Abstract
Tetrahedral network is considered as an effective tool to create the finite element network model of simulation, and many research studies have been investigated. The aim of this paper is to calculate several topological indices of the linear and circle tetrahedral networks. Firstly, the resistance distances of the linear tetrahedral network under different classifications have been calculated. Secondly, according to the above results, two kinds of degree-Kirchhoff indices of the linear tetrahedral network have been achieved. Finally, the exact expressions of Kemeny’s constant, Randic index, and Zagreb index of the linear tetrahedral network have been deduced. By using the same method, the topological indices of circle tetrahedral network have also been obtained.
1. Introduction
In actual life, many problems can be described using graph models. The graph model as a tool to describe network has been widely studied. Each network can be considered as graph. The problems of the vertices in the graph correspond to the points in the network, and the edges in the graph correspond to the network connection relationship between the points. In this paper, only simple, undirected, and connected graphs are considered. Suppose is a graph, and it satisfies and . The degree of vertex in the graph is denoted by . Connecting the two vertices , the distance [1] is defined as the length of the shortest path. And the resistance distance between vertex and vertex is delimited as the effective resistance, which is denoted as [2]. For more terminologies, one can refer to reference [3].
The sum of the resistance distance between each pair of vertices in the graph is defined as the Kirchhoff index [2], as follows:
Similarly, Chen and Zhang [4] put forward the following definition of the multiplicative degree-Kirchhoff index, that is, as follows:
Subsequently, Gutman et al. [5] proposed the following definition of the additive degree-Kirchhoff index, that is, as follows:
For a random walk [6, 7] in a network, the expectation of the average first arrival time [8, 9] from a vertex to another vertex selected according to the stable distribution of Markov process [10–13] is called Kemeny’s constant of the network. Kemeny’s constant is given bywhere is the number of edges in the graph .
In previous studies, several topological indices based on vertex-degree have been applied in research. The following three topological indices (Kemeny’s constant, Randic index, and Zagreb index) are the most widely used:
By the definition of the multiplicative (the additive) degree-Kirchhoff index, the main job is to calculate . From the perspective of practical application, the resistance distance considers all the paths between any two vertices, not just the short path, so the resistance distance can reflect the relationship between any two vertices better than the distance. This paper applies the expressions of the resistance distance between any two vertices of the linear and circle tetrahedral networks to derive the multiplicative (the additive) degree-Kirchhoff index of them, respectively. This kind of linear tetrahedral network is a one-dimensional infinitely extended network which is linked by a series of tetrahedrons. Its structure is morphologically manifested as elongation in one direction (see Figure 1), and is the number of the tetrahedron in the network. The structure of this kind of circle tetrahedral network is a combination of tetrahedrons and octahedrons whose form is a two-dimensional wireless extension. And the octahedrons are connected by common edge (see Figure 2).
The structure of this paper is shown as follows: we introduce several fundamental definitions of the electrical network and give two important lemmas in Section 2. We present some proofs of our main results, namely, the multiplicative (the additive) degree-Kirchhoff index, Kemeny’s constant [14–16], Randic index [17–20], and Zagreb index [21, 22] of the linear and circle tetrahedral networks in Section 3. We summarize this article in Section 4.
2. Preliminaries
In the following section, we will give two important lemmas that will make a tremendous effect on our conclusions.
Lemma 1 (see [23]). Suppose the distance between vertex and vertex is and .
Lemma 2 (see [23]). Suppose the distance between vertex and vertex is and .(1) when and (2) when , , and (3) when and (4) when and
3. Main Results
In this section, the main purpose is to derive the multiplicative (the additive) degree-Kirchhoff index, Kemeny’s constant, Randic index, and Zagreb index of the linear (the circle) network (see Figures 3 and 4).
3.1. The Linear Tetrahedral Network
3.1.1. The Additive Degree-Kirchhoff Index
Theorem 1. Let be a linear tetrahedral network.
Proof. The linear tetrahedral network is shown in Figure 3. The number of vertices in is , and the number of edges in is . When the distance between any two vertices is 1, the number of pairs of degree three and degree three is , the number of pairs of degree three and degree six is , and the number of pairs of degree six and degree six is . Besides, the maximum distance between the vertices of degree three and degree three in is . The maximum distance between the vertices of degree three and degree six in is . The maximum distance between the vertices of degree six and degree six in is . When the distance between any two vertices is , there are pairs of degree three and degree three and pairs of degree three and degree six, where . And when the distance between any two vertices is , the number of pairs of degree six and degree six is , where . Specially, when , there are 9 pairs of vertices between degree three and degree three. The above contents are shown in Table 1.
By using equation (3) and Lemma 1, we calculate the following result:Then, we obtain our desired consequence.
3.1.2. The Multiplicative Degree-Kirchhoff Index
As shown above, we can find the multiplicative degree-Kirchhoff index of the linear network.
Theorem 2. . Let be a linear tetrahedral network.
Proof. By using equation (2) and Lemma 1, we calculate the following result:
3.1.3. Kemeny’s Constant, Randic Index, and Zagreb Index
The relationship between vertices and edges of based on degree is shown in Tables 2 and 3.
By utilizing equations (4)–(7) and Tables 2 and 3, the following three indices can be achieved easily:
3.2. The Circle Tetrahedral Network
3.2.1. The Additive Degree-Kirchhoff Index
Theorem 3. Let be a circle tetrahedral network.
Proof. The labels for all vertices are shown in Figure 2. In order to calculate conveniently, allow if and if . Set and as different vertices in . Results can be divided into the following six cases: Case 1. for even . When , the number of vertices of degree three in is . For every , if , then must be a member of the set , so when the distance between any two vertices is , there are pairs of them. For every , if , then must be a member of the set and thus when distance between any two vertices is , there are pairs. Then, through Lemma 2, we gain Case 2. for odd . When , the number of vertices of degree three in is . For each , if , then , so when , the number of the pairs of the vertex and is . Through Lemma 2, we gain Case 3. for even . For each , if , then , and thus when , the number of the pairs of the vertex and is . For every , if , then , and thus when , there are pairs. Then, by using Lemma 2, we infer Case 4. for odd . When , if , then , and thus when , there are pairs. Then, by using Lemma 2, we deduce Case 5. for even . For each , if , then , and thus when , there are pairs. Then, by using Lemma 2, we gain Case 6. for odd .For every , if , then , so when , the number of the pairs of the vertex and is . For each , if , then must be , and thus when , there are pairs. Then, by using Lemma 2, we attainIf is even, by applying equations (15), (17), and (19) and Lemma 2, thenIf is odd, by applying equations (16), (18), and (20) and Lemma 2, thenConsequently, the proof is completed.
3.2.2. The Multiplicative Degree-Kirchhoff Index
As shown above, the multiplicative degree-Kirchhoff index of the circle network can be derived by classification.
Theorem 4. Let be a circle tetrahedral network.
Proof. The results in six cases are given as follows: Case 1. for even . Case 2. for odd . Case 3. for even . Case 4. for odd . Case 5. for even . Case 6. for odd .If is even, through equations (24), (26), and (28) and Lemma 2, thenIf is odd, through equations (25), (27), and (29) and Lemma 2, thenThis completes the proof.
3.2.3. Kemeny’s Constant, Randic Index, and Zagreb Index
The relationship between vertices and edges of based on degree is shown in Tables 4 and 5.
By utilizing equations (4)–(7) and Tables 4 and 5, the following three indices can be obtained easily:
If is even, then
If is odd, then
And the comparisons of the five topological indices obtained above are shown in Figures 5 and 6.
According to Figures 5 and 6, it is clear that the multiplicative degree-Kirchhoff index of both the linear and circle graph is the fast growing, and the additive degree-Kirchhoff index is the second growing. Other than that, in the circle graph, whenever is odd or even, the Zagreb index is growing faster than Randic index and Kemeny’s constant. And in the linear graph, the Zagreb index is also growing faster than Kemeny’s constant and Randic index.
4. Conclusion
In this paper, two types of tetrahedral networks have been discussed and the resistance distance in different cases has been computed, respectively. Above all, the multiplicative (the additive) degree-Kirchhoff index of the networks has been calculated. Then, Kemeny’s constant, Randic index, and Zagreb index have also been derived. Furthermore, the comparisons of the topological indices for the linear and circle networks have been studied. In future, we will devote ourselves to research more properties for types of stereochemical networks.
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.