Extreme eigenvalues of nonregular graphs

Let λ1 be the greatest eigenvalue and λn the least eigenvalue of the adjacency matrix of a connected graph G with n vertices, m edges and diameter D. We prove that if G is nonregular, thenΔ - λ1 \u3e frac(n Δ - 2 m, n (D (n Δ - 2 m) + 1)) ≥ frac(1, n (D + 1)), where Δ is the maximum degree of G. The inequality improves previous bounds of Stevanović and of Zhang. It also implies that a lower bound on λn obtained by Alon and Sudakov for (possibly regular) connected nonbipartite graphs also holds for connected nonregular graphs. © 2006 Elsevier Inc. All rights reserved