Spectra of weighted rooted graphs having prescribed subgraphs at some levels

Abstract

Let B be a weighted generalized Bethe tree of k levels (k > 1) in which nj is the number of vertices at the level k-j+1 (1 ≤ j ≤ k). Let Δ \subset {1, 2,., k-1} and F={Gj:j \in Δ}, where Gj is a prescribed weighted graph on each set of children of B at the level k-j+1. In this paper, the eigenvalues of a block symmetric tridiagonal matrix of order n1+n2 +...+nk are characterized as the eigenvalues of symmetric tridiagonal matrices of order j, 1≤j≤k, easily constructed from the degrees of the vertices, the weights of the edges, and the eigenvalues of the matrices associated to the family of graphs F. These results are applied to characterize the eigenvalues of the Laplacian matrix, including their multiplicities, of the graph β(F) obtained from β and all the graphs in F={Gj:j \in Δ}; and also of the signless Laplacian and adjacency matrices whenever the graphs of the family F are regular.