Extreme eigenvalues of nonregular graphs

Let λ1λ1 be the greatest eigenvalue and λnλn 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>nΔ−2mn(D(nΔ−2m)+1)⩾1n(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 λnλn obtained by Alon and Sudakov for (possibly regular) connected nonbipartite graphs also holds for connected nonregular graphs.