New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix

The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the {\sc Hoffman} lower bound. Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We show that the conjectured lower bound is tight for several families of graphs. We also performed various searches for a counter-example, but none was found. Our proofs rely on a new technique of converting the adjacency matrix into the zero matrix by conjugating with unitary matrices and use majorization of spectra of self-adjoint matrices. We also show that the above bounds are actually lower bounds on the normalized orthogonal rank of a graph, which is always less than or equal to the chromatic number. The normalized orthogonal rank is the minimum dimension making it possible to assign vectors with entries of modulus one to the vertices such that two such vectors are orthogonal if the corresponding vertices are connected. All these bounds are also valid when we replace the adjacency matrix A by W * A where W is an arbitrary self-adjoint matrix and * denotes the Schur product, that is, entrywise product of W and A

Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number

In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].Comment: 15 pages 1 figure; linear algebra and its applications 201

New measures of graph irregularity

In this paper, we define and compare four new measures of graph irregularity. We use these measures to prove upper bounds for the chromatic number and the Colin de Verdiere parameter. We also strengthen the concise Turan theorem for irregular graphs and investigate to what extent Turan's theorem can be similarly strengthened for generalized r-partite graphs. We conclude by relating these new measures to the Randic index and using the measures to devise new normalised indices of network heterogeneity