The Calculations of Topological Indices on Certain Networks

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 of QSAR 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. 0en, the above indices’ formulas are obtained.


Introduction
e 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][2][3][4][5]. In this paper, we mainly calculate several topological indices, which are invariants that can describe some properties of graph. e topological indices consist of three parts, namely, degreebased indices, spectrum-based indices, and distance-based indices; meanwhile, many indices based on both degree and distance are followed in [6,7]. e 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 (S n ), chain silicate networks (C n ), hexagonal networks (H n ), n-dimensional honeycomb networks (HC n ), cellular networks (CP n ), and Sierpiński networks (s(p, m)). e 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.
All graphs and networks are limited to simple undirected graphs. Let G � (V(G), E(G)) represent the vertex set and edge set of the networks, respectively. e degree of vertex i is the number of edges associated with i, expressed by d i . e 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. e atom-bond connectivity index ABC(G) is written as and posted by Estrada et al. [9]. e geometric-arithmetic index GA(G) is the valency-based topological index denoted by better than other topological indices for predicting the physical and chemical properties of some substances, and more properties of GA(G) index is not introduced and readers can read the literature [11]. In 2010, Furtula et al.
proposed AZI(G) index [12], named as the augmented Zagred index, denoted by (3) e ordinary first Zagreb index of the multiplication version is represented as     For the chemical properties and applications of this index, readers can refer to [13][14][15][16][17][18][19]. e second multiplication Zagreb index [20] is written as On the basis of the Zagreb index, Azari et al. [21] put forward their general form and defined it as In 1972, the F 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 About its related research, the reader may refer to [23][24][25][26].

Main Results and Discussion
In this section, according to the definition of the ABC 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]. S n represents a silicate network, where n is the number of hexagons between the boundary and the center. en, one has |V(S n )| � 15n 2 + 3n, | (S n )| � 36n 2 and the following results.
According to the distribution of networks vertices, there are three sets of vertex division based on valencies, as A 1 , A 2 , and A 3 . e set A 1 consists of 6n edges ij, where d i � 3 and d j � 3. e set A 2 consists of 18n 2 + 6n edges ij, Hexagonal networks is written as H n , 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][29][30][31]. Similarly Proof. Only consider that H n is an n-dimensional hexagonal networks. So, At present, we discuss about another member of the silicate networks and the chain silicate networks, which is a linear combination of n tetrahedrons, referred to as C n . In the same way, the edges of the silicate networks can be divided into three sets of vertex division based on valencies, as A 1 , A 2 , and

Journal of Mathematics
Proof. Let G be a chain silicate network. en, where where where Oxide networks play an important role in silicate networks. When the silicon atoms in the silicate networks are removed, the oxide networks are obtained.
Proof. Let G be an oxide network. en, one has Journal of Mathematics 7 Cellular networks is mainly composed of three parts: mobile station, network subsystem, and base station subsystem, denoted by CP n . 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 A 1 , A 2 , and A 3 .
Proof. Let G be a cellular network. en, one has Next, our step is to study the generalized Sierpiński networks s(K p , m) when its subgraph is a complete graph. By consulting [32], the edges of the s(K p , m) can be divided into two sets of vertex division based on valencies, as A 1 and A 2 . e set A 1 consists of m(m − 1) edges ij, where d i � m and d j � m + 1. e set A 2 consists of (m t+1 − 2m 2 + m)/2 edges ij, where d i � m + 1 and d j � m + 1.
Journal of Mathematics Theorem 6. Suppose G is a Sierpiński networks and its subgraph is a complete graph. en, Proof. Let G be Sierpiński networks with the seed graph being acomplete graph. en, one has Finally, we discuss the Sierpiński networks s(p, m) when the seed graph is a m-regular graph without triangles. Similarly, the edges of the s(p, m) can be divided into three sets of vertex division based on valencies, as A 1 , A 2 , and A 3 . e set A 1 consists of ((p t− 1 m)/2)(p − 2m) edges ij, where d i � m and d j � m. e set A 2 consists of (p t− 1 + ((p t− 1 − p)/(1 − p)))m 2 edges ij, where d i � m and d j � m + 1. e set A 3 consists of (pm/2)((1 − p t− 1 )/(1 − p)) + pm 2 ((1 − p t− 2 )/(1 − p)) edges ij, where d i � m + 1 and d j � m + 1. □ Theorem 7. Suppose G is a Sierpiński networks s(p, m) and its subgraph is a m-regular graph without triangles. en, Proof. Let G be Sierpiński networks and its subgraph be m-regular graph without triangles. en, one has