An Inertial Lower Bound for the Chromatic Number of a Graph

Abstract

It is well known that n/(n − µ), where µ is the spectral radius of a graph with n vertices, is a lower bound for the clique number. We conjecture that µ can be replaced in this bound with √ s + , where s + is the sum of the squares of the positive eigenvalues. We prove this conjecture for various classes of graphs, including triangle-free graphs, and for almost all graphs.

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

“…Finally, we use the facts that the rank of a sum or is at most the sum of the ranks of the summands, and the rank of a product is at most the minimum of the ranks of the factors. This, together with Lemma 2 in [17] (that is, if X, Y ∈ C n×n are positive semidefinite and X Y , then rank( X) ≥ rank(Y )), yields the desired inequality…”

“…1], but using p k (A) instead of A. The proof in [17] relies on the fact that there exist χ − 1 unitary matrices U i such that:…”

Estimating clique size via discarding subgraphs

“…The values of the ϑ function can be computed in polynomial time [3,4], but the degree of the polynomial bound of the running time is high. The bounds on the clique number we have listed so far are the following [8]:…”

Solving Large Maximum Clique Problems on a Quantum Annealer

“…Likewise, we consider the following two lower bounds: It should be noted that there are many other easily computable upper and lower bounds for maximum clique, maximum independent set, and chromatic number that generally apply to any type of graph (Soto et al, 2011;Elphick and Wocjan, 2018). However, in experiments we conducted (not reported in this article) those generally applicable bounds proved to be quite conservative, with the exception of the bound of (Budinich, 2003, eq. (4)) which we employ for high densities.…”