Spectra of weighted generalized Bethe trees joined at the root

A generalized Bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree. Let {Bi:1⩽i⩽m} be a set of trees such that, for i=1,2,…,m,(1)Bi is a generalized Bethe tree of ki levels,(2)the vertices of Bi at the level j have degree di,ki-j+1 for j=1,2,…,ki, and(3)the edges of Bi joining the vertices at the level j with the vertices at the level (j+1) have weight wi,ki-j for j=1,2,…,ki-1.Let v{Bi:1⩽i⩽m} be the tree obtained from the union of the trees Bi joined at their respective root vertices. We give a complete characterization of the eigenvalues of the Laplacian and adjacency matrices of v{Bi:1⩽i⩽m}. Moreover, we derive results concerning their multiplicities. In particular, we characterize the spectral radii, the algebraic conectivity and the second largest Laplacian eigenvalue.