Bounds on graph eigenvalues I


We improve some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n ≥ 2, maximum degree Δ, and girth at least 5, thenμ (G) ≤ min a, sqrt(n - 1)},where μ(G) is the largest eigenvalue of the adjacency matrix of G. Also, if G is a graph of order n ≥ 2 with dominating number γ(G) = γ, then(λ2 (G) ≤ fenced((n, if γ = 1,; n - γ, if γ ≥ 2,)); λn (G) ≥ ⌈ n / γ ⌉,)where 0 = λ1(G) ≤ λ2(G) ≤ ⋯ ≤ λn(G) are the eigenvalues of the Laplacian of G. We also determine all cases of equality in the above inequalities. © 2006 Elsevier Inc. All rights reserved.

Publication Title

Linear Algebra and Its Applications