Several Topological Indices of Two Kinds of Tetrahedral Networks

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.


Introduction
In actual life, many problems can be described using graph models. e graph model as a tool to describe network has been widely studied. Each network can be considered as graph. e 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 G � (V(G), E(G)) is a graph, and it satisfies |V(G)| � n and |E(G)| � m. e degree of vertex p in the graph G is denoted by d p . Connecting the two vertices p, q ∈ V(G), the distance d G (p, q) [1] is defined as the length of the shortest path. And the resistance distance between vertex p and vertex q is delimited as the effective resistance, which is denoted as r G (p, q) [2]. For more terminologies, one can refer to reference [3]. e sum of the resistance distance between each pair of vertices in the graph G 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: Kf For a random walk [6,7] in a network, the expectation of the average first arrival time [8,9] from a vertex p to another vertex q selected according to the stable distribution of Markov process [10][11][12][13] is called Kemeny's constant of the network. Kemeny's constant is given by where a is the number of edges in the graph G.
In previous studies, several topological indices based on vertex-degree have been applied in research. e 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 r G (p, q). 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. is 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. is 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 n is the number of the tetrahedron in the network. e 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). e 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][15][16], Randic index [17][18][19][20], and Zagreb index [21,22] of the linear and circle tetrahedral networks in Section 3. We summarize this article in Section 4.

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 p and vertex q is t and p, q ∈ L(n), (n ≥ 3).
Lemma 2 (see [23]). Suppose the distance between vertex p and vertex q is t and p, q ∈ C(n), (n ≥ 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).

e Linear Tetrahedral Network
Proof. e linear tetrahedral network L(n) is shown in Figure 3. e number of vertices in L(n) is 3n + 1, and the number of edges in L(n) is 6n. When the distance between any two vertices is 1, the number of pairs of degree three and degree three is n + 4, the number of pairs of degree three and degree six is 4n − 2, and the number of pairs of degree six and degree six is n − 2. Besides, the maximum distance between the vertices of degree three and degree three in L(n) is n. e maximum distance between the vertices of degree three and degree six in L(n) is n − 1. e maximum distance between the vertices of degree six and degree six in L(n) is n − 2. When the distance between any two vertices is t, there are 4[n − (t + 1)] + 12 pairs of degree three and degree three and 4[n − (t + 1)] + 6 pairs of degree three and degree six, where t � 2, 3, . . . , n − 1. And when the distance between any two vertices is t, the number of pairs of degree six and degree six is n − (t + 1), where t � 2, 3, . . . , n − 2. Specially, when t � n, there are 9 pairs of vertices between degree three and degree three. e above contents are shown in Table 1. By using equation (3) and Lemma 1, we calculate the following result:   Kf * (L(n)) � 12n 3 + 18n 2 − 3n.
Proof. By using equation (2) and Lemma 1, we calculate the following result: □ Tables 2 and 3. By utilizing equations (4)-(7) and Tables 2 and 3, the following three indices can be achieved easily:
Proof. e labels for all vertices are shown in Figure 2. In order to calculate conveniently, allow p m � p n if m ≡ n(mods) and p mk � p nk if m ≡ n(mods)(s � 1, 2). Set p and q as different vertices in C(n). Results can be divided into the following six cases: r(p, q) for even n.