Chromatic number and the spectral radius

Let G be a graph, χ be its chromatic number, λ be the largest eigenvalue of its Laplacian, and µ be the largest eigenvalue of its adjacency matrix. Then, complementing a well-known result of Hoffman, we show that λ ≥ χ χ − 1 µ with equality holding for regular complete χ-partite graphs. We denote the eigenvalues of a Hermitian matrix A as µ (A) = µ 1 (A) ≥ · · · ≥ µ min (A). Given a graph G, we write A (G) for its adjacency matrix, D (G) for the diagonal matrix of its degree sequence, and set L (G) = D (G) − A (G). Letting χ (G) be the chromatic number of a graph G, we prove that µ (L (G)) ≥ χ (G) χ (G) − 1 µ (A (G)) , (1) closely mimicking the well-known inequality of Hoffman [1] − µ min (A (G)) ≥ 1 χ (G) − 1 µ (A (G)) (2) We deduce inequalities (1) and (2) from a matrix theorem of its own interest. Theorem 1 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 A, µ (B − A) ≥ µ B + 1 r − 1 A .