On the computation of characteristic polynomials and spectra of some balanced rooted trees

A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. In this paper we obtain an explicit formula for computation of the characteristic polynomials of the adjacency matrix and the Laplacian matrix of the rooted trees which is obtained from the unio‎n of some generalized Bethe trees joined at their respective root vertices.