The normalized Laplacian spectrum of n-polygon graphs and applications

Abstract

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