Spectra of weighted generalized Bethe trees joined at the root Original Research Article

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