# The Calculations of Topological Indices on Certain Networks

Authors: Jia-Bao Liu; Ting Zhang; Sakander Hayat
Journal: Journal of Mathematics
 (2021)

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## Abstract

It is one of the core problems in the study of chemical graph theory to study the topological index of molecular graph and the internal relationship between its structural properties and some invariants. In recent years, topological index has been gradually applied to the models ofQSAR and QSPR. In this work, using the definition of the ABC index, AZI index, GA index, the multiplicative version of ordinary first Zagreb index, the second multiplicative Zagreb index, and Zagreb index, we calculate the degree-based topological indices of some networks. Then, the above indices’ formulas are obtained.


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## Body

## 1. Introduction

The topological index is a numerical parameter in the structure graph of molecular compounds, and it can be used to predict the chemical and physical properties of molecules or to predict the biological activity [1–5]. In this paper, we mainly calculate several topological indices, which are invariants that can describe some properties of graph. The topological indices consist of three parts, namely, degree-based indices, spectrum-based indices, and distance-based indices; meanwhile, many indices based on both degree and distance are followed in [6, 7]. The Kirchhoff index is based on the boiling point of kerosene, and other indices can predict the chemical and biological properties of some substances. We are dealing with some degree-based indices, such as, ABC index, AZI index, and GA index. We also calculate some indices for some chemical networks, examples include the first and the second Zagreb index and Zagreb index, by which we can predict the stability or others properties of some networks, such as n-dimensional silicate networks (Sn), chain silicate networks (Cn), hexagonal networks (Hn), n-dimensional honeycomb networks (HCn), cellular networks (CPn), and Sierpiński networks (sp,m). The graph of various networks are shown in Figures 1–6. In the rest of the paper, we made the following arrangements. In Section 1, we introduce some indices and their backgrounds. In Section 2, we show the important results of this paper. In Section 3, we make a summary.Figure 1
The silicate networksSn.Figure 2
The chain silicate networksCn.Figure 3
The hexagonal networksHn.Figure 4
The oxide networksOn.Figure 5
The cellular networksCPn.Figure 6
The generalized Sierpiński networks. (a)t = 1. (b) t = 2. (c) t = 3.
(a)(b)(c)All graphs and networks are limited to simple undirected graphs. LetG=VG,EG represent the vertex set and edge set of the networks, respectively. The degree of vertex i is the number of edges associated with i, expressed by di. The standard notation and topological descriptors are mainly followed in [8].According to the chemical molecules, the molecular graph is made up of atoms and bonds. The atom-bond connectivity indexABCG is written as(1)ABCG=∑ij∈EGdi+dj−2di×dj,and posted by Estrada et al. [9]. The geometric-arithmetic index GAG is the valency-based topological index denoted by(2)GAG=∑ij∈EG2di×djdi+dj,which is proposed by Furtula and Vukicević followed in [10]. Compared with other indices, geometric-arithmetic index is better than other topological indices for predicting the physical and chemical properties of some substances, and more properties of GAG index is not introduced and readers can read the literature [11]. In 2010, Furtula et al. proposed AZIG index [12], named as the augmented Zagred index, denoted by(3)AZIG=∑ij∈EGdi×djdi+dj−23.The ordinary first Zagreb index of the multiplication version is represented as(4)Π1∗G=∏ij∈EGdi+dj.For the chemical properties and applications of this index, readers can refer to [13–19]. The second multiplication Zagreb index [20] is written as(5)Π2G=∏ij∈EGdi×dj.On the basis of the Zagreb index, Azari et al. [21] put forward their general form and defined it as(6)Za,bG=∑ij∈EGdiadjb+dibdja.In 1972, theF index was put forward, but there is little research on it. In 2015, B. Furtula and I. Gutman [22] redefined it as the forgotten topological index, or the F index for short, and defined it as(7)FG=∑ijεEGdi2+dj2.About its related research, the reader may refer to [23–26].


## 2. Main Results and Discussion

In this section, according to the definition of theABC index, AZI index, GA index, multiplicative version of ordinary first Zagreb index, second multiplicative Zagreb index, and Zagreb index, we calculate the correlation index formula of several kinds of networks and get their concrete expressions.Silicate is one of the most abundant minerals in the world. It is a mixture of metal compounds and sand [27]. Sn represents a silicate network, where n is the number of hexagons between the boundary and the center. Then, one has VSn=15n2+3n,Sn=36n2 and the following results.According to the distribution of networks vertices, there are three sets of vertex division based on valencies, asA1,A2,andA3. The set A1 consists of 6n edges ij, where di=3anddj=3. The set A2 consists of 18n2+6n edges ij, where di=3anddj=6. The set A3 consists of 18n2−12n edges ij, where di=6anddj=6.Theorem 1.
SupposeG is a silicate network. Then,(8)GAG=18+122n2+42−6n,ABCG=314+310n2+4+14−210n,AZIG=18473+18453n2+3725+6×18373−12×18353n.Proof.
LetG be a silicate network. Then, one has(9)GAG=∑ij∈EG2di×djdi+dj=6n×23×33+3+18n2+6n×23×63+6+18n2−12n×26×66+6=6n+18n2+6n×223+18n2−12n=18+122n2+42−6n,ABCG=∑ij∈EGdi+dj−2di×dj=6n×3+3−23×3+18n2+6n×3+6−23×6+18n2−12n×6+6−26×6=6n×23+18n2+6n×146+18n2−12n×106=314+310n2+4+14−210n,AZIG=∑ij∈EGdi×djdi+dj−23=6n×3×33+3−23+18n2+6n×3×63+6−23+18n2−12n×6×66+6−23=6n×943+18n2+6n×1873+18n2−6n×1853=18473+18453n2+3725+6×18373−12×18353n.
Hexagonal networks is written asHn, which is composed of n hexagons. According to the relationship of degree series, we mainly calculate the following indices. One can refer to more research on hexagon networks [28–31].
Similarly, according to the degree distribution of the Hexagonal networks vertices, there are five sets of vertex division based on valencies, asA1,A2,A3,A4,andA5. The set A1 consists of 12 edges ij, where di=3anddj=4. The set A2 consists of 6 edges ij, where di=3sdj=6. The set A3 consists of 6n−18 edges ij, where di=4anddj=4. The set A4 consists of 12n−24 edges ij, where di=4anddj=6. The set A5 consists of 9n2−33n+30 edges ij, where di=6anddj=6.Theorem 2.
SupposeG is a hexagonal network. Then,(10)GAG=9n2+2465−27n+12+42+483553−76,ABCG=3102n2+326+43−11102n+215+14−962−83+510,AZIG=9×18353n2+2×839+324−33×18353n+12453+6×18373−2×833+6×18325−648.Proof.
Only consider thatHn is an n-dimensional hexagonal networks. So, VHn=3n2−3n+1,EHn=9n2−15n+6. Thus,(11)GAG=∑ij∈EG2di×djdi+dj=12×2×3×43+4+6×2×3×63+6+6n−18×2×4×44+4+12n−24×2×4×64+6+9n2−33n+30×2×6×66+6=12×437+6×2×329+6n−18×2×48+12n−24×4610+9n2−33n+30×2×6×66+6=9n2+2465−27n+12+42+483553−76,ABCG=∑ij∈EGdi+dj−2di×dj=12×3+4−23×4+6×3+6−23×6+6n−18×4+4−24×4+12n−24×4+6−24×6+9n2−33n+30×6+6−26×6=12×512+6×718+6n−18×616+12n−24×824+9n2−33n+30×1036=3102n2+326+43−11102n+215+14−962−83+510,AZIG=∑ij∈EGdi×djdi+dj−23=12×3×43+3−23+6×3×63+6−23+6n−18×4×44+4−23+12n−24×4×64+6−23+9n2−33n+30×6×66+6−23=12×12353+6×18373+6n−18×16363+12n−24×24383+9n2−33n+30×363103=9×18353n2+2×839+324−33×18353n+12453+6×18373−2×833+6×18325−648.
At present, we discuss about another member of the silicate networks and the chain silicate networks, which is a linear combination ofn tetrahedrons, referred to as Cn. In the same way, the edges of the silicate networks can be divided into three sets of vertex division based on valencies, as A1,A2,andA3. For n=1, the set A1 consists of 6 edges ij, where di=3anddj=3, the set A2 consists of 0 edges ij, where di=3anddj=6, and the set A3 consists of 0 edges ij, where di=6anddj=6. For n≥2, the set A1 consists of n+4 edges ij, where di=3anddj=3, the set A2 consists of 4n−2 edges ij, where di=3anddj=6, and the set A3 consists of n−2 edges ij, where di=6anddj=6.Theorem 3.
SupposeG is a chain silicate network with 6n edges and 3n+1 vertices. Then,(12)GAG=6,n=1,2+823n+2−423,n≥2,ABCG=4,n=1,4+414+106n+8−14−103,n≥2,AZIG=3725,n=1,9343+4×18373+18353n+9316−2×18373−2×18353,n≥2.Proof.
LetG be a chain silicate network. Then,(13)GAG=∑ij∈EG2di×djdi+dj,where(14)n=1,GAG=6×23×33+3=6,n≥2,GAG=n+4×23×33+3+4n−2×23×63+6+n−2×26×66+6=n+4+4n−2×223+n−2=2+823n+2−423,ABCG=∑ij∈EGdi+dj−2di×dj,where(15)n=1,ABCG=6×3+3−23×3=4,n≥2,ABCG=n+4×3+3−23×3+4n−2×3+6−23×6+n−2×6+6−26×6=23n+4+4n−2×146+n−2×106=4+414+106n+8−14−103,AZIG=∑ij∈EGdi×djdi+dj−23,where(16)n=1,AZIG=6×3×33+3−23=3725,n≥2,AZIG=n+4×3×33+3−23+4n−2×3×63+6−23+n−2×6×66+6−23=n+49343+4n−2×18373+n−2×363103=9343+4×18373+18353n+9316−2×18373−2×18353.
Oxide networks play an important role in silicate networks. When the silicon atoms in the silicate networks are removed, the oxide networks are obtained. Then-dimensional oxide networks are defined as On. By observing the edge division of oxide networks, there are two sets of vertex division based on valencies, as A1andA2. The set A1 consists of 12n edges ij, where di=2anddj=4. The set A2 consists of 18n2−12n edges ij, where di=4anddj=4.Theorem 4.
SupposeG is an oxide network with 18n2 edges and 9n2+3n vertices. Then,(17)GAG=18n2+82−12n,ABCG=962n2+62−36n,AZIG=10243n2−11849n.Proof.
LetG be an oxide network. Then, one has(18)GAG=∑ij∈EG2di×djdi+dj=12n×22×42+4+18n2−12n×24×44+4=18n2+82−12n,ABCG=∑ij∈EGdi+dj−2di×dj=12n×2+4−22×4+18n2−12n×4+4−24×4=12n×22+18n2−12n×64=962n2+62−36n,AZIG=∑ij∈EGdi×djdi+dj−23=12n×2×42+4−23+18n2−12n×4×44+4−23=12n×23+18n2−12n×8327=10243n2−11849n.
Cellular networks is mainly composed of three parts: mobile station, network subsystem, and base station subsystem, denoted byCPn. It plays an important role in computer graphics and in chemistry. Meanwhile, it also can be characterized as benzene hydrocarbons. In the same way, the edges of cellular networks can be divided into three sets based on valencies, as A1,A2,andA3. The set A1 consists of 6 edges ij, where di=2anddj=2. The set A2 consists of 12n−12 edges ij, where di=2anddj=3. The set A3 consists of 9n2−15n+6 edges ij, where di=3anddj=3.Theorem 5.
SupposeG is a cellular network with 6n2 edges and 9n2−3n vertices. Then,(19)GAG=9n2+2465−15n+12−2456,ABCG=6n2+62−10n+4−32,AZIG=729649n2−15n+6+96n−48.Proof.
LetG be a cellular network. Then, one has(20)GAG=∑ij∈EG2di×djdi+dj=6×22×22+2+12n−12×22×32+3+9n2−15n+6×23×33+3=6×2×24+12n−12×265+9n2−15n+6×2×36=9n2+2465−15n+12−2456,ABCG=∑ij∈EGdi+dj−2di×dj=6×2+2−22×2+12n−12×2+3−22×3+9n2−15n+6×3+3−23×3=32+6n−62+6n2−10n+4×2=6n2+62−10n+4−32,AZIG=∑ij∈EGdi×djdi+dj−23=6×2×22+2−23+12n−12×2×32+3−23+9n2−15n+6×3×33+3−23=6×23+12n−12×23+9n2−15n+6×943=729649n2−15n+6+96n−48.
Next, our step is to study the generalized Sierpiński networkssKp,m when its subgraph is a complete graph. By consulting [32], the edges of the sKp,m can be divided into two sets of vertex division based on valencies, as A1andA2. The set A1 consists of mm−1 edges ij, where di=manddj=m+1. The set A2 consists of mt+1−2m2+m/2 edges ij, where di=m+1anddj=m+1.Theorem 6.
SupposeG is a Sierpiński networks and its subgraph is a complete graph. Then,(21)Π2G=mmm−1⋅m+1mt+1−m2,Π1∗G=2m+1mm−1⋅2m+2mt+1−2m2+m/2,FG=mt+3+2mt+2+mt+1−3m3−m2,Za,bG=m2−m⋅mam+1b+mbm+1a+m+1a+b⋅mt+1−2m2+m.Proof.
LetG be Sierpiński networks with the seed graph being acomplete graph. Then, one has(22)Π2G=∏ij∈EGdi×dj=m⋅m+1mm−1⋅m+12mt+1−2m2+m/2=m⋅m+1mm−1⋅m+1mt+1−2m2+m=mmm−1⋅m+1mt+1−m2,Π1∗G=∏ij∈EGdi+dj=m+m+1mm−1+m+1+m+1mt+1−2m2+m/2=2m+1mm−1⋅2m+2mt+1−2m2+m/2,FG=∑ij∈EGdi2+dj2=m2+m+12⋅mm−1+m+12+m+12mt+1−2m2+m2=2m2+2m+1⋅m2−m+m2+2m+1⋅mt+1−2m2+m=mt+3+2mt+2+mt+1−3m3−m2,Za,bG=∑ij∈EGdiadjb+dibdja=mam+1b+mbm+1a⋅mm−1+mt+1−2m2+m2⋅2m+1am+1b=m2−m⋅mam+1b+mbm+1a+m+1a+b⋅mt+1−2m2+m.
Finally, we discuss the Sierpiński networkssp,m when the seed graph is a m-regular graph without triangles. Similarly, the edges of the sp,m can be divided into three sets of vertex division based on valencies, as A1,A2,andA3. The set A1 consists of pt−1m/2p−2m edges ij, where di=manddj=m. The set A2 consists of pt−1+pt−1−p/1−pm2 edges ij, where di=manddj=m+1. The set A3 consists of pm/21−pt−1/1−p+pm21−pt−2/1−p edges ij, where di=m+1anddj=m+1.Theorem 7.
SupposeG is a Sierpiński networks sp,m and its subgraph is a m-regular graph without triangles. Then,(23)Π2G=mpt−1mp−m+pt−1−p/1−pm2⋅m+1pt−1m2+m2p−pt−1+pm1−pt−1/1−p,Π1∗G=2pt−1m/2p−2m+pm1−pt−1+2m−2mpt−2/21−p⋅mpt−1m/2p−2m⋅m+1pm/21−pt−1/1−p+pm21−pt−2/1−p⋅2m+1pt−1+pt−1−p/1−pm2,FG=pt−1+pt−1−p1−pm4+pt−1p−2mm3+pt−1m2+m2p−pt−1+pm1−pt−11−pm+12,Za,bG=pt−1p−2m⋅ma+b+1+pt−1+pt−1−p1−pm2mam+1b+mbm+1a+pm1−pt−1+2m−2mpt−21−pm+1a+b.Proof.
LetG be Sierpiński networks and its subgraph be m-regular graph without triangles. Then, one has(24)Π2G=∏ij∈EGdi×dj=m⋅mpt−1m/2p−2m⋅m⋅m+1pt−1+pt−1−p/1−pm2⋅m+1⋅m+1pm/21−pt−1/1−p+pm21−pt−2/1−p=mpt−1mp−2m⋅mpt−1+pt−1−n/1−m2⋅m+1pt−1+pt−1−p/1−pm2+pm1−pt−1/1−p+2pm21−pt−2/1−p=mpt−1mp−m+pt−1−p/1−pm2⋅m+1pt−1m2+m2p−pt−1+pm1−pt−1/1−p,Π1∗G=∏ij∈EGdi+dj=m+mpt−1m/2p−2m⋅m+m+1pt−1+pt−1−p/1−pm2⋅m+1+m+1pm/21−pt−1/1−p+pm21−pt−2/1−n=2pt−1m/2p−2m+pm/21−pt−1/1−p+pm21−pt−2/1−p⋅mpt−1m/2p−2m⋅2m+1pt−1+pt−1−p/1−pm2⋅m+1pm/21−pt−1/1−p+pm21−pt−2/1−p=2pt−1m/2p−2m+pm1−pt−1+2m−2mpt−2/21−p⋅mpt−1m/2p−2m⋅m+1pm/21−pt−1/1−p+pm21−pt−2/1−p⋅2m+1pt−1+pt−1−p/1−pm2,FG=∑ij∈EGdi2+dj2=m2+m2⋅pt−1m2p−2m+pt−1+pt−1−p1−pm2m2+m+12+pm21−pt−11−p+pm21−pt−21−p⋅m+12+m+12=pt−1m3p−2m+pt−1+pt−1−p1−pm4+pt−1+pt−1−p1−pm2m+12+pm1−pt−11−p+2pm21−pt−21−p⋅m+12=pt−1+pt−1−p1−pm4+pt−1p−2mm3+pt−1m2+m2p−pt−1+pm1−pt−11−pm+12,Za,bG=∑ij∈EGdiadjb+dibdja=pt−1m2p−2m⋅mamb+mbma+pt−1+pt−1−p1−pm2⋅mam+1b+mbm+1a+pm21−pt−11−p+pm21−pt−21−p⋅2m+1a+b=pt−1p−2m⋅ma+b+1+pt−1+pt−1−p1−pm2mam+1b+mbm+1a+pm1−pt−1+2m−2mpt−21−pm+1a+b,as desired.


## 3. Conclusion

In this paper, we studied theABC index of some chemical networks and obtained AZI index, GA index, the multiplicative version of ordinary first Zagreb index, the second multiplicative Zagreb indices, and the Zagreb index. By calculating the correlation index of several specific chemical networks, we can get the above indices formulas. This also provides potential help for scholars to study networks characteristics better. For further work, if the corresponding networks are replaced by other networks, we can also calculate and get the corresponding formulas.



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