Cyclic connectivity index of bipolar fuzzy incidence graph

Juanjuan Lu; Linli Zhu; Wei Gao

doi:10.1515/chem-2022-0149

Article Open Access

Cyclic connectivity index of bipolar fuzzy incidence graph

Juanjuan Lu , Linli Zhu and Wei Gao

Published/Copyright: April 19, 2022

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From the journal Open Chemistry Volume 20 Issue 1

Abstract

In the performance characterization of chemical molecular structures, several uncertain properties are often encountered, and fuzzy theory is precisely the tool to characterize these uncertainties. When molecular structures are described by molecular graphs, the corresponding fuzzy graph theory is used to characterize the uncertainty of atoms and atomic bonds. In this study, there is introduced cyclic connectivity index and its average version for bipolar fuzzy incidence graph (BFIG), and several theoretical results are obtained in the light of graph theory and fuzzy theory. Finally, the given new fuzzy index is applied to the testing of anti-aging-related drugs yields average uncertainty data for the corresponding molecular structures.

Keywords: fuzzy graph; cyclic connectivity index; average cyclic connectivity index; fuzzy incidence graph; bipolar fuzzy incidence graph

1 Introduction

Originated from the chemical experiments in the 1950s, chemical graph theory has been an important branch of theoretical chemistry. In the course of experiments, chemists discovered that the characteristics of the compound have a mysterious relationship with the molecular structure. Therefore, from the analysis of the molecular structure characteristics of compounds, drugs, and materials, the physical, chemical, biological, and pharmaceutical characteristics of the substance can be predicted, to some extent. The specific method is to use graphs to represent the molecular structure, atoms corresponding to vertices, and the chemical bonds between atoms corresponding to edges between vertices. It is defined as the topological index on the molecular map, and then the topological index on the molecular map of the specific chemical structure is calculated to achieve the predicted effect. In recent years, this field has received extensive attention from chemists and mathematicians, and a wealth of theoretical results have been obtained (see Horoldagva and Das [1], Javaid et al. [2], Akhter et al. [3], Shanmukha et al. [4], Yasmeen et al. [5], Awais et al. [6], Gao et al. [7,8], Chen and Singh [9], and Imran et al. [10]).

In fact, a large number of results of topological index calculations have been applied in actual chemistry and molecular related engineering and have played an active role. Mondal et al. [11] proposed the regression analysis for the octane data set to check the predictability of the multiplicative degree-based indices and applied it to some anti-COVID-19 chemicals. Zhang et al. [12] studied the hardness of some superhard BCx crystals in terms of topological indices. Zhong et al. [13] investigated QSPR (quantitative structure–property relationship) using topological indices and applied it to Covid-19 drugs. Saeed et al. [14] researched the topological characteristics for Zeolite Socony Mobil-5 by topological indices. Yaseen et al. [15] determined topological indices of the newly synthesized porphyrins and got correlated them with their molar refractivity. Rashid et al. [16] considered the chemical molecular of titanium difluoride TiF2 and the crystallographic structure of Cu₂O. Balaban [17] presented contributions in both experimental organic and theoretical chemistries on topological computing. Wu et al. [18] suggested a highly selective topological index of chemical bonds-bATID. Zheng et al. [19] studied the characteristics of the anticancer drugs by topological index calculation. Qiong and Li [20] computed some topological indices for archimedean lattices.

Fuzzy mathematics is a tool to describe the uncertainty of things, and graphs are used for structure data representation. In computer science, fuzzy graphs are used to describe irregular data with uncertain features. Recently, the definition of the topological index has been derived from general graphs to various settings of fuzzy graphs and bipolar fuzzy graphs. Binu et al. [21] introduced the connectivity index and average connectivity index of graphs in the fuzzy setting. Binu et al. [22] discussed the Wiener index of fuzzy trees and fuzzy cycles. More related contents can be referred to Fang et al. [23], Gong and Hua [24,25], Nazeer et al. [26], Kavitha et al. [27], Arockiaraj et al. [28], and Binu et al. [29,30].

The enlightenment given by the aforementioned research works is that if a certain property about atoms or chemical bonds in a molecular graph has certain uncertainty, then this uncertainty can be described by defining certain membership functions of vertices and edges. Furthermore, the entire molecular map with fuzzy features becomes a fuzzy molecular graph. Under this framework, the topological index defined on the fuzzy molecular graph can function just like the crisp molecular graph.

Although some literature have explored the uncertainty of molecular graph (see Rouvray [31]), the topological index on the fuzzy molecular graph has not yet found a suitable application field. Therefore, we believe that this field still has research values, since theoretical research often goes further and earlier than experiments. Although it is unknown how the theory is applied at present, it does not mean that it will not work in the future. In summary, due to the success of the topological index on crisp molecular graphs, we believe that as an extension of fuzzy molecular graphs in this field, it will also play a related role. In Section 5, a simulation application on the idealized fuzzy molecular graph has been presented, and this example aims to show how the fuzzy topological index can be used in real chemistry engineering.

Very recently, Fang et al. [23] introduced the connectivity index of fuzzy incidence graph (FIG), and then Gong and Hua [24] extended it to bipolar fuzzy incidence graph (BFIG). The article by Binu et al. [30] has also been studied, in which the cyclic connectivity index (CCI) of fuzzy graphs (FGs) was defined. Motivated by Gong and Hua [24] and Binu et al. [30], in this article, CCI of BFIG have been introduced, and furthermore, some properties on cyclic connectivity index in bipolar incidence fuzzy setting are achieved. The remaining contents are arranged as follows: Section 2, we manifest the definitions as preparatory knowledge; then, the new definition and theoretical results are given; then, these new concepts were reduced and conclusions to incidence fuzzy setting have been presented; finally, an example in the application scenario of anti-aging drugs is presented to show how to use CCI of BFIG in the chemical engineering, and the average cyclic connectivity index of resveratrol, rapamycin, and NAD is calculated.

2 Preknowledge on bipolar fuzzy incidence graph

This section presents the concepts of BFIG, which is introduced in ref. [24]. Let G be an incidence graph (IG) with I ⊆ V × E , and we call ( x , x x ′ ) an incidence pair (IP) (or, pair). A sequence x 0 , ( x 0 , x 0 x 1 ) , x 0 x 1 , ( x 1 , x 0 x 1 ) , x 1 ,…, x n − 1 , ( x n − 1 , x n − 1 x n ) , x n − 1 x n , ( x n , x n − 1 x n ) , x n is a walk in IG, and furthermore, it is closed if x 0 = x n . A walk is a path if no two vertices are the same, and an IG is connected if any two vertices are connected by a path. A cycle is a path with x 0 = x n .

A BFIG is formulated by G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) , where ( ϑ P , ϑ N ) : V → [ 0 , 1 ] × [ − 1 , 0 ] , ( ζ P , ζ N ) : E → [ 0 , 1 ] × [ − 1 , 0 ] , and ( Ξ P , Ξ N ) : V × E → [ 0 , 1 ] × [ − 1 , 0 ] . Moreover, Ξ P ( x , e ) ≤ ϑ P ( x ) ∧ ζ P ( e ) and Ξ N ( x , e ) ≥ ϑ N ( x ) ∨ ζ N ( e ) for any x ∈ V and e ∈ E , and ( Ξ P , Ξ N ) is a bipolar fuzzy incidence (BFI) of G. Set ϑ ⁎ as the vertex set of G. The positive incidence strength (PIS) is denoted by the minimum value of Ξ P ( x , x x ′ ) , while negative incidence strength (NIS) is denoted by the maximum value of Ξ N ( x , x x ′ ) .

In BFIG setting, Ξ P and Ξ N express the positive and negative relations between atoms and chemical bonds. The concept of partial subgraph can be referred to the study by Gong and Hua [24].

Let G be a BFIG. An IP ( x , x x ′ ) is strong if Ξ P ( x , x x ′ ) ≥ ICONN G − ( x , x x ′ ) P ( x , x x ′ ) , Ξ N ( x , x x ′ ) ≤ ICONN G − ( x , x x ′ ) N ( x , x x ′ ) , where ICONN G − ( x , x x ' ) P ( x , x x ′ ) is the maximum PIS of a − a b and ICONN G − ( x , x x ' ) N ( x , x x ′ ) is the minimum NIS of a − a b . If Ξ P ( x , x x ′ ) > ICONN G − ( x , x x ′ ) P ( x , x x ′ ) and Ξ N ( x , x x ′ ) < ICONN G − ( x , x x ′ ) N ( x , x x ′ ) , then the pair ( x , x x ′ ) is called α -strong. If Ξ P ( x , x x ′ ) = ICONN G − ( x , x x ′ ) P ( x , x x ′ ) and Ξ N ( x , x x ′ ) = ICONN G − ( x , x x ′ ) N ( x , x x ′ ) , then the pair ( x , x x ′ ) is called β -strong. A path or cycle is strong if all its pairs are strong.

A BFIG G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) is complete if Ξ P ( x , x x ′ ) = ϑ P ( x ) ∧ ζ P ( x x ′ ) and Ξ N ( x , x x ′ ) = ϑ N ( x ) ∨ ζ N ( x x ′ ) for any ( x , x x ′ ) ∈ Ξ ⁎ . Let G 1 = ( ϑ 1 P , ϑ 1 N , ζ 1 P , ζ 1 N , Ξ 1 P , Ξ 1 N ) and G 2 = ( ϑ 2 P , ϑ 2 N , ζ 2 P , ζ 2 N , Ξ 2 P , Ξ 2 N ) be two BFIGs. If there is a bijective mapping (BM) ι : V ( G 1 ) → V ( G 2 ) with ϑ 1 P ( x ) = ϑ 2 P ( ι ( x ) ) , ϑ 1 N ( x ) = ϑ 2 N ( ι ( x ) ) for all x ∈ V ( G 1 ) , ζ 1 P ( x x ′ ) = ζ 2 P ( ι ( x ) ι ( x ′ ) ) , ζ 1 N ( x x ′ ) = ζ 2 N ( ι ( x ) ι ( x ′ ) ) for all x x ′ ∈ E ( G 1 ) , and Ξ 1 P ( x , x x ′ ) = Ξ 2 P ( ι ( x ) , ι ( x ) ι ( x ′ ) ) , Ξ 1 N ( x , x x ′ ) = Ξ 2 N ( ι ( x ) , ι ( x ) ι ( x ′ ) ) for any pair ( x , x x ′ ) . Then, these two BFIG are isomorphism.

3 New definitions and theoretical results

Definition 1

Let G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) be a BFIG. The cyclic connectivity index (CCI) of G is formulated by

CCI BFI ( G ) = ( CCI BFI P ( G ) , CCI BFI N ( G ) )

= ∑ x , y ∈ ϑ ⁎ ϑ P ( x ) ϑ P ( y ) CCONN G P ( x , y ) , ∑ x , y ∈ ϑ ⁎ ϑ N ( x ) ϑ N ( y ) CCONN G N ( x , y ) ,

where CCI BFI P ( G ) and CCI BFI N ( G ) are positive CCI and negative CCI of G, respectively, CCONN G P ( x , y ) is the maximum value of PIS for all the possible strong incidence cycles over x and y, and CCONN G N ( x , y ) is the minimum value of NIS for all the possible strong incidence cycles over x and y.

The drawback of the cyclic connectivity index in Definition 1 is that it is calculated purely cumulatively. That is, it is possible that the cyclic connectivity between atoms in the entire molecular graph is very weak, but because the number of vertices and edges in the molecular graph is very large, the positive cyclic connectivity index of the entire graph is very large, and the negative cyclic connectivity index is very small. To overcome this flaw, the content of the average version of the cyclic connectivity index will be defined later.

$Figure 1 A BFIG with CCI BFI ( G ) = ( 2.6 , − 2.4 ) {\text{CCI}}_{\text{BFI}}(G)=(2.6,-2.4) .$

Figure 1

A BFIG with CCI BFI ( G ) = ( 2.6 , − 2.4 ) .

Example 1

Considering a BFIG as depicted in Figure 1. Set ( ϑ P ( x ) , ϑ N ( x ) ) = ( 1 , − 1 ) for arbitrary x ∈ ϑ ⁎ . We confirm that ζ P ( x 1 x 2 ) = 0.7 , ζ N ( x 1 x 2 ) = − 0.4 , ζ P ( x 2 x 3 ) = 0.5 , ζ N ( x 2 x 3 ) = − 0.6 , ζ P ( x 2 x 5 ) = 0.8 , ζ N ( x 2 x 5 ) = − 0.3 , ζ P ( x 3 x 5 ) = 0.5 , ζ N ( x 3 x 5 ) = − 0.5 , ζ P ( x 3 x 4 ) = 0.3 , ζ N ( x 3 x 4 ) = − 0.7 , ζ P ( x 4 x 1 ) = 0.3 , ζ N ( x 4 x 1 ) = − 0.7 , Ξ P ( x 1 , x 1 x 2 ) = 0.6 , Ξ N ( x 1 , x 1 x 2 ) = − 0.3 , Ξ P ( x 2 , x 2 x 1 ) = 0.7 , Ξ N ( x 2 , x 2 x 1 ) = − 0.4 , Ξ P ( x 2 , x 2 x 3 ) = 0.4 , Ξ N ( x 2 , x 2 x 3 ) = − 0.5 , Ξ P ( x 3 , x 3 x 2 ) = 0.5 , Ξ N ( x 3 , x 3 x 2 ) = − 0.5 , Ξ P ( x 2 , x 2 x 5 ) = 0.8 , Ξ N ( x 2 , x 2 x 5 ) = − 0.2 , Ξ P ( x 5 , x 5 x 2 ) = 0.7 , Ξ N ( x 5 , x 5 x 2 ) = − 0.3 , Ξ P ( x 3 , x 3 x 5 ) = 0.4 , Ξ N ( x 3 , x 3 x 5 ) = − 0.4 , Ξ P ( x 5 , x 5 x 3 ) = 0.5 , Ξ N ( x 5 , x 5 x 3 ) = − 0.5 , Ξ P ( x 3 , x 3 x 4 ) = 0.3 , Ξ N ( x 3 , x 3 x 4 ) = − 0.5 , Ξ P ( x 4 , x 4 x 3 ) = 0.2 , Ξ N ( x 4 , x 4 x 3 ) = − 0.6 , Ξ P ( x 4 , x 4 x 1 ) = 0.2 , Ξ N ( x 4 , x 4 x 1 ) = − 0.5 , Ξ P ( x 1 , x 1 x 4 ) = 0.3 , Ξ N ( x 1 , x 1 x 4 ) = − 0.6 , CCI BFI ( G ) = ( 2.6 , − 2.4 ) .

Let G be a BFIG and H be a partial subgraph (PS) of G. Then, CCI BFI P ( H ) ≤ CCI BFI P ( G ) and CCI BFI N ( H ) ≥ CCI BFI N ( G ) may not be established. The next instance was used to elaborate this phenomenon.

$Figure 2 A BFIG with CCI BFI ( G ) = ( 1.2 , − 0.6 ) {\text{CCI}}_{\text{BFI}}(G)=(1.2,-0.6) .$

Figure 2

A BFIG with CCI BFI ( G ) = ( 1.2 , − 0.6 ) .

Example 2

Consider a bipolar fuzzy incidence graph G in Figure 2, where ϑ ⁎ = { x 1 , x 2 , x 3 , x 4 , x 5 } . Set ( ϑ P ( x ) , ϑ N ( x ) ) = ( 1 , − 1 ) for arbitrary x ∈ ϑ ⁎ . We observe that ζ P ( x 1 x 2 ) = 0.2 , ζ N ( x 1 x 2 ) = − 0.8 , ζ P ( x 2 x 3 ) = 0.95 , ζ N ( x 2 x 3 ) = − 0.1 , ζ P ( x 3 x 4 ) = 1 , ζ N ( x 3 x 4 ) = − 0.2 , ζ P ( x 4 x 1 ) = 0.3 , ζ N ( x 4 x 1 ) = − 0.8 , ζ P ( x 2 x 4 ) = 0.4 , ζ N ( x 2 x 4 ) = − 0.6 , ζ P ( x 3 x 5 ) = 0.4 , ζ N ( x 3 x 5 ) = − 0.7 , ζ P ( x 4 x 5 ) = 0.9 , ζ N ( x 4 x 5 ) = − 0.3 , Ξ P ( x 1 , x 1 x 2 ) = 0.2 , Ξ N ( x 1 , x 1 x 2 ) = − 0.7 , Ξ P ( x 2 , x 2 x 1 ) = 0.2 , Ξ N ( x 2 , x 2 x 1 ) = − 0.8 , Ξ P ( x 2 , x 2 x 3 ) = 0.9 , Ξ N ( x 2 , x 2 x 3 ) = − 0.1 , Ξ P ( x 3 , x 3 x 2 ) = 0.9 , Ξ N ( x 3 , x 3 x 2 ) = − 0.1 , Ξ P ( x 3 , x 3 x 4 ) = 0.9 , Ξ N ( x 3 , x 3 x 4 ) = − 0.1 , Ξ P ( x 4 , x 4 x 3 ) = 0.9 , Ξ N ( x 4 , x 4 x 3 ) = − 0.1 , Ξ P ( x 4 , x 4 x 1 ) = 0.3 , Ξ N ( x 4 , x 4 x 1 ) = − 0.7 , Ξ P ( x 1 , x 1 x 4 ) = 0.2 , Ξ N ( x 1 , x 1 x 4 ) = − 0.7 , Ξ P ( x 4 , x 4 x 2 ) = 0.3 , Ξ N ( x 4 , x 4 x 2 ) = − 0.6 , Ξ P ( x 2 , x 2 x 4 ) = 0.3 , Ξ N ( x 2 , x 2 x 4 ) = − 0.5 , Ξ P ( x 3 , x 3 x 5 ) = 0.4 , Ξ N ( x 3 , x 3 x 5 ) = − 0.6 , Ξ P ( x 5 , x 5 x 3 ) = 0.3 , Ξ N ( x 5 , x 5 x 3 ) = − 0.7 , Ξ P ( x 4 , x 4 x 5 ) = 0.9 , Ξ N ( x 4 , x 4 x 5 ) = − 0.2 , Ξ P ( x 5 , x 5 x 4 ) = 0.9 , Ξ N ( x 5 , x 5 x 4 ) = − 0.1 , and CCI BFI ( G ) = ( 1.2 , − 0.6 ) since pairs x 2 x 4 and x 3 x 5 are not strong.

By deleting edge x 3 x 4 , the following partial graph H of G was obtained as shown in Figure 3. In terms of graph analysis and calculation, CCI BFI ( H ) = ( 2.6 , − 2.2 ) is obtained. Hence, we derive CCI BFI P ( H ) > CCI BFI P ( G ) and CCI BFI N ( H ) < CCI BFI N ( G ) .

$Figure 3 A partial bipolar fuzzy incidence subgraph with CCI BFI ( H ) = ( 2.6 , − 2.2 ) {\text{CCI}}_{\text{BFI}}(H)=(2.6,-2.2) .$

Figure 3

A partial bipolar fuzzy incidence subgraph with CCI BFI ( H ) = ( 2.6 , − 2.2 ) .

The following conclusion reveals the relationship of CCI between BFIG and its PS when G is strong (all pairs are strong).

Theorem 1

If H = ( κ P , κ N , ϕ P , ϕ N , Ω P , Ω N ) is a PS of strong BFIG G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) , then CCI BFI P ( H ) ≤ CCI BFI P ( G ) , CCI BFI N ( H ) ≥ CCI BFI N ( G ) .

Proof

For any z 1 , z 2 ∈ κ ⁎ , we obtain κ P ( z 1 ) ≤ ϑ P ( z 1 ) , κ N ( z 1 ) ≥ ϑ N ( z 1 ) , κ P ( z 2 ) ≤ ϑ P ( z 2 ) , and κ N ( z 2 ) ≥ ϑ N ( z 2 ) by means of partial bipolar fuzzy incidence subgraph. Moreover, we obtain CCONN H P ( z 1 , z 2 ) ≤ CCONN G P ( z 1 , z 2 ) , CCONN H N ( z 1 , z 2 ) ≥ CCONN G N ( z 1 , z 2 ) . Hence, we obtain

∑ z 1 , z 2 ∈ ϑ ⁎ ϑ P ( z 1 ) ϑ P ( z 2 ) CCONN H P ( z 1 , z 2 ) ≤ ∑ z 1 , z 2 ∈ ϑ ⁎ ϑ P ( z 1 ) ϑ P ( z 2 ) CCONN G P ( z 1 , z 2 ) ,

∑ z 1 , z 2 ∈ ϑ ⁎ ϑ N ( z 1 ) ϑ N ( z 2 ) CCONN H N ( z 1 , z 2 ) ≥ ∑ z 1 , z 2 ∈ ϑ ⁎ ϑ N ( z 1 ) ϑ N ( z 2 ) CCONN G N ( z 1 , z 2 ) .

The proof of Theorem 1 is completed.

Theorem 2 describes the relationship on CCI between the original bipolar fuzzy graph and its subgraph under the connected framework.

Table 1

The value of ACCI for BFIGs in Figures 1–4

Value of ACCI	BFIG
( 0.26 , − 0.24 )	Figure 1
( 0.12 , − 0.06 )	Figure 2
( 0.26 , − 0.22 )	Figure 3
( 0.25 , − 0.3 )	Figure 4

Theorem 2

If H is a subgraph of connected BFIG G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) , where V ( H ) = V ( G ) − { v } , then CCI BFI P ( H ) < CCI BFI P ( G ) , CCI BFI N ( H ) > CCI BFI N ( G ) .

Example 3

Considering a BFIG as in Figure 4, which is yielded from Figure 2 by removing x 5 . It was confirmed that ϑ ⁎ = { x 1 , x 2 , x 3 , x 4 } and CCI BFI ( G ) = ( 1.5 , − 1.8 ) .

Theorem 3 describes the equivalence of CCI values between isomorphic bipolar fuzzy graphs.

$Figure 4 Subgraph of BFIG with CCI BFI ( G ) = ( 1.5 , − 1.8 ) {\text{CCI}}_{\text{BFI}}(G)=(1.5,-1.8) .$

Figure 4

Subgraph of BFIG with CCI BFI ( G ) = ( 1.5 , − 1.8 ) .

Theorem 3

Let G 1 = ( ϑ 1 P , ϑ 1 N , ζ 1 P , ζ 1 N , Ξ 1 P , Ξ 1 N ) and G 2 = ( ϑ 2 P , ϑ 2 N , ζ 2 P , ζ 2 N , Ξ 2 P , Ξ 2 N ) be two isomorphic BFIGs. Then, CCI BFI ( G 1 ) = CCI BFI ( G 2 ) , i.e., CCI BFI P ( G 1 ) = CCI BFI P ( G 2 ) and CCI BFI N ( G 1 ) = CCI BFI N ( G 2 ) .

Proof

Let ι be a BM between two BFIGs. We deduce CCONN G 1 P ( z 1 , z 2 ) = CCONN G 2 P ( ι ( z 1 ) , ι ( z 2 ) ) and CCONN G 1 N ( z 1 , z 2 ) = CCONN G 2 N ( ι ( z 1 ) , ι ( z 2 ) ) for arbitrary z 1 , z 2 ∈ η ⁎ . Hence,

∑ z 1 , z 2 ∈ ϑ 1 ⁎ ϑ P ( z 1 ) ϑ P ( z 2 ) CCONN G 1 P ( z 1 , z 2 ) = ∑ z 1 , z 2 ∈ ϑ 2 ⁎ ϑ P ( z 1 ) ϑ P ( z 2 ) CCONN G 2 P ( ι ( z 1 ) , ι ( z 2 ) ) ,

∑ z 1 , z 2 ∈ ϑ 1 ⁎ ϑ N ( z 1 ) ϑ N ( z 2 ) CCONN G 1 N ( z 1 , z 2 ) = ∑ z 1 , z 2 ∈ ϑ 2 ⁎ ϑ N ( z 1 ) ϑ N ( z 2 ) CCONN G 2 N ( ι ( z 1 ) , ι ( z 2 ) ) .

As a result, we confirm CCI BFI P ( G 1 ) = CCI BFI P ( G 2 ) and CCI BFI N ( G 1 ) = CCI BFI N ( G 2 ) .

Definition 2

Let G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) be a BFIG. The average cyclic connectivity index (ACCI) of G is given by

ACCI BFI ( G ) = ( ACCI BFI P ( G ) , ACCI BFI N ( G ) )

= 1 ∣ ϑ ⁎ ∣ 2 ∑ z 1 , z 2 ∈ ϑ ⁎ ϑ P ( z 1 ) ϑ P ( z 2 ) CCONN G P ( z 1 , z 2 ) , × ∑ z 1 , z 2 ∈ ϑ ⁎ ϑ N ( z 1 ) ϑ N ( z 2 ) CCONN G N ( z 1 , z 2 ) .

Example 4

Consider BFIGs manifested in Figures 1–4, we have the following conclusion, which is presented in Table 1.

Definition 3

Let G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) be a BFIG and x ∈ ϑ ⁎ .

If ACCI BFI P ( G − { x } ) < ACCI BFI P ( G ) and ACCI BFI N ( G − { x } ) > ACCI BFI N ( G ) , then x is a BFICC-reducing vertex.
If ACI BFI P ( G − { x } ) > ACI BFI P ( G ) and ACI BFI N ( G − { x } ) < ACI BFI N ( G ) , then x is a BFICC-enhancing vertex.
If ACI BFI P ( G − { x } ) = ACI BFI P ( G ) and ACI BFI N ( G − { x } ) = ACI BFI N ( G ) , then x is a BFICC-neutral vertex.

The vertex classification in Definition 3 describes the position and influence of the vertex in the entire molecular graph to a certain extent, as well as its influence on the bipolar cyclic connectivity of the entire bipolar fuzzy graph.

Theorem 4

Let G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) be a BFIG, ∣ ϑ ⁎ ∣ ≥ 3 , x ∈ ϑ ⁎ , a = CCI BFI P ( G ) CCI BFI P ( G − { x } ) and b = CCI BFI N ( G ) CCI BFI N ( G − { x } ) . Then,

x is a BFICC -enhancing vertex ⇔ a < ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 and b > ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 ;
x is a BFICC-reducing vertex ⇔ a > ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 and b < ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 ;
x is a BFICC- neutral vertex ⇔ a = ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 and b = ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 .

Proof

For convenience, the proof of the last part of the result is only shown. Similar methods can be used for other parts.

Let x is a BFICCI-reducing vertex. In terms of its definition, we derive

ACCI BFI P ( G − { x } ) < ACCI BFI P ( G ) and ACCI BFI N ( G − { x } ) > ACCI BFI N ( G )

⇒ CCI BFI P ( G − { x } ) ( ∣ ϑ ⁎ ∣ − 1 ) ( ∣ ϑ ⁎ ∣ − 2 ) < CCI BFI P ( G ) ∣ ϑ ⁎ ∣ ( ∣ ϑ ⁎ ∣ − 1 ) and CCI BFI N ( G − { x } ) ( ∣ ϑ ⁎ ∣ − 1 ) ( ∣ ϑ ⁎ ∣ − 2 ) > CCI BFI N ( G ) ( ∣ ϑ ⁎ ∣ − 1 ) ∣ ϑ ⁎ ∣

⇒ ( ∣ ϑ ⁎ ∣ − 1 ) ∣ ϑ ⁎ ∣ ( ∣ ϑ ⁎ ∣ − 1 ) ( ∣ ϑ ⁎ ∣ − 2 ) < CCI BFI P ( G ) CCI BFI P ( G − { x } ) and ( ∣ ϑ ⁎ ∣ − 1 ) ∣ ϑ ⁎ ∣ ( ∣ ϑ ⁎ ∣ − 1 ) ( ∣ ϑ ⁎ ∣ − 2 ) > CCI BFI N ( G ) CCI BFI N ( G − { x } )

⇒ a > ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 and b < ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 .

Suppose a > ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 and b < ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2

⇒ CCI BFI P ( G ) CCI BFI P ( G − { x } ) > ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 and CCI BFI N ( G ) CCI BFI N ( G − { x } ) < ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2

⇒ CCI BFI P ( G − { x } ) ( ∣ ϑ ⁎ ∣ − 1 ) ( ∣ ϑ ⁎ ∣ − 2 ) < CCI BFI P ( G ) ( ∣ ϑ ⁎ ∣ − 1 ) ∣ ϑ ⁎ ∣ and CCI BFI N ( G − { x } ) ( ∣ ϑ ⁎ ∣ − 1 ) ( ∣ ϑ ⁎ ∣ − 2 ) > CCI BFI N ( G ) ( ∣ ϑ ⁎ ∣ − 1 ) ∣ ϑ ⁎ ∣

⇒ ACCI BFI P ( G − { x } ) < ACCI BFI P ( G ) and ACCI BFI N ( G − { x } ) > ACCI BFI N ( G )

⇒ x is a BFICC-reducing vertex. □

Definition 4

Let G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N ) be a connected BFIG with ∣ ϑ ⁎ ∣ ≥ 3 .

If G has at least one BFICC-enhancing vertex, then G is called BFICC enhancing graph.
If G has no BFICC-reducing vertex, then G is called BFICC reducing graph.
If all the vertices in G are BFICC-neutral vertices, then G is called BFICC neutral graph.

4 Reduce to FIG setting

It is noticed that the fuzzy incidence graph setting is in light of eliminating the relevant part of the negative membership function. Hence, the definitions and theorems are directly listed here, and the detailed explanations are skipped.

Definition 5

Let G = ( ϑ , ζ , Ξ ) be a FIG. The CCI of G is formulated by

CCI FI ( G ) = ∑ z 1 , z 2 ∈ ϑ ⁎ ϑ ( z 1 ) ϑ ( z 2 ) CCONN G ( z 1 , z 2 ) ,

where CCONN G ( z 1 , z 2 ) is the maximum value of IS for all the possible strong incidence cycles over z 1 and z 2 .

Let G be a FIG and H be a partial subgraph of G. Although CCI FI ( H ) ≤ CCI FI ( G ) may not be established, the fact below is shown.

Theorem 5

If H = ( κ , ϕ , Ω ) is a PS of strong FIG G = ( ϑ , ζ , Ξ ) , then CCI FI ( H ) ≤ CCI FI ( G ) .

Theorem 6

If H is a subgraph of connected FIG G = ( η , ζ , Ξ ) by deleting one vertex from G, then CCI FI ( H ) < CCI FI ( G ) .

Theorem 7

CCI FI of isomorphic FIGs are the same.

Definition 6

Let G = ( ϑ , ζ , Ξ ) be an FIG. The ACCI of G is formulated by

ACCI FI ( G ) = 1 ∣ ϑ ⁎ ∣ 2 ∑ z 1 , z 2 ∈ ϑ ⁎ ϑ ( z 1 ) ϑ ( z 2 ) CCONN G ( z 1 , z 2 ) .

Definition 7

Let G = ( ϑ , ζ , Ξ ) be a FIG and x ∈ ϑ ⁎ .

If ACCI FI ( G − { x } ) < ACCI FI ( G ) , then x is a FICC-reducing vertex.
If ACI FI ( G − { v } ) > ACI FI ( G ) , then v is a FICC--enhancing vertex.
If ACI FI ( G − { v } ) = ACI FI ( G ) , then v is a FICC--neutral vertex.

Theorem 8

Let G = ( ϑ , ζ , Ξ ) be an FIG, ∣ ϑ ⁎ ∣ ≥ 3 , x ∈ ϑ ⁎ , a = CCI FI ( G ) CCI FI ( G − { x } ) . Then,

x is a FICC-enhancing vertex ⇔ a < ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 ;
x is a FICC -reducing vertex ⇔ a > ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 ;
x is a FICC – neutral vertex ⇔ a = ∣ ϑ ⁎ ∣ ∣ ϑ ⁎ ∣ − 2 .

Definition 8

Let G = ( ϑ , ζ , Ξ ) be a connected FIG with ∣ ϑ ⁎ ∣ ≥ 3 .

If G has at least one FICC-enhancing vertex, then G is an FICC enhancing graph.
If G has no FICC-reducing vertex, then G is an FICC reducing graph.
If all the vertices in G are FICC-neutral vertices, then G is an FICC neutral graph.

5 Anti-aging drug simulation

In this section, the application of the CCI of BFIG to a specific application has been conducted, and as an instance, it presents an idea on how to apply CCI calculation to actual engineering. MATLAB software has been used to complete the computational tasks.

Studies in modern medicine and biology have shown that the aging mechanism has both irresistible external environmental factors (i.e., the irreversible damage to human cells and molecules caused by the environment, and the limited survival resources in the environment where organisms are located), as well as genetic internal factors (i.e., the self-repair ability of molecules is controlled by genes and performance genetics). When a person is a teenager, the repair mechanism works at its best. Then, after the age of 40, the repair molecules in the human body begin to decrease or fail. Repair molecules that have the dual functions of repairing cells and controlling genes are lost due to too much work and cannot reach the designated positions. At this time, the molecule is restricted by certain functions and is not trying to repair it. On the other hand, the gene related to the repair molecule has not changed and has not been reduced, but it seems to be turned off or turned down. In this respect, the essence of anti-aging drugs is to activate these repair molecules and make them work as they did when the human body was young.

Controlling the diet, keeping a proper low temperature, and doing proper exercises can all put a certain amount of pressure on the body. The principle of anti-aging drugs is to achieve the same effect through drugs. For example, rapamycin (see Figure 5) and resveratrol (C₁₄H₁₂O₃, see Figure 6) can simulate calorie restriction in the body and bring stress signals to the body. Nicotinamide adenine dinucleotide (NAD and its molecular structure are shown in Figure 7) is easily combined with H+ to become NADH, and the chemical reaction is reversible. Therefore, the role of NAD is to transfer hydrogen ions (protons) to various chemical reactions in cells, thereby providing support for chemical reactions in cells. It is used for the metabolism and construction of new cells, at the same time it transmits information within the cell and is also a necessary raw material for mitochondria to convert food into energy. More importantly, it is the raw material for various repair molecules and more NAD can make various repair molecules complete the repair work faster. In other words, the essence of the methods of dieting, keeping a proper low temperature, and proper exercise is to stimulate the increase of NAD production in the human body, thereby enhancing the repairability.

Figure 5

The structure of rapamycin [32].

Figure 6

The structure of resveratrol [33].

Figure 7

The structure of nicotinamide adenine dinucleotide.

However, the cell membrane of the cell prohibits the passage of NAD, that is, the body cannot absorb foreign NAD. Therefore, anti-aging drugs are the raw materials for the manufacture of various NADs, that is, the precursors of NAD, which are molecules with a similar structure to NAD and can be absorbed by the human body.

For the anti-aging drugs rapamycin and resveratrol, as well as drugs similar in structure to NAD, further stability is expected. Shown in the molecular structure, if from the perspective of graph theory, its CCI needs to be guaranteed to a certain extent. For example, drugs with similar NAD structures are the raw materials for repairing molecules, and repairing molecules need to be combined with damaged DNA molecules. Therefore, in terms of molecular structure, repair molecules often have two arms, which can be linked to the damaged part (see Figure 8). During the repair process, the repair molecules and damaged parts need to resist external attacks and cannot be broken before the repair is completed. Therefore, the repair molecule itself needs to have a certain vulnerability, and its CCI has certain requirements.

Figure 8

Repair molecules to connect damaged DNA by arm.

It inspires us to study the CCI of these anti-aging drugs and their related structures, and the molecular structures of rapamycin, resveratrol, and NAD were calculated in this section. Before calculating, the following points should be taken into account:

First, the real molecular structures of these anti-aging drugs are very complicated. Fortunately, there are several published works, and experimental and ab-initio computations that present more detailed information on the molecular structures. Kessler et al. [32] studied the structure of rapamycin by nuclear magnetic resonance and molecular dynamics and reported a well-defined conformation of the rapamycin molecule in solution. Caruso et al. [33] pointed out that resveratrol has shown activity as an effective anticarcinogenic agent against several tumor cells in vitro and in vivo, and they studied resveratrol and its derivatives from the experimental and ab-initio calculations and described the structural features. Guillot et al. [34] studied NAD from the theoretical and experimental prospective and determined the structure of NAD from X-ray and neutron diffraction experiments by Miwa et al. [35]. NAD itself is not absorbed by cell membranes, and the real drug structure is a modified molecule of NAD. However, verification of the specific manifestations could not be performed. They are often trade secrets. Therefore, the molecular structure calculated in this article is only an approximate structure of the real anti-aging drugs.
Second, filtering the approximate structure is required, as well as extracting the main part of the molecular structure to participate in the calculation.
Finally, although the setting of the membership function has been roughly explained, in fact, the statue and role of each atom and each chemical bond between the atoms in the entire molecular structure are still unknown. Whether the value of the membership function can really reflect the actual uncertainty could not be precisely confirmed. In the following calculations, only a possible uncertainty is simulated, and the value of the membership function under the simulated conditions is given to calculate the estimated value of ACCI.

The detailed procedures to determine the ACCI of given bipolar fuzzy incidence molecular graph are listed below:

Input: a BFIG G = ( ϑ P , ϑ N , ζ P , ζ N , Ξ P , Ξ N )

Step 1: Determine all strong IP ( x , x x ′ ) in G and set G 1 = ( ϑ 1 P , ϑ 1 N , ζ 1 P , ζ 1 N , Ξ 1 P , Ξ 1 N ) as the strong bipolar fuzzy incidence molecular subgraph of G (consisting of all strong IP of G).

Step 2: Determine all cycles in G 1 and denote C = { C 1 , C 2 , ⋯ , C m } by the set of these cycles.

Step 3: Compute CCONN G P ( x , x ′ ) and CCONN G N ( x , x ′ ) for each pair of x , x ′ ∈ ϑ ⁎ .

Step 4: Calculate CCI BFI P ( G ) and CCI BFI N ( G ) , and thus determine the CCI of G.

Step 5: Derive ACCI BFI ( G ) = 1 ∣ ϑ ⁎ ∣ 2 ( CCI BFI P ( G ) , CCI BFI N ( G ) ) .

Output: return ACCI of bipolar fuzzy incidence molecular graph G.

To facilitate the calculation, the model has been simplified and ( ϑ P ( v ) , ϑ N ( v ) ) = ( 1 , − 1 ) is assumed for each atom. The uncertainty of ionic bond, covalent bond, and metal bond in the drug are (0.9, −0.1), (0.85, −0.15), and (0.8, −0.2), respectively. For the values of Ξ P and Ξ N of pairs, It is considered that the stability of the connection between the atom and the atom bond, that is, the quality of the excuse between the atom and the bond to which it is connected. The detailed ACCI values are listed in Table 2.

From Table 2, it can be concluded that rapamycin has the strongest connectivity among three molecular structures in the aforementioned ideal framework. However, the aforementioned calculation results have two major limitations: (1) the monocular structure is approximate; (2) the membership function representing uncertainty is simulated by ourselves. In the actual chemical engineering field, it is necessary to define the uncertainty according to the actual situation and to choose the membership function to describe the uncertainty.

Table 2

Comparison of ACCI of three molecular structures

Molecular structure	ACCI
Rapamycin	(0.64, −0.57)
Resveratrol	(0.59, −0.59)
NAD structure-liked	(0.53, −0.71)

6 Conclusion

This contribution introduces the CCI and ACCI of BFIGs, the characteristics of this topological index in bipolar fuzzy graph setting are derived, and finally, cyclic connectivity index is applied to uncertainty calculation of anti-aging drugs. Although there is an example showing the last section to imitate the usage of CCI computing for specific field, so far it is still unclear how to apply the theory in this article to the real chemical engineering field. However, it is perceived that in the future, these results have potential applications:

The calculation of the topological index of molecular graphs has proved to have broad application prospects in the field of chemistry, such as the prediction of the boiling point and melting point of compounds, the prediction of the activity of anti-aging drugs, the evaluation of the performance of nanotubes and nanomaterials, the defense of COVID-19, and so on. The fuzzy molecular graph is an enhanced version of the original molecular graph, which combines the uncertainties of vertices and edges. When a certain feature in the molecular structure contains uncertainty, fuzzy molecular graphs can play a role.
In the field of information science, graph is a structured representation or irregular representation of data. It has a wide range of applications in molecular networks, gene expression human skeleton characterization, and compound knowledge structure representation (GO ontology and PO ontology). The fuzzy graph framework combines the uncertainty associated with the graph combination and can handle fuzzy information and fuzzy relations and has a wider application background.

In conclusion, it is optimistic that the theoretical results of this article have potential chemical applications in the near future.

Acknowledgments

We are very grateful to the reviewers for their constructive comments.

Funding information: This work was supported by Natural Science Foundation of Jiangsu Province, China (BK20191032) and University Philosophy and Social Science research projects of Jiangsu Province (2020SJA1195).
Author contributions: Juanjuan Lu: writing original draft; Linli Zhu: writing review and editing; Wei Gao: writing review and editing.
Conflict of interest: The authors declare that 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: There is no experiment data in this work.

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Received: 2021-12-05

Revised: 2022-03-13

Accepted: 2022-03-28

Published Online: 2022-04-19

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

Articles in the same Issue

https://doi.org/10.1515/chem-2022-0149

Keywords for this article

fuzzy graph; cyclic connectivity index; average cyclic connectivity index; fuzzy incidence graph; bipolar fuzzy incidence graph

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