Applications of recurrence relations for the characteristic polynomials of Bethe trees

A Bethe tree of k levels, Bk.d/, is a rooted tree such that the root vertex has degree d, the vertices from level 2 to k 􀀀 1 have degree d C 1 and the vertices at level k are leaves. In this paper, we obtain a recurrence relation for the characteristic polynomial and the Laplacian characteristic polynomial of Bethe trees. As an application, we prove that there are no integral Bethe trees except for the star B2.n2/, and we obtain a recurrence relation for the Laplacian-energy-like invariant of Bethe trees.