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.

Let L 8,4 n represent a linear octagonal-quadrilateral network, consisting of n eightmember rings and n four-member rings. Such a graph contains a unique pair of opposite edges. The Möbius graph Q n (8, 4) is constructed by reverse identifying these opposite edges, whereas the cylinder graph Q n 0 (8, 4) identifies the opposite edges in the natural manner. In this paper, the explicit formulas for the Kirchhoff index and complexity of Q n (8, 4) and Q n 0 (8, 4) are deduced from Laplacian characteristic polynomials using to decomposition theorem and Vieta's theorem. A consequence is the surprising fact that the Kirchhoff index of Q n (8, 4) (resp. Q n 0 (8, 4)) is approximately a third (resp. half) of its Wiener index as n ! ∞.