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, then Δ − λ1> nΔ − 2m n(D(nΔ − 2m)+1) ≥ 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