Spectra of weighted rooted graphs having prescribed subgraphs at some levels

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.CIDMAFCTFEDER/POCI 2010PTDC/MAT/112276/2009Fondecyt - IC Project 11090211Fondecyt Regular 110007

On the Laplacian and signless Laplacian spectra of complete multipartite graphs

Let G be a finite simple graph with vertex set V(G) = {v1, v2, v3, …, vn} and edge set E(G). The adjacency matrix of G is an (nn)-matrix A(G) = [aij] where aij = 1 if vivj E(G) and aij = 0 elsewhere, and the degree matrix of G is a diagonal (nn)-matrix D(G) = [dij] where dii = degG(vi) and dij = 0 for i ≠ j. The Laplacian matrix of G is L(G) = D(G) – A(G) and the signless Laplacian matrix of G is Q(G) = D(G) + A(G). The study of spectrum of Laplacian and signless Laplacian matrix of graph are interesting topic till today. In this paper, we determine the Laplacian and signless Laplacian spectra of complete multipartite graphs

Computing the Laplacian spectra of some graphs

In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of a regular graph or a path.Center for Research and Development in Mathematics and ApplicationsFCTFEDER/POCI 2010Fondecyt-IC Project 11090211CNPq—Grants 309531/2009-8CNPq—Grants 473815/2010-