Chromatic number and spectral radius

Write (A) = 1 (A) min (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 A; (B A) B + 1 r 1 : Let G be a nonempty graph, (G) be its chromatic number, A be its adjacency matrix, and L be its Laplacian. The above inequality implies the well-known result of A.J. Ho¤man (G) 1 + (A) min (A) ; and also, (G) 1 + (A) (L) (A) : Equality holds in the latter inequality if and only if every two color classes of G induce a j min (A)j-regular subgraph. Keywords: graph Laplacian; largest eigenvalue; least eigenvalue; k-partite graph; chromatic number. AMS classi…cation: 05C50. Write (A) = 1 (A) min (A) for the eigenvalues of a Hermitian matrix A. Given a graph G; let (G) be its chromatic number, A (G) be its adjacency matrix, and D (G) be the diagonal matrix of its degree sequence; set L (G) = D (G) A (G) : Letting G be a nonempty graph with L (G) = L and A (G) = A; we prove that (G) 1 + (A) (L) (A) ; (1)