Spectral characterization of some weighted rooted graphs with cliques

Abstract

The level of a vertex in a rooted graph is one more than its distance from the root vertex. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree. We characterize completely the eigenvalues of the Laplacian, signless Laplacian and adjacency matrices of a weighted rooted graph G obtained from a weighted generalized Bethe tree of k levels and weighted cliques in which (1) the edges connecting vertices at consecutive levels have the same weight, (2) each set of children, in one or more levels, defines a weighted clique, and (3) cliques at the same level are isomorphic. These eigenvalues are the eigenvalues of symmetric tridiagonal matrices of order j × j , 1 ⩽ j ⩽ k . Moreover, we give results on the multiplicity of the eigenvalues, on the spectral radii and on the algebraic conectivity. Finally, we apply the results to the unweighted case and some particular graphs are studied.