Chromatic number and spectral radius

Abstract

Write μ (A) = μ1 (A) ≥ ⋯ ≥ μmin (A) for the eigenvalues of a Hermitian matrix A. Our main result is:. Let A be a Hermitian matrix partitioned into r × r blocks so that all diagonal blocks are zero. Then for every real diagonal matrix B of the same size as Aμ (B - A) ≥ μ fenced(B + frac(1, r - 1) A) .Let G be a nonempty graph, χ (G) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality implies the well-known result of Hoffmanχ (G) ≥ 1 + frac(μ (A), - μmin (A)),and also,χ (G) ≥ 1 + frac(μ (A), μ (L) - μ (A)) .Equality holds in the latter inequality if and only if every two color classes of G induce a {divides} μmin (A) {divides}-regular subgraph. © 2007 Elsevier Inc. All rights reserved.