The normalized Laplacian spectrum of n-polygon graphs and applications.

Given an arbitrary connected graph G, the n-polygon graph $ \tau _n(G) $ τ n (G) is obtained by adding a path with length n $ (n\geq ~2) $ (n ≥ 2) to each edge of graph G, and the iterated n-polygon graphs $ \tau _n^g(G) $ τ n g (G) ( $ g\geq ~0 $ g ≥ 0) are obtained from the iteration $ \tau _n^g(G)=\tau _n(\tau _n^{g-1}(G)) $ τ n g (G) = τ n (τ n g − 1 (G)) , with the initial condition $ \tau _n^0(G)=G $ τ n 0 (G) = G. In this paper, a method for calculating the eigenvalues of the normalized Laplacian matrix for graph $ \tau _n(G) $ 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 $ \tau _n(G) $ τ n (G) and graphs $ \tau _n^g(G) $ τ n g (G) ( $ g\geq ~0 $ 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 $ \tau _n(G) $ τ n (G) and graphs $ \tau _n^g(G) $ τ n g (G) by exploring their connections with the normalized Laplacian spectrum, and obtain exact results for these quantities. [ABSTRACT FROM AUTHOR]