 , 1.0 ] , [ 0.7 , 1.5 ] , [ 1.0 , 2.1 ] ) .

OPEN IN VIEWER

Fig. 2

Closed neighborhood degree of a 3-PIVFG G.

Thus, G is a regular m-PIVFG as well as totally regular m-PIVFG.

Definition 3.5. An m-PIVFG G = (V, A, B) of G^* = (V, E) is said to be complete if p i ∘ μ B l ( xy ) = min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } and p i ∘ μ B u ( xy ) = min { p i ∘ μ A u ( x ) , p i ∘ μ A u ( y ) } , for every pair of vertices x, y ∈ V.

By definition it is obvious that every complete m-PIVFG is totally regular, but the converse is not true.

Example 3.3. From Example 3.2, as seen in Fig. 2, it is observed that G is totally regular but not complete.

Theorem 3.1. Let G = (V, A, B) be an m-PIVFG of G^* = (V, E). Then A = < [ p 1 ∘ μ A l , p 1 ∘ μ A u ] , [ p 2 ∘ μ A l , p 2 ∘ μ A u ] , . . . , [ p i ∘ μ A l , p i ∘ μ A u ] , . . . , [ p m ∘ μ A l , p m ∘ μ A u ] > , for i = 1, 2, . . . , m is a constant mapping if and only if the subsequent properties are equivalent:

G is a regular m-PIVFG.

Proof. Suppose that A = < [ p 1 ∘ μ A l , p 1 ∘ μ A u ] , [ p 2 ∘ μ A l , p 2 ∘ μ A u ] , . . . , [ p i ∘ μ A l , p i ∘ μ A u ] , . . . , [ p m ∘ μ A l , p m ∘ μ A u ] > . Let p i ∘ μ A l ( x ) = g i and p i ∘ μ A u ( x ) = h i ; for i = 1, 2, . . . , m and ∀x ∈ V.

Now, (i) ⇒ (ii)

Let us assume that G is an m-PIVFG. Let p i ∘ deg μ l ( x ) = r i and p i ∘ deg μ u ( x ) = s i ; for i = 1, 2, . . . , m and ∀x ∈ V. So, p 1 ∘ deg μ l [ x ] = p 1 ∘ deg μ l ( x ) + p 1 ∘ μ A l ( x ) = r 1 + g 1 and p 1 ∘ deg μ u [ x ] = p 1 ∘ deg μ u ( x ) + p 1 ∘ μ A u ( x ) = s 1 + h 1 ; ∀x ∈ V. Similarly for others. In general, p i ∘ deg μ l [ x ] = r i + g i , p i ∘ deg μ u [ x ] = s i + h i , for i = 1, 2, . . . , m and ∀x ∈ V. Thus, G is a totally regular m-PIVFG.

Again, (ii) ⇒ (i)

Let us consider that G is totally regular m-PIVFG. Afterwards, p i ∘ deg μ l [ x ] = c i , p i ∘ deg μ u [ x ] = k i ; ∀x ∈ V. Now, for i=1, p 1 ∘ deg μ l [ x ] = c 1 or, p 1 ∘ deg μ l ( x ) + p 1 ∘ μ A l ( x ) = c 1 or, p 1 ∘ deg μ l ( x ) + g 1 = c 1 or, p 1 ∘ deg μ l ( x ) = c 1 - g 1 . Also, p 1 ∘ deg μ u [ x ] = k 1 or, p 1 ∘ deg μ u ( x ) + p 1 ∘ μ A u ( x ) = k 1 or, p 1 ∘ deg μ u ( x ) + h 1 = k 1 or, p 1 ∘ deg μ u ( x ) = k 1 - h 1 . In general, p i ∘ deg μ l ( x ) = c i - g i , p i ∘ deg μ u ( x ) = k i - h i ∀ x ∈ V . Therefore, G is a regular m-PIVFG. Consequently, (i) and (ii) are equivalent.

Theorem 3.2. Let G = (V, A, B) be an m-PIVFG where G^* is a crisp graph with an odd cycle. Then, G is a regular m-PIVFG if and only if B is a constant map.

Proof. Let B = < [ p 1 ∘ μ B l , p 1 ∘ μ B u ] , [ p 2 ∘ μ B l , p 2 ∘ μ B u ] , . . . , [ p i ∘ μ B l , p i ∘ μ B u ] , . . . , [ p m ∘ μ B l , p m ∘ μ B u ] > is a constant map, for i = 1, 2, . . . , m. Let, p i ∘ μ B l ( xy ) = g i , p i ∘ μ B u ( xy ) = h i , ∀xy ∈ E. Then, p i ∘ deg μ l ( x ) = 2 g i , p i ∘ deg μ u ( x ) = 2 h i , ∀x ∈ V. Thus, G is a regular m-PIVFG. Conversely, let G be a regular m-PIVFG. Thereafter, p i ∘ deg μ l ( x ) = r i , p i ∘ deg μ u ( x ) = s i , ∀x ∈ V. Also, let e₁, e₂, . . . , e_2n+1 be the edges of G in that order. Let p i ∘ μ B l ( e 1 ) = c i , p i ∘ μ B u ( e 1 ) = k i . Then, p i ∘ μ B l ( e 2 ) = r i - c i , p i ∘ μ B l ( e 3 ) = r i - ( r i - c i ) = c i , p i ∘ μ B l ( e 4 ) = r i - c i , p i ∘ μ B l ( e 5 ) = r i - ( r i - c i ) = c i and so on. Therefore, p i ∘ μ B l ( e j ) = { c i if j is odd r i - c i if j is even for i = 1, 2, . . . , m and j = 1, 2, . . . , (2n + 1). Thus p i ∘ μ B l ( e 1 ) = c i and p i ∘ μ B l ( e 2 n + 1 ) = c i . So, e₁ and e_2n+1 incident at a vertex v₁. Then, p i ∘ deg μ l ( v 1 ) = r i or, p i ∘ deg μ l ( e 1 ) + p i ∘ deg μ l ( e 2 n + 1 ) = r i or, c _i  + c _i  = r _i or, c _i  = r _i /2. This proves that p i ∘ μ B l ( xy ) is a regular function.

Similarly, p i ∘ μ B u ( e 2 ) = s i - k i , p i ∘ μ B u ( e 3 ) = s i - ( s i - k i ) = k i , p i ∘ μ B u ( e 4 ) = s i - k i , p i ∘ μ B u ( e 5 ) = s i - ( s i - k i ) = k i and so on. Therefore, p i ∘ μ B u ( e j ) = { k i if j is odd s i - k i if j is even for i = 1, 2, . . . , m and j = 1, 2, . . . , (2n + 1). Thus p i ∘ μ B u ( e 1 ) = k i and p i ∘ μ B u ( e 2 n + 1 ) = k i . So, e₁ and e_2n+1 incident at a vertex v₁. Now, p i ∘ deg μ u ( v 1 ) = s i or, p i ∘ deg μ u ( e 1 ) + p i ∘ deg μ u ( e 2 n + 1 ) = s i or, k _i  + k _i  = s _i Thus, c _i  = r _i /2. This proves that p i ∘ μ B u ( xy ) is a regular function ∀xy ∈ E. Therefore, B is a constant map.

Corollary 1. Let G be an m-PIVFG, where G^* is a usual graph having even cycle. Thereafter, G is a regular m-PIVFG if and only if B = < [ p 1 ∘ μ B l , p 1 ∘ μ B u ] , [ p 2 ∘ μ B l , p 2 ∘ μ B u ] , . . . , [ p i ∘ μ B l , p i ∘ μ B u ] , . . . , [ p m ∘ μ B l , p m ∘ μ B u ] > for i = 1, 2, . . . , m is a constant map or alternate edges have the same membership degrees.

Definition 3.6. The density of an m-PIVFG G = (V, A, B) is D (G) = ([p₁ ∘ D ^l  (G) , p₁ ∘ D ^u  (G)] , [p₂ ∘ D ^l  (G) , p₂ ∘ D ^u  (G)] , . . . , [p _i  ∘ D ^l  (G) , p _i  ∘ D ^u  (G)] , . . . , [p _m  ∘ D ^l  (G) , p _m  ∘ D ^u  (G)]); where p i ∘ D l ( G ) = 2 ∑ xy ∈ E p i ∘ μ B l ( xy ) ∑ x , y ∈ V min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } p i ∘ D u ( G ) = 2 ∑ xy ∈ E p i ∘ μ B u ( xy ) ∑ x , y ∈ V min { p i ∘ μ A u ( x ) , p i ∘ μ A u ( y ) } for x, y ∈ V, provided D ^l  (G) < D ^u  (G). If the condition is not true then we have to define the density by other way.

Definition 3.7. An m-PIVFG G is balanced if D (H) ≤ D (G) i.e, p _i  ∘ D ^l  (H) ≤ p _i  ∘ D ^l  (G) and p _i  ∘ D ^u  (H) ≤ p _i  ∘ D ^u  (G) for all subgraphs H of G.

Definition 3.8. An m-PIVFG G is strictly balanced if for every x, y ∈ V, D (H) = D (G) for all non-empty subgraphs.

Theorem 3.3. Every complete m-PIVFG is balanced.

Proof. Let G = (V, A, B) be a complete m-PIVFG. Thereafter, p i ∘ μ B l ( xy ) = min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } and p i ∘ μ B u ( xy ) = min { p i ∘ μ A u ( x ) , p i ∘ μ A u ( y ) } for every pair of vertices x, y ∈ V and ∀ i. Consequently, ∑ xy ∈ E p i ∘ μ B l ( xy ) = ∑ x , y ∈ V min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } and ∑ xy ∈ E p i ∘ μ B u ( xy ) = ∑ x , y ∈ V min { p i ∘ μ A u ( x ) , p i ∘ μ A u ( y ) } . p i ∘ D l ( G ) = 2 ∑ xy ∈ E p i ∘ μ B l ( xy ) ∑ x , y ∈ V min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } = 2 ∑ x , y ∈ V min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } ∑ x , y ∈ V min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } = 2 .

Similarly we get, p _i  ∘ D ^u  (G) =2, ∀ i. Also, every subgraph of a complete m-PIVFG is complete. Hence, it can be easily checked that D (H) = ([2, 2] , [2, 2] , . . . , [2, 2]) for m pole and for every H ⊆ G. Thus G is balanced m-PIVFG.

Note 1. Converse of above theorem is not true, that is every balanced m-PIVFG is not necessary complete.

Note 2. Let G be a self-complementary m-PIVFG. Then, D [G] = ([1, 1] , [1, 1] , . . . , [1, 1]).

Theorem 3.4. Let G be strictly balanced m-PIVFG and G ¯ be its complement; thereafter, D ( G ) + D ( G ¯ ) = ( [ 2 , 2 ] , [ 2 , 2 ] , ⋯ , [ 2 , 2 ] ) .

Proof. Let G be a strictly balanced m-PIVFG and G ¯ be the complement. Let H be any non-empty sub graph of G. Since, G is a strictly balanced m-PIVFG then D (G) = D (H) for any H ⊆ G and x, y ∈ V. In G ¯ ,

p i ∘ μ B ¯ l ( xy ) = min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } - p i ∘ μ B l ( xy )

p i ∘ μ B ¯ l ( xy ) min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } = 1 - p i ∘ μ B l ( xy ) min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } . Taking summation both sides,

∑ p i ∘ μ B ¯ l ( xy ) min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } = 1 - ∑ p i ∘ μ B l ( xy ) min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } , for every x, y  ∈  V. Multiplying by 2 both sides, 2 ∑ p i ∘ μ B ¯ l ( xy ) min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) } = 2 - 2 ∑ p i ∘ μ B l ( xy ) min { p i ∘ μ A l ( x ) , p i ∘ μ A l ( y ) }

i.e, p i ∘ D i l ( G ¯ ) = 2 - p i ∘ D i l ( G ) . Similarly, p i ∘ D u ( G ¯ ) = 2 - p i ∘ D u ( G ) . Again, D ( G ) + D ( G ¯ ) = ( [ p 1 ∘ D l ( G ) , p 1 ∘ D u ( G ) ] , [ p 2 ∘ D l ( G ) , p 2 ∘ D u ( G ) ] , ⋯ , [ p m ∘ D l ( G ) , p m ∘ D u ( G ) ] ) + ( [ p 1 ∘ D l ( G ¯ ) , p 1 ∘ D u ( G ¯ ) ] , [ p 2 ∘ D l ( G ¯ ) , p 2 ∘ D u ( G ¯ ) ] , . . . , [ p m ∘ D l ( G ¯ ) , p m ∘ D u ( G ¯ ) ] ) = ( [ p 1 ∘ D l ( G ) + p 1 ∘ D l ( G ¯ ) , p 1 ∘ D u ( G ) + p 1 ∘ D u ( G ¯ ) ] , [ p 2 ∘ D l ( G ) + p 2 ∘ D l ( G ¯ ) , p 2 ∘ D u ( G ) + p 2 ∘ D u ( G ¯ ) ] , . . . , [ p m ∘ D l ( G ) + p m ∘ D l ( G ¯ ) , p m ∘ D u ( G ) + p m ∘ D u ( G ¯ ) ] ) = ( [ 2 , 2 ] , [ 2 , 2 ] , ⋯ , [ 2 , 2 ] ) . This completes the proof.

Note 3. The complement of a strictly balanced m-PIVFG is also strictly balanced.

Note 4. Let G₁ and G₂ be isomorphic m-PIVFG. If G₂ is balanced then G₁ is also balanced.

In this present section, first, the irregularity in m-PIVFG with suitable examples are defined and some theorems related to that are described.

Definition 4.1. Let G = (V, A, B) be an m-PIVFG on G^* = (V, E). If there exists a vertex in G which is adjacent to vertices with distinct neighborhood degree, then G is called an irregular m-PIVFG.

Example 4.1. Let us assume an m-PIVFG G = (V, A, B) (Fig. 3), where A = ( a < [ 0.1 , 0.3 ] , [ 0.2 , 0.7 ] , [ 0.3 , 0.4 ] > , b [ 0.3 , 0.4 ] , [ 0.2 , 0.5 ] , [ 0.2 , 0.5 ] , c [ 0.2 , 0.3 ] , [ 0.4 , 0.5 ] , [ 0.1 , 0.2 ] ) B = ( ab < [ 0.1 , 0.2 ] , [ 0.2 , 0.5 ] , [ 0.1 , 0.2 ] > , bc < [ 0.2 , 0.4 ] , [ 0.3 , 0.4 ] , [ 0.3 , 0.6 ] > , ca < [ 0.1 , 0.3 ] , [ 0.2 , 0.3 ] , [ 0.1 , 0.3 ] > )

OPEN IN VIEWER

Fig. 3

An irregular 3-PIVFG G.

Here, deg (a) = ([0.2, 0.5] , [0.4, 0.8] , [0.2, 0.5]),

deg (b) = ([0.3, 0.6] , [0.5, 0.9] , [0.4, 0.8]),

deg (c) = ([0.3, 0.7] , [0.5, 0.7] , [0.4, 0.9]).

Thus, deg (a) ≠ deg (b) and deg (a) ≠ deg (c). Therefore, G is an irregular graph.

Definition 4.2. Let G be an m-PIVFG of a crisp graph G^*. If there exits a vertex in G which is adjacent to vertices with distinct closed neighborhood degree, then, it is said to be a totally irregular m-PIVFG.

Example 4.2. Let us think about an m-PIVFG G = (V, A, B) (Fig. 4), where A = ( u < [ 0.1 , 0.3 ] , [ 0.2 , 0.7 ] , [ 0.3 , 0.4 ] > , v [ 0.3 , 0.4 ] , [ 0.2 , 0.5 ] , [ 0.2 , 0.5 ] , w [ 0.2 , 0.3 ] , [ 0.4 , 0.5 ] , [ 0.1 , 0.2 ] ) B = ( uv < [ 0.1 , 0.2 ] , [ 0.2 , 0.6 ] , [ 0.3 , 0.5 ] > , uw < [ 0.2 , 0.3 ] , [ 0.1 , 0.4 ] , [ 0.3 , 0.4 ] > ) . Here, deg [u] = ([0.4, 0.9] , [0.6, 1.8] , [0.6, 1.4]),

deg [v] = ([0.4, 0.7] , [0.4, 1.0] , [0.6, 1.2]),

deg [w] = ([0.2, 0.6] , [0.6, 1.2] , [0.6, 1.4]).

OPEN IN VIEWER

Fig. 4

A 3-PIVFG G which is totally irregular.

Hence, there exists a vertex u which has distinct closed neighbourhood degrees with the vertices v and w. Therefore, the graph shown in Fig. 4 is a totally irregular m-PIVFG.

Definition 4.3. A connected m-PIVFG G = (V, A, B) is called a neighborly irregular m-PIVFG if every two adjacent vertices of G have distinct open neighborhood values in G.

Example 4.3. Let us think about an m-PIVFG G = (V, A, B), where

A = ( u < [ 0.2 , 0.3 ] , [ 0.5 , 0.7 ] , [ 0.1 , 0.4 ] > , v < [ 0.1 , 0.5 ] , [ 0.4 , 0.6 ] , [ 0.2 , 0.7 ] > , w [ 0.2 , 0.3 ] , [ 0.4 , 0.5 ] , [ 0.1 , 0.2 ] , x < [ 0.1 , 0.5 ] , [ 0.4 , 0.6 ] , [ 0.2 , 0.7 ] > )

B = ( uv < [ 0.1 , 0.3 ] , [ 0.4 , 0.6 ] , [ 0.1 , 0.4 ] > , vw < [ 0.1 , 0.2 ] , [ 0.3 , 0.6 ] , [ 0.1 , 0.3 ] > , wx < [ 0.1 , 0.3 ] , [ 0.2 , 0.5 ] , [ 0.1 , 0.2 ] > , xu < [ 0.1 , 0.3 ] , [ 0.4 , 0.5 ] , [ 0.1 , 0.2 ] > )

Here, deg (u) = ([0.2, 0.5] , [0.4, 1.0] , [0.3, 0.7]),

deg (v) = ([0.2, 0.5] , [0.4, 1.2] , [0.3, 0.7]),

deg (w) = ([0.2, 0.5] , [0.7, 1.0] , [0.2, 0.7]),

deg (x) = ([0.2, 0.5] , [0.6, 0.8] , [0.2, 0.7]).

Thus, G is neighborly irregular.

Definition 4.4. A connected m-PIVFG G = (V, A, B) is said to be a neighborly totally irregular m-PIVFG if every two adjacent vertices of G have distinct closed neighborhood degree.

Example 4.4. From Fig. 5, we get,

OPEN IN VIEWER

Fig. 5

A 3-PIVFG G which is neighborly irregular.

deg [u] = ([0.4, 0.8] , [0.9, 1.7] , [0.4, 1.1]),

deg [w] = ([0.4, 0.8] , [1.2, 1.7] , [0.3, 1.1]),

deg [v] = ([0.3, 1.0] , [0.8, 1.8] , [0.5, 1.4]),

deg [x] = ([0.3, 1.0] , [1.0, 1.4] , [0.4, 1.4]) .

Since, every two adjacent vertices of G having different closed neighborhood grades G is neighborly totally irregular m-PIVFG.

Definition 4.5. Let G be connected m-PIVFG. G is said to be a highly irregular m-PIVFG if every vertex of G is adjacent to vertices in G with distinct neighborhood values.

Example 4.5. It is clearly observed that G of example 4.3 is highly irregular.

Theorem 4.1. An m-PIVFG G is highly neighborly irregular m-PIVFG iff neighborhood degrees of all vertices of G are different.

Proof. Let an m-PIVFG G is highly irregular as well as neighborly irregular m-PIVFG. Also let deg ( v j ) = ( [ s j 1 , t j 1 ] , [ s j 2 , t j 2 ] , ⋯ , [ s j m , t j m ] ) and adjacent vertices of v₁ be v₂, v₃, ⋯ v _n with neighborhood degrees ( [ s 2 1 , t 2 1 ] , [ s 2 2 , t 2 2 ] , ⋯ , [ s 2 m , t 2 m ] ) , ( [ s 3 1 , t 3 1 ] , [ s 3 2 , t 3 2 ] , ⋯ , [ s 3 m , t 3 m ] ) ,⋯, ( [ s n 1 , t n 1 ] , [ s n 2 , t n 2 ] , ⋯ , [ s n m , t n m ] ) respectively. Then, since G is highly irregular, s 2 i ≠ s 3 j ≠ ⋯ ≠ s n j and t 2 i ≠ t 3 j ≠ ⋯ ≠ t n j . Again since G is neighborly irregular, then s 1 j ≠ s 2 i ≠ s 3 j ≠ ⋯ ≠ s n j and t 1 j ≠ t 2 i ≠ t 3 j ≠ ⋯ ≠ t n j , for each i = 1, 2, ⋯ , m. Therefore, the neighborhood grades of all the vertices of G are different.

Conversely, let the neighborhood grades of all the vertices of G are different, then s 1 j ≠ s 2 i ≠ s 3 j ≠ ⋯ ≠ s n j and t 1 j ≠ t 2 i ≠ t 3 j ≠ ⋯ ≠ t n j . So, G is highly irregular as well as neighborly irregular.

Theorem 4.2. An m-PIVFG G = (V, A, B) of some crisp graph G^* = (V, E) with a cycle of 3 vertices is highly irregular as well as neighborly irregular m-PIVFG iff for each pole of the vertices both the membership values upper and lower between every pair of vertices are all distinct.

Proof. Let us consider that for each pole of the vertices both the membership values lower and upper are all different.

Case I:

Let x, y, z ∈ V.

Given that, p i ∘ μ A l ( x ) ≠ p i ∘ μ A l ( y ) ≠ p i ∘ μ A l ( z ) and p i ∘ μ A u ( x ) ≠ p i ∘ μ A u ( y ) ≠ p i ∘ μ A u ( z ) , for each i = 1, 2, ⋯ , m. Then, ∑ x ∈ N ( x ) p i ∘ μ A l ( x ) ≠ ∑ x ∈ N ( x ) p i ∘ μ A l ( y ) ≠ ∑ x ∈ N ( x ) p i ∘ μ A l ( z ) , i.e., p i ∘ deg μ l ( x ) ≠ p i ∘ deg μ l ( y ) ≠ p i ∘ deg μ l ( z ) . Similarly p i ∘ deg μ u ( x ) ≠ p i ∘ deg μ u ( y ) ≠ p i ∘ deg μ u ( z ) . Hence, G is neighborly irregular and highly irregular.

Case II:

Let G be highly irregular and neighborly irregular. Let deg (x) = (p₁ ∘ deg(x) , p _i  ∘ deg(x) , ⋯ , p _i  ∘ deg(x)) = ([s¹, t¹] , [s², t²] , ⋯ , [s ^m , t ^m ]). Now let us assume both the membership values lower and upper of any two distinct vertices are same for each pole. Let x, y ∈ V. Let p i ∘ μ A l ( x ) = p i ∘ μ A l ( y ) and p i ∘ μ A u ( x ) = p i ∘ μ A u ( y ) , for i = 1, 2, ⋯ , m. Then, deg (x) = deg (y). Since G^* is cyclic, it contradicts the idea that G is a neighborly irregular as well as highly irregular m-PIVFG. Hence our assumption is wrong. Hence for each pole, the lower and upper membership values of all the vertices of G are all different.

Note 5. A neighborly irregular m-PIVFG may not be neighborly total irregular.

Example 4.6. Let us consider an m-PIVFG G = (V, A, B) (Fig. 6), where A = ( a < [ 0.2 , 0.5 ] , [ 0.7 , 0.8 ] , [ 0.2 , 0.3 ] > , b [ 0.1 , 0.4 ] , [ 0.6 , 0.7 ] , [ 0.4 , 0.9 ] , c [ 0.1 , 0.3 ] , [ 0.5 , 1.0 ] , [ 0.4 , 0.6 ] , d < [ 0.1 , 0.2 ] , [ 0.2 , 0.3 ] , [ 0.4 , 0.8 ] > , e < [ 0.2 , 0.4 ] , [ 0.6 , 0.7 ] , [ 0.3 , 0.9 ] > ) B = ( ab < [ 0.1 , 0.3 ] , [ 0.2 , 0.6 ] , [ 0.1 , 0.3 ] > , bc < [ 0.0 , 0.1 ] , [ 0.3 , 0.5 ] , [ 0.2 , 0.6 ] > , cd < [ 0.1 , 0.2 ] , [ 0.3 , 0.3 ] , [ 0.1 , 0.6 ] > , da < [ 0.0 , 0.1 ] , [ 0.2 , 0.3 ] , [ 0.1 , 0.3 ] > , de < [ 0.1 , 0.2 ] , [ 0.1 , 0.3 ] , [ 0.2 , 0.7 ] > ) .

OPEN IN VIEWER

Fig. 6

Neighborly irregular but not neighborly total irregular 3-PIVFG G.

Here, deg (a) = ([0.1, 0.4] , [0.4, 0.9] , [0.2, 0.6]) ,

deg [a] = ([0.3, 0.9] , [1.1, 1.7] , [0.4, 0.9]),

deg (b) = ([0.1, 0.4] , [0.5, 1.1] , [0.3, 0.9]) ,

deg [b] = ([0.2, 0.6] , [1.1, 1.8] , [0.7, 1.8]),

deg (c) = ([0.1, 0.3] , [0.6, 0.8] , [0.3, 1.2]) ,

deg [c] = ([0.2, 0.6] , [1.1, 1.8] , [0.7, 1.8]),

deg (d) = ([0.2, 0.5] , [0.6, 0.9] , [0.4, 1.6]) ,

deg [d] = ([0.3, 0.7] , [0.8, 1.2] , [0.8, 2.4]),

deg (e) = ([0.1, 0.2] , [0.1, 0.3] , [0.2, 0.7]) ,

deg [e] = ([0.1, 0.3] , [0.5, 1.0] , [0.5, 1.6]) .

Here, deg (b)= deg (c) but deg [b]=deg [c]. That means two adjacent vertices b and c have same closed neighborhood degree. Thus, G is neighborly irregular but not neighborly total irregular.

Theorem 4.3. If an m-PIVFG G is neighborly irregular m-PIVFG and ( [ p 1 ∘ μ A l , p 1 ∘ μ A u ] , [ p 2 ∘ μ A l , p 2 ∘ μ A u ] , ⋯ , [ p m ∘ μ A l , p m ∘ μ A u ] ) is a constant map, then it is neighborly totally irregular m-PIVFG.

Proof. Let G be a neighborly irregular m-PIVFG. Then, the open neighborhood degrees of every two adjacent vertices are distinct. Let x, y ∈ V be adjacent vertices with distinct open neighborhood degrees, i.e., deg (x) = ([s₁, t₁] , [s₂, t₂] , ⋯ , [s _m , t _m ]) and deg (y) = ([k₁, l₁] , [k₂, l₂] , ⋯ , [k _m , l _m ]), where s + i ≠ t _i and k _i  ≠ l _i . Let p _i  ∘ μ ^l  (x) = c₁ and p _i  ∘ μ ^u  (x) = c₂. Since [ p i ∘ μ A l , p i ∘ μ A u ] is constant function, p i ∘ deg μ l [ x ] = p i ∘ deg μ l ( x ) + p i ∘ μ l ( x ) = s i + c 1 and p i ∘ deg μ u [ x ] = p i ∘ deg μ u ( x ) + p i ∘ μ u ( x ) = t i + c 2 . Similarly, p i ∘ deg μ l [ y ] = p i ∘ deg μ l ( y ) + p i ∘ μ l ( y ) = k i + c 1 and p i ∘ deg μ u [ y ] = p i ∘ deg μ u ( y ) + p i ∘ μ u ( y ) = l i + c 2 . Now let us claim that p i ∘ deg μ l [ x ] = p i ∘ deg μ l [ y ] and p i ∘ deg μ u [ x ] = p i ∘ deg μ u [ y ] . Consider p i ∘ deg μ l [ x ] = p i ∘ deg μ l [ y ] or, s _i  + c₁ = k _i  + c₁ or, s _i  = k _i , for all i = 1, 2, ⋯ , m. This contradicts the fact that s _i  ≠ k _i or p i ∘ deg μ l [ x ] ≠ p i ∘ deg μ l [ y ] . Similarly for p i ∘ deg μ u [ x ] = p i ∘ deg μ u [ y ] or, t _i  + c₂ = l _i  + c₂ or, t _i  = l _i , for all i = 1, 2, ⋯ , m. Which contradicts the fact that t _i  ≠ l _i or, p i ∘ deg μ u [ x ] ≠ p i ∘ deg μ u [ y ] . Thus, G is a highly totally irregular m-PIVFG.