New Spectral Bounds on the Chromatic Number Encompassing all Eigenvalues of the Adjacency Matrix

Abstract

The purpose of this article is to improve existing lower bounds on the chromatic number $\chi$. Let $\mu_1,\ldots,\mu_n$ be the eigenvalues of the adjacency matrix sorted in non-increasing order.First, we prove the lower bound $\chi \ge 1 + \max_m\{\sum_{i=1}^m \mu_i / -\sum_{i=1}^m \mu_{n - i +1}\}$ for $m=1,\ldots,n-1$. 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 Hoffman lower bound.Second, we conjecture the lower bound $\chi \ge 1 + s^+ / s^-$, where $s^+$ and $s^-$ are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the bound $\chi \ge s^+/s^-$. We show that the conjectured lower bound is true 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 considering a family of conjugates of the adjacency matrix, which add to the zero matrix, 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 $A$.