Eigenvalues and extremal degrees of graphs
Abstract
Let G be a graph with n vertices, μ1 (G) ≥ ⋯ ≥ μn (G) be the eigenvalues of its adjacency matrix, and 0 = λ1 (G) ≤ ⋯ ≤ λn (G) be the eigenvalues of its Laplacian. We show thatδ (G) ≤ μk (G) + λk (G) ≤ Δ (G) for all 1 ≤ k ≤ nandμk (G) + μn - k + 2 (over(G, -)) ≥ δ (G) - Δ (G) - 1 for all 2 ≤ k ≤ n . Let G be an infinite family of graphs. We prove that G is quasi-random if and only if μn (G) + μn (over(G, -)) = o (n) for every G ∈ G of order n. This also implies that if λn (G) + λn (over(G, -)) = n + o (n) (or equivalently λ2 (G) + λ2 (over(G, -)) = o (n)) for every G ∈ G of order n, then G is quasi-random. © 2006 Elsevier Inc. All rights reserved.
Publication Title
Linear Algebra and Its Applications
Recommended Citation
Nikiforov, V. (2006). Eigenvalues and extremal degrees of graphs. Linear Algebra and Its Applications, 419 (2022-02-03), 735-738. https://doi.org/10.1016/j.laa.2006.06.013
