Spectra of generalized Bethe trees attached to a path

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let P_m be a path of m vertices. Let {B_i:1?i?m} be a set of generalized Bethe trees. Let P_m{B_i:1?i?m} be the tree obtained from P_m and the trees B₁,B₂,…,B_m by identifying the root vertex of B_i with the i-th vertex of P_m. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of P_m{B_i:1?i?m}. In particular, we characterize their spectral radii and the algebraic conectivity. Moreover, we derive results concerning their multiplicities. Finally, we apply the results to the case B₁=B₂=…=B_m.