The normalized Laplacian polynomial of rooted product of graphs

Let H be a simple graph with a vertices and G be a sequence of n rooted graphs G(1), G(2), ..., G(n). The rooted product H(G), of H by G is constructed by identifying the root vertex of G(i) with the ith vertex of H. In this paper, the characteristic polynomial of the normalized Laplacian matrix of H(G) is obtained. As an application of our results, we obtain the normalized Laplacian polynomial and spectrum of the generalized Bethe trees.