Applications of recurrence relations for the characteristic polynomials of Bethe trees

A Bethe tree of kk levels, Bk(d)Bk(d), is a rooted tree such that the root vertex has degree dd, the vertices from level 22 to k−1k−1 have degree d+1d+1 and the vertices at level kk 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)B2(n2), and we obtain a recurrence relation for the Laplacian-energy-like invariant of Bethe trees.