Topological indices of bipolar fuzzy incidence graph

Shu Gong; Gang Hua

doi:10.1515/chem-2021-0082

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Topological indices of bipolar fuzzy incidence graph

Shu Gong and Gang Hua

Published/Copyright: September 2, 2021

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From the journal Open Chemistry Volume 19 Issue 1

Abstract

The topological index of graph has a wide range of applications in theoretical chemistry, network design, data transmission, etc. In fuzzy graph settings, these topological indices have completely different definitions and connotations. In this work, we define new Wiener index and connectivity index for bipolar fuzzy incidence graphs, and obtain the characteristics of these indices by means of the definition of fuzzy membership functions. Furthermore, the interrelationship between Wiener index and connectivity index is considered.

Keywords: fuzzy graph; topology index; bipolar fuzzy incidence graph

1 Introduction

Compared with regular data, such as images, other tools are often needed to describe irregular data. Among them, the graph model is an effective tool for describing irregular data, such as human skeleton, molecular structure, meteorological data, traffic flow data, etc., all of which can be described and analyzed using graphic models. Therefore, topological indices are defined on the graph to describe the characteristics of the graph. For example, the topological indices defined on molecular structure are often used to describe the physical and chemical characteristics of the compound, such as PI index, Wiener index, Szeged index, and so on (Ghorbani et al. [1] and Jafarzadeh and Iranmanesh [2]).

Fuzzy theory is used to describe things and data that have fuzzy relationships and it is introduced by Zhang [3] into bipolar fuzzy setting, to record the bipolar behavior of the object. Since the topological index is widely studied in various settings [4,5,6,7,8,9,10,11], it is natural to study the general characteristics of important topological indices in special bipolar fuzzy graph (BFG) setting.

In fact, BFGs have been widely used in the field of chemistry. The following example illustrates the application of fuzzy graph theory in compound storage management.

Figure 1

Model example of chemical management.

Example 1

Chemical management is a very meticulous task, since many chemicals are incompatible with each other and they will react when mixed. In this case, the recyclables can be dangerous, and must be taken care when trying to mix or store these chemicals. This example uses a fuzzy graph to describe the fuzzy characteristics of the compounds that are compatible with each other. Each vertex represents a type of compound. If there is an edge between two vertices, it means that the corresponding kinds of compounds are relatively safe to mix. If there is no edge between adjacent vertices, it means that the corresponding types of compounds are relatively dangerous to mix. Figure 1 shows a fuzzy graph of some common kinds of compounds mixed with each other. The membership function of each vertex indicates how the same type of compounds react positively or negatively when mixed.

For example, the degree of membership (0.95, −0.05) at the vertex of “Inorganic acids” means that these chemicals are approximately 95% compatible, and approximately 5% have a chance to explode. Here the negative degree of membership describes the incompatibility of the group of chemicals. In Figure 1, due to space constraint, we did not give the values of membership function of the edges. It is just an example to illustrate the application of BFGs in the field of chemistry.

In chemical graph theory, we use vertices to represent atoms, and the edges between vertices are used to denote the chemical bonds between the atoms. In the standard model, the positions of all the vertices and edges are the same, and only the topological structure of the molecular graph can distinguish the topological characteristics of each vertex. Obviously, this model cannot represent chemical properties, for example, the kind of atom, the kind of chemical bond the edge represents, the kind of bonding force between the chemical bond and the atom, and so on. All of these need to introduce a certain measure in the molecular graph, and the fuzzy graph theory solves this problem to a certain extent. The uncertainty in the molecular graph is represented by the membership function of the vertices or edges. The fuzzy index is a topological index defined on the fuzzy graph, which plays a positive role in characterizing the fuzzy molecular graph.

All of these need to introduce a certain measures in the molecular graph, and some developments have already been made in the literature [12]. In this direction, we now introduce fuzzy graph theory to solve the abovementioned problems to a certain extent. In this work, we mainly study the Wiener index and connectivity index of particular BFGs, hence new definitions are introduced and several characteristics are determined.

2 Definitions in BFG setting

Let V be a set with at least one element. The set A = { ( v , μ A P ( v ) , μ A N ( v ) ) : v ∈ V } is a bipolar fuzzy set in V, if two maps satisfy μ A P : V → [ 0 , 1 ] and μ A N : V → [ − 1 , 0 ] . A mapping B = ( μ B P , μ B N ) : V × V → [ 0 , 1 ] × [ − 1 , 0 ] is a bipolar fuzzy relation on V, if μ B P ( v , v ′ ) ∈ [ 0 , 1 ] , μ B N ( v , v ′ ) ∈ [ − 1 , 0 ] , μ B P ( v , v ′ ) ≤ m i n { μ A P ( v ) , μ A P ( v ′ ) } , and μ B N ( v , v ′ ) ≥ max { μ A N ( v ) , μ A N ( v ′ ) } for any ( v , v ′ ) ∈ V × V . Fixed the set V, ∼ is an equivalence relation on V × V − { ( v , v ) : v ∈ V } which is defined as ( v 1 , v 2 ) ∼ ( v 1 ′ , v 2 ′ ) , if and only if ( v 1 , v 2 ) = ( v 1 ′ , v 2 ′ ) or ( v 1 , v 2 ) = ( v 2 ′ , v 1 ′ ) . The quotient set yielded here is denoted by V ˜ 2 , and we denote v v ′ or v ′ v by the equivalent class which contains the element ( v , v ′ ) .

Yang et al. [13] proposed the generalized BFGs as follows. If A = { ( v , μ A P ( v ) , μ A N ( v ) ) : v ∈ V } is a bipolar fuzzy set on an underlying set V and B = ( μ B P , μ B N ) is a bipolar fuzzy set in V ˜ 2 , where μ B P ( v , v ′ ) ≤ m i n { μ A P ( v ) , μ A P ( v ′ ) } and μ B N ( v , v ′ ) ≥ max { μ A N ( v ) , μ A N ( v ′ ) } for any ( v , v ′ ) ∈ V ˜ 2 and μ B P ( v , v ′ ) = μ B N ( v , v ′ ) = 0 for any ( v , v ′ ) ∈ V ˜ 2 − E , then G = ( V , A , B ) is a BFG of the graph G ⁎ = ( V , E ) .

Mathew et al. [14] further gave the following concepts: a BFG G ' = ( V , A ′ , B ′ ) is a partial bipolar fuzzy subgraph of G, if μ A ′ P ( v ) ≤ μ A P ( v ) and μ A ′ N ( v ) ≥ μ A N ( v ) for any v ∈ V and μ B ' P ( v , v ′ ) ≤ μ B P ( v , v ′ ) and μ B ' N ( v , v ′ ) ≥ μ B N ( v , v ′ ) for any ( v , v ′ ) ∈ E ( G ) ; a BFG G ′ = ( V ′ , A ′ , B ′ ) is a bipolar fuzzy subgraph of G, if μ A ′ P ( v ) = μ A P ( v ) and μ A ′ N ( v ) = μ A N ( v ) for any v ∈ V ′ and μ B ′ P ( v , v ′ ) = μ B P ( v , v ′ ) and μ B ′ N ( v , v ′ ) = μ B N ( v , v ′ ) for any ( v , v ′ ) ∈ E ( G ′ ) .

The following concepts are introduced by Akram [15] and Akram and Karunambigai [16]. Let G = ( V , A , B ) be a BFG. P = v 0 v 1 ⋯ v k − 1 v k is a path with length k from x = v 0 to y = v k in G and is a sequence of different vertices, where ( μ B P ( v i − 1 , v i ) > 0 , μ B N ( v i − 1 , v i ) < 0 ) for i ∈ ( 1 , ⋯ , k ) . Then (throughout the article, ∧ means minimum and ∨ implies maximum),

( μ B P ( x , y ) ) k = sup { μ B P ( x , v 1 ) ∧ μ B P ( v 1 , v 2 ) ∧ ⋯ ∧ μ B P ( v k − 1 , y ) }

and

( μ B N ( x , y ) ) k = inf { μ B N ( x , v 1 ) ∨ μ B N ( v 1 , v 2 ) ∨ ⋯ ∨ μ B N ( v k − 1 , y ) } .

Set

CONN G P ( x , y ) = sup k ∈ ℕ { ( μ B P ( x , y ) ) k } ,

CONN G N ( x , y ) = inf k ∈ ℕ { ( μ B N ( x , y ) ) k } .

Then, the strength of connectedness between arbitrary x , y ∈ V ( G ) in BFG G is denoted by

CONN G ( x , y ) = ( CONN G P ( x , y ) , CONN G N ( x , y ) ) = ( ( μ B P ( x , y ) ) ∞ , ( μ B N ( x , y ) ) ∞ ) .

If μ B P ( v , v ′ ) > 0 and μ B N ( v , v ′ ) < 0 for any ( v , v ′ ) in G, then the BFG G is connected. If μ B P ( v , v ′ ) = m i n { μ A P ( v ) , μ A P ( v ′ ) } and μ B N ( v , v ′ ) = max { μ A N ( v ) , μ A N ( v ′ ) } for any v , v ′ ∈ V , then the BFG G is called a complete BFG.

The following concepts were introduced by Karunambigai et al. [17], Akram and Farooq [18], Singh and Kumar [19], and Yang et al. [20].

For a BFG G, if μ B P ( v , v ′ ) ≥ ( μ B P ( v , v ′ ) ) ∞ and μ B N ( v , v ′ ) ≥ ( μ B N ( v , v ′ ) ) ∞ , then arc ( v , v ′ ) is called a strong arc of G. A path v − v ′ is a strong path, if all arcs on the path are strong. The ( α , β ) -cut of a BFG G is G ( α , β ) = ( V ′ , A ′ , B ′ ) with μ A P ( v ) ≥ α and μ A N ( v ) ≤ β for all v ∈ V ( G ( α , β ) ) and μ A P ( v , v ′ ) ≥ α and μ A N ( v , v ′ ) ≤ β for all edges v v ′ ∈ E ( G ( α , β ) ) , where α , β ∈ [ 0 , 1 ] .
An edge ( v , v ′ ) is a bipolar fuzzy bridge, if ( μ B ′ P ( v , v ′ ) ) ∞ < ( μ B P ( v , v ′ ) ) ∞ and ( μ B ′ N ( v , v ′ ) ) ∞ > ( μ B N ( v , v ′ ) ) ∞ (i.e., an edge v v ′ is a bipolar fuzzy bridge if the deletion of v v ′ reduces the positive connectedness and increases the negative connectedness between certain pairs of vertices in G).
Let G = ( V , A , B ) and G ′ = ( V ′ , A ′ , B ′ ) be two BFGs of the graph G ⁎ = ( V , E ) and G ′ ⁎ = ( V ′ , E ′ ) . If there is a bijective mapping f : V → V ′ with μ A P ( v ) = μ A ′ P ( f ( v ) ) , μ A N ( v ) = μ A ′ N ( f ( v ) ) for all v ∈ V ( G ) and μ B P ( v , v ′ ) = μ B ′ P ( f ( v ) , f ( v ′ ) ) and μ B N ( v , v ′ ) = μ B ′ N ( f ( v ) , f ( v ′ ) ) for all ( v , v ′ ) ∈ V ˜ 2 , then the mapping f : V → V ′ is called an isomorphism. In this case, G ≅ G ′ .

The connectivity index of BFG is defined by Poulik and Ghorai [21], which is formulated by

CI BF (G) = (CI BF P (G),CI BF N (G)) = ∑ x , y ∈ V μ A P ( x ) μ A P ( y ) CONN G P ( x , y ) , × ∑ x , y ∈ V μ A N ( x ) μ A N ( y ) CONN G N ( x , y ) ,

where CI BF P ( G ) and CI BF N ( G ) are positive connectivity index and negative connectivity index of G, respectively. Poulik and Ghorai [22,23] defined the geodesic distance and Wiener index as follows: the bipolar fuzzy geodesic distance from v to v′ is the any smallest strong path between them; the Wiener index of a bipolar fuzzy graph is denoted by

WI BF ( G ) = ( WI BF P ( G ) , WI BF N ( G ) ) = ∑ x , y ∈ V μ A P ( x ) μ A P ( y ) d s P ( x , y ) , × ∑ x , y ∈ V μ A N ( x ) μ A N ( y ) d s N ( x , y ) ,

where WI BF P ( G ) and WI BF N ( G ) are the positive and negative Wiener indices of G and d s P ( x , y ) and d s N ( x , y ) are the minimum and maximum values of the sums of all positive and negative membership values of all edges on the strong geodesics from x to y.

3 Bipolar fuzzy incidence graph

The aim of this Section is to introduce the concepts of bipolar fuzzy incidence graph, and the main notations are followed from Fang et al. [24], but extended to bipolar fuzzy setting. An incidence graph of graph G = ( V , E ) is denoted by G = ( V , E , I ) with I ⊆ V × E . If ( v , v v ′ ) in the incidence graph, then ( v , v v ′ ) is an incidence pair or shortly pair. A sequence v 0 , ( v 0 , v 0 v 1 ) , v 0 v 1 , ( v 1 , v 0 v 1 ) , v 1 , …, v n − 1 , ( v n − 1 , v n − 1 v n ) , v n − 1 v n , ( v n , v n − 1 v n ) , v n is a walk in the incidence graph. If v 0 = v n , then it is said to be closed. A walk is a path, if all the vertices are different, and an incidence graph is connected, if all the pairs of vertices are connected by a path.

Definition 1

Let G = ( V , E ) be a graph, η P : V → [ 0 , 1 ] , η N : V → [ − 1 , 0 ] , θ P : E → [ 0 , 1 ] , θ N : E → [ − 1 , 0 ] , Ψ P : V × E → [ 0 , 1 ] , and Ψ N : V × E → [ − 1 , 0 ] . If Ψ P ( v , e ) ≤ η P ( v ) ∧ θ P ( e ) and Ψ N ( v , e ) ≥ η N ( v ) ∨ θ N ( e ) for any v ∈ V and e ∈ E , then ( Ψ P , Ψ N ) is called a bipolar fuzzy incidence of G.

In the fuzzy molecular graph, the relationship between atoms and chemical bonds can be expressed to describe the stability of the molecular structure. The bipolar fuzzy membership function can describe the relative properties of atoms and edges in molecular structures from two aspects. In addition, because in each edge of the incidence molecular graph, the chemical bond needs to be associated with the vertices on the two sides to perform the relationship calculations and then the relationship between the two atoms and the chemical bond can be described separately.

Definition 2

Let G = ( V , E ) be a graph, and ( η P , η N , θ P , θ N ) be a bipolar fuzzy subgraph of G. If ( Ψ P , Ψ N ) is a bipolar fuzzy incidence of G, then G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) is called a bipolar fuzzy incidence graph of G.

Remark 1

Here we explain that the bipolar fuzzy incidence graph is different from directed graph with double weighted values [25,26,27]. First, the meaning of the expression is completely different. The membership function expresses a kind of uncertainty, which has a fundamentally different motivation from the edge weight function. Second, in the bipolar fuzzy incidence graph, the fuzzy value of the edge is determined by θ P and θ N , while Ψ P and Ψ N denote the fuzzy relationship between the vertex and the edge from two different directions. Finally, fuzzy graphs have strict restrictions on the values of the membership functions of vertices and edges.

Definition 3

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph. Then, H = ( κ P , κ N , ϕ P , ϕ N , Ω P , Ω N ) is a bipolar fuzzy incidence subgraph of G if κ P ( v ) ≤ η P ( v ) and κ N ( v ) ≥ η N ( v ) for any v ∈ V , ϕ P ( e ) ≤ θ P ( e ) and ϕ N ( e ) ≥ θ N ( e ) for any e ∈ E , and Ω P ( v , e ) ≤ Ψ P ( v , e ) and Ω N ( v , e ) ≥ Ψ N ( v , e ) for any ( v , e ) ∈ V × E .

Definition 4

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph. Set η ⁎ , θ ⁎ , and Ψ ⁎ be the vertex set, edge set, and incidence set (set of V × E ) of G (these three sets represent the specific component elements of bipolar fuzzy incidence graph G), respectively. A fuzzy incidence path λ from v to v′ ( v v ′ ∈ η ⁎ ∪ θ ⁎ ) is a sequence of elements η ⁎ , θ ⁎ , and Ψ ⁎ . The minimum value of Ψ P ( v , v v ′ ) is called the positive incidence strength and the maximum value of Ψ N ( v , v v ′ ) is called the negative incidence strength, where ( v , v ′ ) ∈ λ .

Note that in bipolar fuzzy incidence graph, the incidence paths can take different forms.

Definition 5

Let G be a bipolar fuzzy incidence graph. An incidence pair ( v , v v ′ ) is strong if Ψ P ( v , v v ′ ) ≥ ICONN G − ( v , v v ′ ) P ( v , v v ′ ) and Ψ N ( v , v v ′ ) ≤ ICONN G − ( v , v v ′ ) N ( v , v v ′ ) , where ICONN G − ( v , v v ′ ) P ( v , v v ′ ) is the maximum positive incidence strength of a − a b and ICONN G − ( v , v v ′ ) N ( v , v v ′ ) is the minimum negative incidence strength of a − a b . If Ψ P ( v , v v ′ ) > ICONN G − ( v , v v ′ ) P ( v , v v ′ ) and Ψ N ( v , v v ′ ) < ICONN G − ( v , v v ′ ) N ( v , v v ′ ) , then the pair ( v , v v ′ ) is called α -strong. If Ψ P ( v , v v ′ ) = ICONN G − ( v , v v ′ ) P ( v , v v ′ ) and Ψ N ( v , v v ′ ) = ICONN G − ( v , v v ′ ) N ( v , v v ′ ) , then the pair ( v , v v ′ ) is called β -strong.

Remark 2

Here ICONN is different from CONN . The former is defined in bipolar incidence fuzzy graph and latter is defined in BFG.

Definition 6

Let G be a bipolar fuzzy incidence graph. If all pairs of λ are strong, then an incidence path λ in G is called the strong incidence path.

Definition 7

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph. Then, H = ( κ P , κ N , ϕ P , ϕ N , Ω P , Ω N ) is a subgraph of G if κ P ( v ) = η P ( v ) and κ N ( v ) = η N ( v ) for any v ∈ κ ⁎ , ϕ P ( e ) = θ P ( e ) , ϕ N ( e ) = θ N ( e ) for any e ∈ ϕ ⁎ , and Ω P ( v , e ) = Ψ P ( v , e ) and Ω N ( v , e ) = Ψ N ( v , e ) for any ( v , e ) ∈ Ω ⁎ .

Proposition 1

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph and H = ( κ P , κ N , ϕ P , ϕ N , Ω P , Ω N ) is a bipolar fuzzy incidence subgraph of G. Then, we have

ICONN H ( v , v v ′ ) P ( v , v v ′ ) ≤ ICONN G ( v , v v ′ ) P ( v , v v ′ )

and

ICONN H ( v , v v ′ ) N ( v , v v ′ ) ≥ ICONN G ( v , v v ′ ) N ( v , v v ′ ) .

Definition 8

A bipolar fuzzy incidence graph G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) is said to be complete if Ψ P ( v , v v ′ ) = η P ( v ) ∧ θ P ( v v ′ ) and Ψ N ( v , v v ′ ) = η N ( v ) ∨ θ N ( v v ′ ) for any ( v , v v ′ ) ∈ ψ ⁎ .

Definition 9

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph. The connectivity index of G is formulated by

CI BFI ( G ) = ( CI BFI P ( G ) , CI BFI N ( G ) ) = ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN G P ( x , y ) , × ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN G N ( x , y ) ,

where CI BFI P ( G ) and CI BFI N ( G ) are positive connectivity index and negative connectivity index of G, ICONN G P ( x , y ) is the maximum value of positive incidence strength for all the possible incidence paths between x and y, and ICONN G N ( x , y ) is the minimum value of negative incidence strength for all the possible incidence paths between x and y.

For convenience, in the calculations below, we take η P ( v ) = 1 and η N ( v ) = − 1 for all v ∈ η ⁎ .

$Figure 2 A bipolar fuzzy incidence graph with CI BFI ( G ) = ( 1.6 , − 2 ) {\text{CI}}_{\text{BFI}}(G)=(1.6,-2) .$

Figure 2

A bipolar fuzzy incidence graph with CI BFI ( G ) = ( 1.6 , − 2 ) .

Example 2

Considering a bipolar fuzzy incidence graph as depicted in Figure 2, we confirm that η ⁎ = { v 1 , v 2 , v 3 , v 4 } , θ P ( v 1 v 2 ) = 0.8 , θ N ( v 1 v 2 ) = − 0.6 , θ P ( v 1 v 3 ) = 0.5 , θ N ( v 1 v 3 ) = − 0.7 , θ P ( v 2 v 3 ) = 0.7 , θ N ( v 2 v 3 ) = − 0.6 , θ P ( v 3 v 4 ) = 0.9 , θ N ( v 3 v 4 ) = − 0.8 , Ψ P ( v 1 , v 1 v 2 ) = 0.8 , Ψ N ( v 1 , v 1 v 2 ) = − 0.5 , Ψ P ( v 2 , v 2 v 1 ) = 0.4 , Ψ N ( v 2 , v 2 v 1 ) = − 0.3 , Ψ P ( v 2 , v 2 v 3 ) = 0.3 , Ψ N ( v 2 , v 2 v 3 ) = − 0.2 , Ψ P ( v 3 , v 3 v 2 ) = 0.6 , Ψ N ( v 3 , v 3 v 2 ) = − 0.5 , Ψ P ( v 1 , v 1 v 3 ) = 0.5 , Ψ N ( v 1 , v 1 v 3 ) = − 0.3 , Ψ P ( v 3 , v 3 v 1 ) = 0.3 , Ψ N ( v 3 , v 3 v 1 ) = − 0.7 , Ψ P ( v 3 , v 3 v 4 ) = 0.4 , Ψ N ( v 3 , v 3 v 4 ) = − 0.5 , Ψ P ( v 4 , v 4 v 3 ) = 0.2 , Ψ N ( v 4 , v 4 v 3 ) = − 0.7 , and CI BFI ( G ) = ( 1.6 , − 2 ) .

The following proposition shows the relationship of connectivity index between bipolar fuzzy incidence graph and its subgraph.

Proposition 2

If H = ( κ P , κ N , ϕ P , ϕ N , Ω P , Ω N ) is a bipolar fuzzy incidence subgraph of G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) as defined in Definition 3, then CI BFI P ( H ) ≤ CI BFI P ( G ) , CI BFI N ( H ) ≥ CI BFI N ( G ) .

Proof

For any v , v ′ ∈ κ ⁎ , we get κ P ( v ) ≤ η P ( v ) , κ N ( v ) ≥ η N ( v ) , κ P ( v ′ ) ≤ η P ( v ′ ) , and κ N ( v ′ ) ≥ η N ( v ′ ) by means of Definition 3. Moreover, we obtain ICONN H P ( v , v ′ ) ≤ ICONN G P ( v , v ′ ) and ICONN H N ( v , v ′ ) ≥ ICONN G N ( v , v ′ ) . Therefore, we infer

∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN H P ( x , y ) ≤ ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN G P ( x , y ) ,

∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN H N ( x , y ) ≥ ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN G N ( x , y ) .

It implies the conclusion.□

Proposition 3

If H = ( κ P , κ N , ϕ P , ϕ N , Ω P , Ω N ) is a subgraph of G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) as defined in Definition 7, then CI BFI P ( H ) ≤ CI BFI P ( G ) and CI BFI N ( H ) ≥ CI BFI N ( G ) .

This is an obvious conclusion, so we skip the proof here.

Proposition 4

If H is a subgraph of connected bipolar fuzzy incidence graph G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) , where V ( H ) = V ( G ) − { v } , then CI BFI P ( H ) < CI BFI P ( G ) and CI BFI N ( H ) > CI BFI N ( G ) .

$Figure 3 Subgraph of bipolar fuzzy incidence graph with CI BFI ( G ) = ( 1 , − 0.9 ) {\text{CI}}_{\text{BFI}}(G)=(1,-0.9) .$

Figure 3

Subgraph of bipolar fuzzy incidence graph with CI BFI ( G ) = ( 1 , − 0.9 ) .

Example 3

Considering a bipolar fuzzy incidence graph as depicted in Figure 3, we confirm that η ⁎ = { v 1 , v 2 , v 3 } , θ P ( v 1 v 2 ) = 0.8 , θ N ( v 1 v 2 ) = − 0.6 , θ P ( v 1 v 3 ) = 0.5 , θ N ( v 1 v 3 ) = − 0.7 , θ P ( v 2 v 3 ) = 0.7 , θ N ( v 2 v 3 ) = − 0.6 , Ψ P ( v 1 , v 1 v 2 ) = 0.8 , Ψ N ( v 1 , v 1 v 2 ) = − 0.5 , Ψ P ( v 2 , v 2 v 1 ) = 0.4 , Ψ N ( v 2 , v 2 v 1 ) = − 0.3 , Ψ P ( v 2 , v 2 v 3 ) = 0.3 , Ψ N ( v 2 , v 2 v 3 ) = − 0.2 , Ψ P ( v 3 , v 3 v 2 ) = 0.6 , Ψ N ( v 3 , v 3 v 2 ) = − 0.5 , Ψ P ( v 1 , v 1 v 3 ) = 0.5 , Ψ N ( v 1 , v 1 v 3 ) = − 0.3 , Ψ P ( v 3 , v 3 v 1 ) = 0.3 , Ψ N ( v 3 , v 3 v 1 ) = − 0.7 , and CI BFI ( G ) = ( 1 , − 0.9 ) .

Definition 10

Let G 1 = ( η 1 P , η 1 N , θ 1 P , θ 1 N , Ψ 1 P , Ψ 1 N ) and G 2 = ( η 2 P , η 2 N , θ 2 P , θ 2 N , Ψ 2 P , Ψ 2 N ) be two bipolar fuzzy incidence graphs. If there is a bijective mapping f : V ( G 1 ) → V ( G 2 ) with η 1 P ( v ) = η 2 P ( f ( v ) ) and η 1 N ( v ) = η 2 N ( f ( v ) ) for all v ∈ V ( G 1 ) , θ 1 P ( v v ′ ) = θ 2 P ( f ( v ) f ( v ′ ) ) and θ 1 N ( v v ′ ) = θ 2 N ( f ( v ) f ( v ′ ) ) for all v v ′ ∈ E ( G 1 ) , and Ψ 1 P ( v , v v ′ ) = Ψ 2 P ( f ( v ) , f ( v ) f ( v ′ ) ) and Ψ 1 N ( v , v v ′ ) = Ψ 2 N ( f ( v ) , f ( v ) f ( v ′ ) ) for any pair ( v , v v ′ ) , then the mapping f is called an isomorphism and we denote G 1 ≅ G 2 .

Definition 11

The Wiener index of a bipolar fuzzy incidence graph G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) is denoted by

WI BFI ( G ) = ( WI BFI P ( G ) , WI BFI N ( G ) ) = ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) d s P ( x , y ) , × ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) d s N ( x , y ) .

4 Main results and proofs

Our first main conclusion reveals the bounds for connectivity index of bipolar fuzzy incidence graph which is manifested as follows.

Theorem 1

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph with ∣ η ⁎ ∣ = n and G 1 = ( η 1 P , η 1 N , θ 1 P , θ 1 N , Ψ 1 P , Ψ 1 N ) be the completion of the bipolar fuzzy incidence graph G. Then, we have 0 ≤ CI BFI P ( G ) ≤ CI BFI P ( G 1 ) and CI BFI N ( G 1 ) ≤ CI BFI N ( G ) ≤ 0 .

Proof

If n = 1 , then 0 = CI BFI P ( G ) = CI BFI P ( G 1 ) and CI BFI N ( G 1 ) = CI BFI N ( G ) = 0 .□

Considering n ≥ 2 , we confirm that η 1 P ( v ) = η P ( v ) and η 1 N ( v ) = η N ( v ) for every v ∈ η ⁎ and θ 1 P ( e ) ≥ θ P ( e ) , θ 1 N ( e ) ≤ θ N ( e ) for every e ∈ θ ⁎ . Also, we have ICONN G P ( x , y ) ≤ ICONN G 1 P ( x , y ) and ICONN G N ( x , y ) ≥ ICONN G 1 N ( x , y ) for any x , y ∈ η ⁎ . Therefore, we have 0 ≤ CI BFI P ( G ) ≤ CI BFI P ( G 1 ) and CI BFI N ( G 1 ) ≤ CI BFI N ( G ) ≤ 0 .

Theorem 1 reveals that the complete bipolar fuzzy incidence graph has the largest positive connectivity index and the lowest negative connectivity index.

$Figure 4 Complete bipolar fuzzy incidence graph with CI BFI ( G ) = ( 2.2 , − 1.9 ) {\text{CI}}_{\text{BFI}}(G)=(2.2,-1.9) .$

Figure 4

Complete bipolar fuzzy incidence graph with CI BFI ( G ) = ( 2.2 , − 1.9 ) .

Example 4

The bipolar fuzzy incidence graph in Figure 4 is a complete bipolar fuzzy incidence graph. We deduce η ⁎ = { v 1 , v 2 , v 3 } , θ P ( v 1 v 2 ) = 0.8 , θ N ( v 1 v 2 ) = − 0.6 , θ P ( v 1 v 3 ) = 0.5 , θ N ( v 1 v 3 ) = − 0.7 , θ P ( v 2 v 3 ) = 0.7 , θ N ( v 2 v 3 ) = − 0.6 , Ψ P ( v 1 , v 1 v 2 ) = 0.8 , Ψ N ( v 1 , v 1 v 2 ) = − 0.6 , Ψ P ( v 2 , v 2 v 1 ) = 0.8 , Ψ N ( v 2 , v 2 v 1 ) = − 0.6 , Ψ P ( v 2 , v 2 v 3 ) = 0.7 , Ψ N ( v 2 , v 2 v 3 ) = − 0.6 , Ψ P ( v 3 , v 3 v 2 ) = 0.7 , Ψ N ( v 3 , v 3 v 2 ) = − 0.6 , Ψ P ( v 1 , v 1 v 3 ) = 0.5 , Ψ N ( v 1 , v 1 v 3 ) = − 0.7 , Ψ P ( v 3 , v 3 v 1 ) = 0.5 , Ψ N ( v 3 , v 3 v 1 ) = − 0.7 , and CI BFI ( G ) = ( 2.2 , − 1.9 ) .

The next theorem shows the characteristics of vertex deleting and edge removing bipolar fuzzy incidence subgraphs with connectivity index.

Theorem 2

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph and G 1 = ( η 1 P , η 1 N , θ 1 P , θ 1 N , Ψ 1 P , Ψ 1 N ) be the bipolar fuzzy incidence subgraph of G by removing an incidence pair ( v , v v ′ ) ∈ Ψ ⁎ . Then, CI BFI P ( G 1 ) < CI BFI P ( G ) and CI BFI N ( G 1 ) > CI BFI N ( G ) , if and only if ( v , v v ′ ) is a fuzzy bridge.

Proof

Set pair ( v , v v ′ ) is a fuzzy bridge of G. Then, in view of its definition, we yield□

ICONN G − ( v , v v ′ ) P ( v , v ′ ) < ICONN G P ( v , v ′ ) ,

ICONN G − ( v , v v ′ ) N ( v , v ′ ) > ICONN G N ( v , v ′ ) ,

which implies CI BFI P ( G 1 ) < CI BFI P ( G ) and CI BFI N ( G ) < CI BFI N ( G 1 ) .

On the contrary, we assume that CI BFI P ( G 1 ) < CI BFI P ( G ) and CI BFI N ( G 1 ) > CI BFI N ( G ) . We further suppose that ( v , v v ′ ) is not a fuzzy bridge. Then, for any pair of ( x , y ) in G, we have ICONN G P ( v , v ′ ) ≤ ICONN G 1 P ( v , v ′ ) or ICONN G N ( v , v ′ ) ≥ ICONN G 1 N ( v , v ′ ) ⇒ ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN G P ( x , y ) ≤ ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN G 1 P ( x , y ) or ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN G N ( x , y ) ≥ ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN G 1 N ( x , y ) or both ⇒ CI BFI P ( G ) ≤ CI BFI P ( G 1 ) or CI BFI N ( G 1 ) ≤ CI BFI N ( G ) or both, which lead to a contradiction. Therefore, ( v , v v ′ ) is a fuzzy bridge.

Theorem 3

Let G 1 = ( η 1 P , η 1 N , θ 1 P , θ 1 N , Ψ 1 P , Ψ 1 N ) and G 2 = ( η 2 P , η 2 N , θ 2 P , θ 2 N , Ψ 2 P , Ψ 2 N ) be two isomorphic bipolar fuzzy incidence graphs. Then, CI BFI ( G 1 ) = CI BFI ( G 2 ) , i.e., CI BFI P ( G 1 ) = CI BFI P ( G 2 ) and CI BFI N ( G 1 ) = CI BFI N ( G 2 ) .

Proof

Let f be the bijection between two bipolar fuzzy incidence graphs as described in Definition 10. We have ICONN G 1 P ( x , y ) = ICONN G 2 P ( f ( x ) , f ( y ) ) and ICONN G 1 N ( x , y ) = ICONN G 2 N ( f ( x ) , f ( y ) ) for any x , y ∈ η ⁎ . Hence,

∑ x , y ∈ η 1 ⁎ η P ( x ) η P ( y ) ICONN G 1 P ( x , y ) = ∑ x , y ∈ η 2 ⁎ η P ( x ) η P ( y ) ICONN G 2 P ( f ( x ) , f ( y ) ) and

∑ x , y ∈ η 1 ⁎ η N ( x ) η N ( y ) ICONN G 1 N ( x , y ) = ∑ x , y ∈ η 2 ⁎ η N ( x ) η N ( y ) ICONN G 2 N ( f ( x ) , f ( y ) ) .

□

Thus, we have CI BFI P ( G 1 ) = CI BFI P ( G 2 ) and CI BFI N ( G 1 ) = CI BFI N ( G 2 ) .

Definition 12

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph with n vertices. The average connectivity index of G is given by

ACI BFI ( G ) = ( ACI BFI P ( G ) , ACI BFI N ( G ) ) = 1 n 2 ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN G P ( x , y ) , × ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN G N ( x , y ) .

Example 5

Considering the bipolar fuzzy incidence graph in Figure 2, we have ACI BFI ( G ) = 4 15 , − 1 3 . For the bipolar fuzzy incidence graph in Figure 3, we have ACI BFI ( G ) = 1 3 , − 0.3 .

Definition 13

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph and v ∈ η ⁎ .

If ACI BFI P ( G − { v } ) < ACI BFI P ( G ) and ACI BFI N ( G − { v } ) > ACI BFI N ( G ) , then v is called a bipolar fuzzy incidence connectivity-reducing vertex;
If ACI BFI P ( G − { v } ) > ACI BFI P ( G ) and ACI BFI N ( G − { v } ) < ACI BFI N ( G ) , then v is called a bipolar fuzzy incidence connectivity-enhancing vertex;
If ACI BFI P ( G − { v } ) = ACI BFI P ( G ) and ACI BFI N ( G − { v } ) = ACI BFI N ( G ) , then v is called a bipolar fuzzy incidence connectivity-neutral vertex.

Theorem 4

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph, n = ∣ η ⁎ ∣ ≥ 3 , v ∈ η ⁎ , a = CI BFI P ( G ) CI BFI P ( G − { v } ) , and b = CI BFI N ( G ) CI BFI N ( G − { v } ) . Then,

v is a bipolar fuzzy incidence connectivity-enhancing vertex, if and only if a < n n − 2 and b > n n − 2 ;
v is a bipolar fuzzy incidence connectivity-reducing vertex, if and only if a > n n − 2 and b < n n − 2 ;
v is a bipolar fuzzy incidence connectivity-neutral vertex, if and only if a = n n − 2 and b = n n − 2 .

Proof

We only prove the second part of the theorem, and the first and the third parts can be done by means of similar tricks.□

Let v be a bipolar fuzzy incidence connectivity-reducing vertex. Using its definition, we infer

ACI BFI P ( G − { v } ) < ACI BFI P ( G ) and ACI BFI N ( G − { v } ) > ACI BFI N ( G )

⇒ CI BFI P ( G − { v } ) n − 1 2 < CI BFI P ( G ) n 2 and CI BFI N ( G − { v } ) n − 1 2 > CI BFI N ( G ) n 2

⇒ n 2 n − 1 2 < CI BFI P ( G ) CI BFI P ( G − { v } ) and n 2 n − 1 2 > CI BFI N ( G ) CI BFI N ( G − { v } )

⇒ a > n n − 2 and b < n n − 2 .

On the contrary, let a > n n − 2 and b < n n − 2

⇒ CI BFI P ( G ) CI BFI P ( G − { v } ) > n n − 2 and CI BFI N ( G ) CI BFI N ( G − { v } ) < n n − 2

⇒ CI BFI P ( G − { v } ) n − 1 2 < CI BFI P ( G ) n 2 and CI BFI N ( G − { v } ) n − 1 2 > CI BFI N ( G ) n 2

⇒ ACI BFI P ( G − { v } ) < ACI BFI P ( G ) and ACI BFI N ( G − { v } ) > ACI BFI N ( G )

⇒ v is a bipolar fuzzy incidence connectivity-reducing vertex.

From Theorem 4, we directly get the following corollary.

Corollary 1

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph, n = ∣ η ⁎ ∣ ≥ 3 , v ∈ η ⁎ , a = CI BFI P ( G ) CI BFI P ( G − { v }) , and b = CI BFI N ( G ) CI BFI N ( G − { v } ) .

If v is a bipolar fuzzy incidence connectivity-enhancing vertex, then a < b ;
If v is a bipolar fuzzy incidence connectivity-reducing vertex, then a > b ;
If v is a bipolar fuzzy incidence connectivity-neutral vertex, then a = b .

Definition 14

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a connected bipolar fuzzy incidence graph with n = ∣ η ⁎ ∣ ≥ 3 .

If G has at least one bipolar fuzzy incidence connectivity-enhancing vertex, then G is called bipolar fuzzy incidence connectivity enhancing graph;
If G has one bipolar fuzzy incidence connectivity-reducing vertex, then G is called bipolar fuzzy incidence connectivity reducing graph;
If all the vertices in G are bipolar fuzzy incidence connectivity-neutral vertices, then G is called bipolar fuzzy incidence connectivity neutral graph.

We now consider the Wiener index of a bipolar fuzzy incidence graph. Considering G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) is a bipolar fuzzy incidence graph and H is a subgraph of G, note that WI BFI P ( H ) ≤ WI BFI P ( G ) and WI BFI N ( H ) ≥ WI BFI N ( G ) may not hold, and we give the following example to explain this point.

$Figure 5 Bipolar fuzzy incidence graph with WI BFI ( G ) = ( WI BFI P ( G ) , WI BFI N ( G ) ) = ( 4 , − 4 ) {\text{WI}}_{\text{BFI}}(G)=({\text{WI}}_{\text{BFI}}^{P}(G),{\text{WI}}_{\text{BFI}}^{N}(G))=(4,-4) .$

Figure 5

Bipolar fuzzy incidence graph with WI BFI ( G ) = ( WI BFI P ( G ) , WI BFI N ( G ) ) = ( 4 , − 4 ) .

Example 6

Considering a bipolar fuzzy incidence graph is depicted in Figure 5, we have η ⁎ = { v 1 , v 2 , v 3 } , θ P ( v 1 v 2 ) = 0.7 , θ N ( v 1 v 2 ) = − 0.7 , θ P ( v 2 v 3 ) = 0.7 , θ N ( v 2 v 3 ) = − 0.7 , θ P ( v 1 v 3 ) = 0.8 , θ N ( v 1 v 3 ) = − 0.8 , Ψ P ( v 1 , v 1 v 2 ) = 0.6 , Ψ N ( v 1 , v 1 v 2 ) = − 0.6 , Ψ P ( v 2 , v 2 v 1 ) = 0.6 , Ψ N ( v 2 , v 2 v 1 ) = − 0.6 , Ψ P ( v 2 , v 2 v 3 ) = 0.6 , Ψ N ( v 2 , v 2 v 3 ) = − 0.6 , Ψ P ( v 3 , v 3 v 2 ) = 0.6 , Ψ N ( v 3 , v 3 v 2 ) = − 0.6 , Ψ P ( v 1 , v 1 v 3 ) = 0.8 , Ψ N ( v 1 , v 1 v 3 ) = − 0.8 , Ψ P ( v 3 , v 3 v 1 ) = 0.8 , and Ψ N ( v 3 , v 3 v 1 ) = − 0.8 . Clearly, each incidence pair is a strong pair and by simple computation we get WI BFI P ( G ) = 4 and WI BFI N ( G ) = − 4 .

$Figure 6 Bipolar fuzzy incidence graph with WI BFI ( H ) = ( WI BFI P ( H ) , WI BFI N ( H ) ) = ( 4.8 , − 4.8 ) {\text{WI}}_{\text{BFI}}(H)=({\text{WI}}_{\text{BFI}}^{P}(H),{\text{WI}}_{\text{BFI}}^{N}(H))=(4.8,-4.8) .$

Figure 6

Bipolar fuzzy incidence graph with WI BFI ( H ) = ( WI BFI P ( H ) , WI BFI N ( H ) ) = ( 4.8 , − 4.8 ) .

Now remove the edge v 1 v 3 from G, we construct a subgraph H of G which is presented in Figure 6. The strong geodesic from v 1 to v 3 is v 1 v 2 v 3 , d s P ( v 1 , v 2 ) = Ψ P ( v 1 , v 1 v 2 ) + Ψ P ( v 2 , v 2 v 1 ) = 1.2 and d s N ( v 1 , v 2 ) = Ψ N ( v 1 , v 1 v 2 ) + Ψ N ( v 2 , v 2 v 1 ) = − 1.2 . In terms of the same trick, we confirm d s P ( v 1 , v 3 ) = 2.4 , d s N ( v 1 , v 3 ) = − 2.4 , d s P ( v 2 , v 3 ) = 1.2 , and d s N ( v 2 , v 3 ) = − 1.2 . Thus, WI BFI P ( H ) = 4.8 and WI BFI N ( H ) = − 4.8 , which imply that WI BFI P ( G ) < WI BFI P ( H ) and WI BFI N ( G ) > WI BFI N ( H ) .

The following result reveals the inner connection between the Wiener index and connectivity index of bipolar fuzzy incidence graph.

Theorem 5

Let G = ( η P , η N , θ P , θ N , Ψ P , Ψ N ) be a bipolar fuzzy incidence graph with ∣ η ⁎ ∣ ≥ 3 . Then, WI BFI P ( G ) > CI BFI P ( G ) and WI BFI N ( G ) < CI BFI N ( G ) .

Proof

Consider a bipolar fuzzy incidence graph with ∣ η ⁎ ∣ ≥ 3 . For any x , y ∈ η ⁎ , the sum of positive membership values and negative membership values of each incidence pairs connecting x and y are d s P ( x , y ) and d s N ( x , y ) , respectively. While the minimum positive membership value and the maximum negative membership value of all strong incidence pairs are ICONN G P ( x , y ) and ICONN G N ( x , y ) , respectively. Hence, we have ICONN G P ( x , y ) < d s P ( x , y ) and ICONN G N ( x , y ) > d s N ( x , y ) , which imply

∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) ICONN G P ( x , y ) < ∑ x , y ∈ η ⁎ η P ( x ) η P ( y ) d s P ( x , y ) ,

∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) ICONN G N ( x , y ) > ∑ x , y ∈ η ⁎ η N ( x ) η N ( y ) d s N ( x , y ) .

□

Therefore, we get WI BFI P ( G ) > CI BFI P ( G ) and WI BFI N ( G ) < CI BFI N ( G ) .

Theorem 6

Let G be a complete bipolar fuzzy incidence graph with two vertices v 1 and v 2 . Assume η P ( v 1 ) = η P ( v 2 ) = 1 and η N ( v 1 ) = η N ( v 2 ) = − 1 . Then, WI BFI P ( G ) = 2 CI BFI P ( G ) and WI BFI N ( G ) = 2 CI BFI N ( G ) .

Proof

According to the assumption of the above theorem, the sum of positive membership values and negative membership values of each incidence pairs connecting v 1 and v 2 are d s P ( v 1 , v 2 ) and d s N ( v 1 , v 2 ) , respectively. While the minimum positive membership value and the maximum negative membership value of all strong incidence pairs are ICONN G P ( v 1 , v 2 ) and ICONN G N ( v 1 , v 2 ) , respectively. This reveals that 2 ICONN G P ( v 1 , v 2 ) = d s P ( v 1 , v 2 ) and 2 ICONN G N ( v 1 , v 2 ) = d s N ( v 1 , v 2 ) . Hence,

2 η P ( v 1 ) η P ( v 2 ) ICONN G P ( v 1 , v 2 ) = η P ( v 1 ) η P ( v 2 ) d s P ( v 1 , v 2 ) ,

2 η N ( v 1 ) η N ( v 2 ) ICONN G N ( v 1 , v 2 ) = η N ( v 1 ) η N ( v 2 ) d s N ( v 1 , v 2 ) .

Therefore, the desired conclusion holds.□

$Figure 7 A complete bipolar fuzzy incidence graph with WI BFI P ( G ) = 2 CI BFI P ( G ) {\text{WI}}_{\text{BFI}}^{P}(G)=2{\text{CI}}_{\text{BFI}}^{P}(G) and WI BFI N ( G ) = 2 CI BFI N ( G ) {\text{WI}}_{\text{BFI}}^{N}(G)=2{\text{CI}}_{\text{BFI}}^{N}(G) .$

Figure 7

A complete bipolar fuzzy incidence graph with WI BFI P ( G ) = 2 CI BFI P ( G ) and WI BFI N ( G ) = 2 CI BFI N ( G ) .

Example 7

Considering the following complete bipolar fuzzy incidence graph with two vertices denoted by v 1 and v 2 which is depicted in Figure 7, we check that θ P ( v 1 v 2 ) = 0.4 , θ N ( v 1 v 2 ) = − 0.4 , Ψ P ( v 1 , v 1 v 2 ) = 0.4 , Ψ N ( v 1 , v 1 v 2 ) = − 0.4 , Ψ P ( v 2 , v 2 v 1 ) = 0.4 , Ψ N ( v 2 , v 2 v 1 ) = − 0.4 , CI BFI P ( G ) = 0.4 , CI BFI N ( G ) = − 0.4 , WI BFI P ( G ) = 0.8 , and WI BFI N ( G ) = − 0.8 .

When it comes to the molecular graph setting, we need to define the incidence membership function on the molecular fuzzy graph, which reflects to a certain extent the closeness between the atoms and the chemical bonds which connect the forces between the molecules, the respective differences between the atoms and the chemical bonds, and the related attributes. Hence, such definition of the membership function requires the participation of a chemist. On the other hand, not all graphs are molecular graphs [28]. Only when the graph structure can represent a certain chemical substance, the graph can be called a molecular graph. Due to the interaction between atoms and chemical bonds, spins and other physical effects are produced and many chemical structures are unstable. Such unstable chemical structures can be characterized by membership functions of atoms, chemical bonds, and their mutual relations. This part of the content requires chemical data or the assistance of a chemist and can be considered for future work.

5 Conclusion

In this article, we first introduce the BFG and two kinds of topological index, then define concepts of bipolar fuzzy incidence graphs, and finally we deduce the characteristics of Wiener index and connectivity index on bipolar fuzzy incidence graphs. The results obtained in this study have potential applications in many fields such as fuzzy reasoning and intelligent decision systems.

Acknowledgments

Special thanks to the reviewers for their meticulous review and constructive comments.

Funding information: This work is supported by Natural Science Foundation of Guangdong Province of China (No. 2020A1515010784), Guangdong University of Science and Technology University Major Scientific Research Achievement Cultivation Program Project 2020 (No. GKY-2020CQPY-2), and Provincial Key Platforms and Major Scientific Research Projects in Guangdong Universities in 2018 (No. 2018KTSCX261).
Author contributions: S.G. – Writing original draft; G.H. – writing review and editing.
Conflict of interest: The authors declare no conflict of interests on publishing this paper.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: We state that data have been used partially in ref. [29]: ‘Remarks on Wiener Index of Bipolar Fuzzy Incidence Graphs’. Gong S and Hua G. Front Phys. 2021; 9:677882.

References

[1] Ghorbani M , Deng B , Hakimi-Nezhaad M , Li X . A survey on borderenergetic graphs. MATCH Commun Math Comput Chem. 2020;84:293–322.Search in Google Scholar

[2] Jafarzadeh N , Iranmanesh A . On graph-based data structures to multiple genome alignment. MATCH Commun Math Comput Chem. 2020;83:33–62.Search in Google Scholar

[3] Zhang WR . Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis. Proceedings of IEEE Conference. Piscataway, NJ: IEEE; 1994. p. 305–9.Search in Google Scholar

[4] Awais HM , Javaid M , Ali A . First general Zagreb index of generalized F-sum graphs. Discrete Dyn Nat Soc. 2020;2020:2954975. 10.1155/2020/2954975.Search in Google Scholar

[5] Nandini GK , Rajan RS , Shantrinal AA , Rajalaxmi TM , Rajasingh I , Balasubramanian K . Topological and thermodynamic entropy measures for COVID-19 pandemic through graph theory. Symmetry-Basel. 2020;12(12):1992. 10.3390/sym12121992.Search in Google Scholar

[6] Sahin B . On Hosoya index and Merrifield-Simmons index of trees with given domination number. Numer Meth Part D E. 2021. 10.1002/num.22712.Search in Google Scholar

[7] Das KC , Rodriguez JM , Sigarreta JM . On the maximal general ABC index of graphs with given maximum degree. Appl Math Comput. 2020;386:125531. 10.1016/j.amc.2020.125531.Search in Google Scholar

[8] Das KC . On the Balaban index of chain graphs. B Malays Math Sci Soc. 2021;44:2123–38. 10.1007/s40840-020-01054-y.Search in Google Scholar

[9] Aslam A , Ahmad S , Binyamin MA , Gao W . Calculating topological indices of certain OTIS Interconnection networks. Open Chem. 2019;17:220–8.10.1515/chem-2019-0029Search in Google Scholar

[10] Gao W , Wang WF , Chen YJ . Tight bounds for the existence of path factors in network vulnerability parameter settings. Int J Intell Syst. 2021;36(3):1133–58.10.1002/int.22335Search in Google Scholar

[11] Gao W , Chen YJ , Wang YQ . Network vulnerability parameter and results on two surfaces. Int J Intell Syst. 2021;36(8):4392–414.10.1002/int.22464Search in Google Scholar

[12] Arockiaraj M , Kavitha SRJ , Mushtaq S , Balasubramanian K . Relativistic topological molecular descriptors of metal trihalides. J Mol Struc. 2020;1217:128368.10.1016/j.molstruc.2020.128368Search in Google Scholar

[13] Yang HL , Li SG , Yang WH , Lu Y . Notes on bipolar fuzzy graphs. Inform Sci. 2013;242:113–21.10.1016/j.ins.2013.03.049Search in Google Scholar

[14] Mathew S , Sunitha MS , Anjali N . Some connectivity concepts in bipolar fuzzy graphs. An Pure Appl Math. 2014;7(2):98–100.Search in Google Scholar

[15] Akram M . Bipolar fuzzy graphs. Inform Sci. 2011;181(24):5548–64.10.1016/j.ins.2011.07.037Search in Google Scholar

[16] Akram M , Karunambigai MG . Metric in bipolar fuzzy graphs. World Appl Sci J. 2011;14(12):1920–7.Search in Google Scholar

[17] Karunambigai MG , Akram M , Palanivel K , Sivasankar S . Domination in bipolar fuzzy graphs. FUZZ-IEEE International Conference on Fuzzy Systems. Piscataway, NJ: IEEE; 2013. 10.1109/FUZZ-IEEE.2013.6622326.Search in Google Scholar

[18] Akram M , Farooq A . Bipolar fuzzy tree. New Trends Math Sci. 2016;4(3):58–72.10.20852/ntmsci.2016318805Search in Google Scholar

[19] Singh PK , Kumar CA . Bipolar fuzzy graph representation of concept lattice. Inform Sci. 2014;288:437–48.10.1016/j.ins.2014.07.038Search in Google Scholar

[20] Yang HL , Li SG , Yang WH , Lu Y . Notes on bipolar fuzzy graphs. Inform Sci. 2013;242:113–21.10.1016/j.ins.2013.03.049Search in Google Scholar

[21] Poulik S , Ghorai G . Certain indices of graphs under bipolar fuzzy environment with applications. Soft Comput. 2020;24:5119–31.10.1007/s00500-019-04265-zSearch in Google Scholar

[22] Poulik S , Ghorai G . Detour g-interior nodes and detour g-boundary nodes in bipolar fuzzy graph with applications. Hacet J Math Stat. 2020;49(1):106–19.10.15672/HJMS.2019.666Search in Google Scholar

[23] Poulik S , Ghorai G . Determination of journeys order based on graph’s Wiener absolute index with bipolar fuzzy information. Inform Sci. 2021;545:608–19.10.1016/j.ins.2020.09.050Search in Google Scholar

[24] Fang J , Nazeer I , Rashid T , Liu J . Connectivity and Wiener index of fuzzy incidence graphs. Math Probl Eng. 2021;2021:6682966. 10.1155/2021/6682966.Search in Google Scholar

[25] Arockiaraj M , Kavitha SRJ , Balasubramanian K , Rajasingh I , Clement J . Topological characterization of coronoid polycyclic aromatic hydrocarbons. Polycycl Aromat Comp. 2020;40(3):784–802.10.1080/10406638.2018.1484778Search in Google Scholar

[26] Arockiaraj M , Clement J , Balasubramanian K . Topological properties of carbon nanocones Polycycl Aromat Comp. 2020;40(5):1332–46.10.1080/10406638.2018.1544156Search in Google Scholar

[27] Arockiaraj M , Klavžar S , Clement J , Mushtaq S , Balasubramanian K . Edge distance-based topological indices of strength-weighted graphs and their application to coronoid systems, carbon nanocones and SiO2 nanostructures. Mol Inform. 2019;38(11–12):1900039.10.1002/minf.201900039Search in Google Scholar PubMed

[28] Dobrowolski JC . The structural formula version of graph theory. MATCH Commun Math Comput Chem. 2019;81:527–55.Search in Google Scholar

[29] Gong S , Hua G . Remarks on Wiener index of bipolar fuzzy incidence graphs. Front Phys. 2021;9:677882. 10.3389/fphy.2021.677882.Search in Google Scholar

Received: 2021-03-18

Revised: 2021-08-07

Accepted: 2021-08-08

Published Online: 2021-09-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/chem-2021-0082

Keywords for this article

fuzzy graph; topology index; bipolar fuzzy incidence graph

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