The normalized Laplacian spectrum of $n$-polygon graphs and its applications

TL;DR

This paper develops a method to compute the normalized Laplacian spectrum of n-polygon graphs and their iterates, enabling precise calculation of various graph invariants like spanning trees and Kemeny's constant.

Contribution

It introduces a novel approach to determine the normalized Laplacian eigenvalues for n-polygon graphs and their iterates, facilitating exact calculations of related graph invariants.

Findings

Eigenvalues of normalized Laplacian for n-polygon graphs are explicitly derived.

Exact formulas for the multiplicative degree-Kirchhoff index, Kemeny's constant, and spanning trees are obtained.

The method applies to iterated n-polygon graphs, extending spectral analysis to complex graph structures.

Abstract

Given an arbitrary connected $G$, the $n$-polygon graph $\tau_n(G)$ is obtained by adding a path with length $n$ $(n\geq 2)$ to each edge of graph $G$, and the iterated $n$-polygon graphs $\tau_n^g(G)$ ($g\geq 0$), is obtained from the iteration $\tau_n^g(G)=\tau_n(\tau_n^{g-1}(G))$, with initial condition $\tau_n^0(G)=G$. In this paper, a method for calculating the eigenvalues of normalized Laplacian matrix for graph $\tau_n(G)$ is presented if the eigenvalues of normalized Laplacian matrix for graph $G$ is given firstly. Then, the normalized Laplacian spectrums for the graph $\tau_n(G)$ and the graphs $\tau_n^g(G)$ ($g\geq 0$) can also be derived. Finally, as applications, we calculate the multiplicative degree-Kirchhoff index, Kemeny's constant and the number of spanning trees for the graph $\tau_n(G)$ and the graphs $\tau_n^g(G)$ by exploring their connections with the normalized Laplacian spectrum, exact results for these quantities are obtained.