Graphs and Hermitian matrices: eigenvalue interlacing

Our first aim in this note is to prove some inequalities relating the eigenvalues of a Hermitian matrix with the eigenvalues of its principal matrices induced by a partition of the index set. One of these inequalities extends an inequality proved by Hoffman in [9]. Secondly, we apply our inequalities to estimate the eigenvalues of the adjacency matrix of a graph, and prove, in particular, that for every r ≥ 3, c > 0 there exists β = β(c, r) such that for every Kr-free graph G = G(n,m) with m > cn, the smallest eigenvalue μn of G satisfies μn ≤ −βn. Similarly for every r ≥ 3, c < 1/2 there exists γ = γ(c, r) such that for every graph G = G(n,m) with m < cn and independence number α (G) < r, the second eigenvalue μ2 of G satisfies μ2 > γn for sufficiently large n.