Upper and lower bounds are obtained for the spread λ1 − λn of the eigenvalues λ1 λ2 · · · λn of the adjacency matrix of a simple graph. © 2001 Elsevier Science Inc. All rights reserved. AMS classification: 05C50; 15A42; 15A36

a concept related to the spectrum of a graph is that of energy. the energy e(g) of a graph g is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of g . the laplacian energy of a graph g is equal to the sum of distances of the laplacian eigenvalues of g and the average degree d(g) of g. in this paper we introduce the concept of laplacian energy of fuzzy graphs. ...

A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 2...

A graph is said to be $I$-eigenvalue free if it has no eigenvalues in the interval $I$ with respect adjacency matrix $A$. In this paper we present twoalgorithms for generating threshold graphs.

The kernel polynomial method (KPM) is a standard tool in condensed matter physics to estimate the density of states for a quantum system. We use the KPM to instead estimate the eigenvalue densities of the normalized adjacency matrices of “natural” graphs. Because natural graph spectra often include high-multiplicity eigenvalues corresponding to certain motifs in the graph, we introduce a pre-pr...

We examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs. We find that when the degree sequence of the graph slowly increases to infinity with the number of vertices, the empirical spectral distribution converges to the semicircle law. Moreover, we prove concentration estimates on the number of eigenvalues over progressi...

Mathematical properties of extreme eigenfunctions of popular geographic weights matrices used in spatial statistics are explored, and applications of these properties are presented. Three theorems are proposed and proved. These theorems pertain to the popular binary geographic weights matrix––an adjacency matrix––based upon a planar graph. They uncover relationships between the determinant of t...

The D-eigenvalues {μ1, μ2, . . . , μn} of a graph G are the eigenvalues of its distance matrix D and form the D-spectrum of G denoted by specD(G) . The D-energy ED(G) of the graph G is the sum of the absolute values of its D-eigenvalues. We describe here the distance spectrum of some self-complementary graphs in the terms of their adjacency spectrum. These results are used to show that there ex...

If the adjacency matrix contains degenerate eigenvalues, we must modify the approach using non-degenerate eigenvalues. We denote the eigenvalues as λki, where the index k runs over different eigenvalues and the index i runs over M associated eigenvectors of the same eigenvalue. Note that there is no unique way of choosing a basis for the eigenvectors of the unperturbed network since any linear ...