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

Abstract

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 ${\mathbb{N}}_n$ (resp., ${\mathbb{N}}_n^{\prime }$). 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 ${\mathbb{N}}_n$. As an application of the Laplacian spectra, we reduce the number of spanning trees and the global mean-first passage time for ${\mathbb{N}}_n$. Furthermore, we show that the Kirchhoff index of ${\mathbb{N}}_n$ is approximately to $\frac{1}{3}$ of its Wiener index. In view of our obtained results, all the corresponding results are considered for ${\mathbb{N}}_n^{\prime }$.

Keywords:

• Rounded networks with 2n-pentagons
• Kirchhoff index
• spanning tree
• global mean-first passage time

2010 AMS Subject Classifications:

• 05C35
• 05C12
• 05C50

Acknowledgments

The author would like to express his sincere gratitude to the referees for their insightful comments and valuable suggestions, which led to a number of improvements in this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).