The normalized Laplacian spectrum of <i>n</i>-polygon graphs and applications

Given an arbitrary connected graph G, the n-polygon τn(G) is obtained by adding a path with length n (n≥2) to each edge of and iterated graphs τng(G) (g≥0) are from iteration τng(G)=τn(τng−1(G)), initial condition τn0(G)=G. In this paper, method for calculating eigenvalues normalized Laplacian matrix presented if G first given. The spectra can also then be derived. Finally, as applications, we calculate multiplicative degree-Kirchhoff index, Kemeny's constant, number spanning trees exploring their connections spectrum, obtain exact results these quantities.