Abstract

The Bethe tree Bd,k is the rooted tree of k levels whose root vertex has degree d, the vertices from level 2 to level k -1 have degree d+1, and the vertices at level k have degree 1. This paper gives a decomposition of the characteristic polynomial of the adjacency matrix of the tree T(d, k, r), obtained by attaching copies of B(d, k) to the vertices of the r-vertex path. Moreover, lower and upper bounds for the energy of T(d, k, r) are obtained.