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

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.