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 of and . In this work, using the definition of the index, index, 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.
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, index, index, and 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 -dimensional silicate networks (), chain silicate networks (), hexagonal networks (), -dimensional honeycomb networks (), cellular networks (), and Sierpiński networks (). 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.
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All graphs and networks are limited to simple undirected graphs. Let represent the vertex set and edge set of the networks, respectively. The degree of vertex is the number of edges associated with , expressed by . 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 index is written asand posted by Estrada et al. [9]. The geometric-arithmetic index is the valency-based topological index denoted bywhich 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 index is not introduced and readers can read the literature [11]. In 2010, Furtula et al. proposed index [12], named as the augmented Zagred index, denoted by
The ordinary first Zagreb index of the multiplication version is represented as
For the chemical properties and applications of this index, readers can refer to [13–19]. The 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 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 index for short, and defined it as
About its related research, the reader may refer to [23–26].
2. Main Results and Discussion
In this section, according to the definition of the index, index, 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]. represents a silicate network, where is the number of hexagons between the boundary and the center. Then, one has and the following results.
According to the distribution of networks vertices, there are three sets of vertex division based on valencies, as . The set consists of edges , where . The set consists of edges , where . The set consists of edges , where .
Theorem 1. Suppose is a silicate network. Then,
Proof. Let be a silicate network. Then, one hasHexagonal networks is written as , which is composed of 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, as . The set consists of 12 edges , where . The set consists of 6 edges , where . The set consists of edges , where . The set consists of edges , where . The set consists of edges , where .
Theorem 2. Suppose is a hexagonal network. Then,
Proof. Only consider that is an n-dimensional hexagonal networks. So, . Thus,At present, we discuss about another member of the silicate networks and the chain silicate networks, which is a linear combination of tetrahedrons, referred to as . In the same way, the edges of the silicate networks can be divided into three sets of vertex division based on valencies, as . For , the set consists of 6 edges , where , the set consists of 0 edges , where , and the set consists of 0 edges , where . For , the set consists of edges , where , the set consists of edges , where , and the set consists of edges , where .
Theorem 3. Suppose is a chain silicate network with edges and vertices. Then,
Proof. Let be a chain silicate network. Then,wherewherewhereOxide networks play an important role in silicate networks. When the silicon atoms in the silicate networks are removed, the oxide networks are obtained. The -dimensional oxide networks are defined as . By observing the edge division of oxide networks, there are two sets of vertex division based on valencies, as . The set consists of edges , where . The set consists of edges , where .
Theorem 4. Suppose is an oxide network with edges and vertices. Then,
Proof. Let be an oxide network. Then, one hasCellular networks is mainly composed of three parts: mobile station, network subsystem, and base station subsystem, denoted by . 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 . The set consists of 6 edges , where . The set consists of edges , where . The set consists of edges , where .
Theorem 5. Suppose is a cellular network with edges and vertices. Then,
Proof. Let be a cellular network. Then, one hasNext, our step is to study the generalized Sierpiński networks when its subgraph is a complete graph. By consulting [32], the edges of the can be divided into two sets of vertex division based on valencies, as . The set consists of edges , where . The set consists of edges , where .
Theorem 6. Suppose is a Sierpiński networks and its subgraph is a complete graph. Then,
Proof. Let be Sierpiński networks with the seed graph being acomplete graph. Then, one hasFinally, we discuss the Sierpiński networks when the seed graph is a -regular graph without triangles. Similarly, the edges of the can be divided into three sets of vertex division based on valencies, as . The set consists of edges , where . The set consists of edges , where . The set consists of edges , where
Theorem 7. Suppose is a Sierpiński networks and its subgraph is a -regular graph without triangles. Then,
Proof. Let be Sierpiński networks and its subgraph be -regular graph without triangles. Then, one hasas desired.
3. Conclusion
In this paper, we studied the index of some chemical networks and obtained index, 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.
Data Availability
The figures, tables, and other data used to support this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported in part by Anhui Provincial Natural Science Foundation under Grant 2008085J01 and Natural Science Fund of Education Department of Anhui Province under Grant KJ2020A0478.