On spectra of weighted graphs of order ≤5 (original) (raw)
Linear Algebra and its Applications, 2002
This paper presents a variety of results on graph spectra. The number of main eigenvalues of a graph is shown to be equal to the rank of an associated matrix. We establish a condition for a graph to have exactly two main eigenvalues and then show how to evaluate them and their associated eigenvectors. It is shown that the main eigenvalues and corresponding eigenvectors of a graph determine those of its complement. We generalize to any eigenvalue a condition for 0 and −1 to be eigenvalues of a graph and its complement, respectively. Finally, we generalize to non-simple eigenvalues a result on the components of an eigenvector associated with a simple eigenvalue.
Spectra of families of matrices described by graphs, digraphs, and sign patterns
2006
The workshop Spectra of Families of Matrices described by Graphs, Digraphs, and Sign Patterns, held at the American Institute of Mathematics Research Conference Center on Oct. 23-27, 2006, focused on three problems: • Determination of the minimum rank, or equivalently maximum multiplicity of an eigenvalue, of real symmetric matrices described by a graph. • The 2n-conjecture for spectrally arbitrary sign patterns. • The energy of graphs.
Developments on spectral characterizations of graphs
Discrete Mathematics, 2009
In [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.
Spectra of weighted rooted graphs having prescribed subgraphs at some levels
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.
The new upper bounds on the spectral radius of weighted graphs
Applied Mathematics and Computation, 2012
We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain another upper bound which is sharp on the spectral radius of the adjacency matrix and compare with some known upper bounds with the help of some examples of graphs. We also characterize graphs for which the bound is attained.
Spectral characterization of some weighted rooted graphs with cliques
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.
On the N-spectrum of oriented graphs
Open Mathematics, 2020
Given any digraph D, its non-negative spectrum (or N-spectrum, shortly) consists of the eigenvalues of the matrix AA T , where A is the adjacency matrix of D. In this study, we relate the classical spectrum of undirected graphs to the N-spectrum of their oriented counterparts, permitting us to derive spectral bounds. Moreover, we study the spectral effects caused by certain modifications of a given digraph.
Spectral characterization of graphs whose second largest eigenvalue is less than 1
Linear Algebra and its Applications, 2011
Graphs with second largest eigenvalue λ 2 1 are extensively studied, however, whether they are determined by their adjacency spectra or not is less considered. In this paper we completely characterize all the connected bipartite graphs with λ 2 < 1 that are determined by their adjacency spectra. In addition, we prove that all the connected non-bipartite graphs with girth no less than 4 and λ 2 < 1 are determined by their adjacency spectra.
arXiv (Cornell University), 2015
In this paper we completely characterize the graphs which have an edge weighted adjacency matrix belonging to the class of n×n involutions with spectrum equal to {λ n-2 1 , λ 2 2 } for some λ 1 and some λ 2 . The connected graphs turn out to be the cographs constructed as the join of at least two unions of pairs of complete graphs, and possibly joined with one other complete graph.