In this paper, all connected graphs with the fourth largest Laplacian eigenvalue less than two are determined, which are used to characterize all connected graphs with exactly three Laplacian eigenvalues no less than two. Moreover, we determine bipartite graphs such that the adjacency matrices of their line graphs have exactly three nonnegative eigenvalues. © 2003 Elsevier Ltd. All rights reser...

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

We revisit Hoffman relation involving chromatic number χ and eigenvalues. We construct some graphs and weighted graphs such that the largest and smallest eigenvalues λ dan μ satisfy λ = (1 − χ)μ. We study in particular the eigenvalues of the integer simplex T 2 m, a 3-chromatic graph on ( m+2 2 ) vertices.

We determine connected bipartite regular graphs with four distinct adjacency eigenvalues that induce periodic Grover walks, and show it is only C6. also there are three kinds of the second largest five eigenvalues. Using walk-regularity, we enumerate feasible spectra for such graphs.

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 a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained. 2011 Elsevier Inc. All rights reserved.

We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all rectagraphs (triangle-free ( 0 , 2 ) $(0,2)$ -graphs) vertex degree at most 6 precisely two ± λ $\pm \lambda $ . Next, we consider what extent induced subgraphs graph are determined by their spectra. Lastly, classify spectrum three and give ...