Spectra of copies of Bethe trees attached to path and applications

María,Robbiano, I.,Gutman, R.,Jiménez

The Bethe tree $B_{d,k}$ 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 adacency 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.