On the spectra of some graphs like weighted rooted trees

Electronic Journal of Linear Algebra, 2011

Let B be a weighted generalized Bethe tree of k levels (k > 1) in which n j is the number of vertices at the level k − j + 1 (1 ≤ j ≤ k). Let ∆ ⊆ {1, 2, . . . , k − 1} and F = {G j : j ∈ ∆}, where G j 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 n 1 + n 2 + · · · + n k 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 B (F) obtained from B and all the graphs in F = {G j : j ∈ ∆} ; and also of the signless Laplacian and adjacency matrices whenever the graphs of the family F are regular.

Linear Algebra and its Applications, 2005

Let T be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let d k−j +1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of T. They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for both matrices are d j − 1, 2 j k − 1, and √ d k , while the diagonal entries are zeros, in the case of the adjacency matrix, and d j , 1 j k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues.

Linear Algebra and Its Applications - LINEAR ALGEBRA APPL, 2010

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.

Discrete Mathematics, 2005

Let G be a simple graph, its Laplacian matrix is the difference of the diagonal matrix of its degrees and its adjacency matrix. Denote its eigenvalues by (G) = 1 (G) 2 (G) · · · n (G) = 0. A vertex of degree one is called a pendant vertex. Let T n,k be a tree with n vertices, which is obtained by adding paths P 1 , P 2 , . . . , P k of almost equal the number of its vertices to the pendant vertices of the star K 1,k . In this paper, the following results are given:

Linear Algebra and its Applications, 2007

Let T be a weighted rooted tree of k levels such that (1) the vertices in level j have a degree equal to d k−j +1 for j = 1, 2, . . . , k, and (2) the edges joining the vertices in level j with the vertices in level (j + 1) have a weight equal to w k−j for j = 1, 2, . . . , k − 1.

Linear Algebra and its Applications, 2008

Let H be a simple graph with n vertices and G be a sequence of n rooted graphs G

Linear Algebra and Its Applications, 2003

Let L(B k ) be the Laplacian matrix of an unweighted balanced binary tree B k of k levels.

2011

This paper contains results in the field of algebraic graph theory and specifically concerns the spectral radius of the Laplacian matrix of a tree. Let A(G) denote the adjacency matrix of a simple graph G. Then, the Laplacian matrix of G is given by L(G) = D(G) − A(G) where D is the diagonal matrix whose diagonal entries are the vertex degrees. The main result provides an upper bound for the spectral radius of any tree with n vertices and k pendant vertices.

Linear Algebra and Its Applications, 2011

Given an n-vertex graph G = (V, E), the Laplacian spectrum of G is the set of eigenvalues of the Laplacian matrix L = D − A, where D and A denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. In this paper, we study the Laplacian spectrum of trees. More precisely, we find a new upper bound on the sum of the k largest Laplacian eigenvalues of every n-vertex tree, where k ∈ {1, . . . , n}. This result is used to establish that the n-vertex star has the highest Laplacian energy over all n-vertex trees, which answers affirmatively to a question raised by Radenković and Gutman .

Computational and Applied Mathematics

An eigenvalue of the adjacency matrix of a graph is said to be main if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph G to have -1λ min as an eigenvalue of its complement, where λ min denotes the least eigenvalue of G. Also, we prove that among connected bipartite graphs, K r,r is the unique graph for which the index of the complement is equal to -1λ min . Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.