Spectral analysis of three invariants associated to random walks on rounded networks with 2 n -pentagons

A network is defined as an abstract structure that consists of nodes that are connected by links. In this paper, we study two types of rounded networks Nn (resp., Nn′). By using the recursive relation, we obtain all the eigenvalues and their multiplicities with regards to the associated Laplacian matrix. Meanwhile, we utilize the corresponding relationship between the roots and the coefficients of the characteristic polynomial. Based on these relationships, we obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. By the decomposition theorem of Laplacian polynomial, we obtained an explicit closed-form formula of the Kirchhoff index for Nn. As an application of the Laplacian spectra, we reduce the number of spanning trees and the global mean-first passage time for Nn. Furthermore, we show that the Kirchhoff index of Nn is approximately to 13 of its Wiener index. In view of our obtained results, all the corresponding results are considered for Nn′.