Spectra of copies of a generalized Bethe tree attached to any graph

Abstract

A generalized Bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree. Let G be any connected graph. Let G { B } be the graph obtained from G by attaching a generalized Bethe tree B , by its root, to each vertex of G . We characterize completely the eigenvalues of the signless Laplacian, Laplacian and adjacency matrices of the graph G { B } including results on the eigenvalue multiplicities. Finally, for the Laplacian and signless Laplacian matrices, we recall a procedure to compute a tight upper bound on the algebraic connectivity of G { B } as well as on the smallest eigenvalue of the signless Laplacian matrix of G { B } whenever G is a non-bipartite graph.