  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.
