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.

• Let G be a graph. For subset X of V ( ), the switching σ is signed graph obtained from by reversing signs all edges between and ) ∖ . A( adjacency matrix An eigenvalue called main if it has an eigenvector sum whose entries not equal to zero. Two equivalent graphs share same spectrum, while they may have different eigenvalues. Akbari et al. (2021) conjectured that let ≠ K 2 , 4 { e } Then ther...

Abstract. This mostly expository note surveys and recovers a lower bound for the number of distinct eigenvalues of real symmetric matrices associated with a graph. The relation is established with the length of some paths of the underlying graph, using an improvement of an inequality involving the multiplicities of the eigenvalues. An interesting use of that number is observed. Some application...

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconne...

The energy of a graph G, often denoted E(G), is defined to be the sum of the absolute value of the eigenvalues of its adjacency matrix. This graph invariant is very closely connected to a chemical quantity known as the total π-electron energy of conjugated hydrocarbon molecules. Very recently graph energy has become a quantity of interest to mathematicians, and several variations have been intr...

This volume continues a series of monographs in algebraic graph theory that specialize to spectral graph theory — the study of interconnections between the properties of a graph and the eigen-values and eigenvectors of the associated adja-cency matrix. The common thread between the two previous works, Spectra of Graphs — Theory and Applications [3] and Recent Results in the Theory of Graph Spec...

4.1 Eigenvalues and Optimization I cannot believe that I have managed to teach three lectures on spectral graph theory without giving the characterization of eigenvalues as solutions to optimization problems. It is one of the most useful ways of understanding eigenvalues of symmetric matrices. To begin, let A be a symmetric matrix with eigenvalues α1 ≥ α2 ≥ · · · ≥ αn, and corresponding orthono...

Given a graph G, let G be an oriented graph of G with the orientation σ and skewadjacency matrix S(G). Then the spectrum of S(G) consisting of all the eigenvalues of S(G) is called the skew-spectrum of G, denoted by Sp(G). The skew energy of the oriented graph G, denoted by ES(G), is defined as the sum of the norms of all the eigenvalues of S(G). In this paper, we give orientations of the Krone...

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...

The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c...