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

Abstract

Given an arbitrary connected $G$, the $n$-polygon graph ${\tau }_n(G)$ is obtained by adding a path with length $n$ $(n\ge 2)$ to each edge of graph $G$, and the iterated $n$-polygon graphs ${\tau }_n^g(G)$ ($g\ge 0$), 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 ${\tau }_n^g(G)$ ($g\ge 0$) 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.

arXiv comment44pages, 1 figure

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