Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/The k-thpower of a graph G=(V,E), G^k, is the graph whose vertex set is V and in which two distinct vertices are adjacent if and only if their distance in G is at most k. This article proves various eigenvalue bounds for the independence number and chromatic number of G^k which purely depend on the spectrum of G, together with a method to optimize them. Our bounds for the k-independence number also work for its quantum counterpart, which is not known to be a computable parameter in general, thus justifying the use of integer programming to optimize them. Some of the bounds previously known in the literature follow as a corollary of our main results. Infinite families of graphs where the bounds are sharp are presented as well.The research of A. Abiad is partially supported by the FWO grant 1285921N. A. Abiad and M.A. Fiol gratefully acknowledge the support from DIAMANT. This research of M.A. Fiol has been partially supported by AGAUR from the Catalan Government under project 2017SGR1087 and by MICINN from the Spanish Government under project PGC2018-095471-B-I00. B. Nogueira acknowledges grant PRPQ/ADRC from UFMG. The authors would also like to thank Anurag Bishnoi for noticing a tight family for our bound (19).Peer ReviewedPostprint (author's final draft

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

Coloring curves that cross a fixed curve

We prove that for every integer $t\geq 1$, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most $t$ points is $\chi$-bounded. This is essentially the strongest $\chi$-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.Comment: Small corrections, improved presentatio