Graphs and Hermitian matrices: Eigenvalue interlacing
In this note we discuss interlacing inequalities relating the eigenvalues of a partitioned Hermitian matrix and the eigenvalues of its blocks. We apply such inequalities to estimate the eigenvalues of the adjacency matrix and the Laplacian of a graph. In particular, we prove 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 > cn2, the smallest eigenvalue μ n of G satisfies μn ≤ - βn. Similarly, for every r ≥ 3, c < 5, there exists γ = γ(c, r) such that for every graph G = G(n, m) with m < cn2 and independence number α(G) < r, the second eigenvalue μ2 of G satisfies μ2 > γn for sufficiently large n. © 2004 Elsevier B.V. All rights reserved.
Bollobás, B., & Nikiforov, V. (2004). Graphs and Hermitian matrices: Eigenvalue interlacing. Discrete Mathematics, 289 (1-3), 119-127. https://doi.org/10.1016/j.disc.2004.07.011