Extreme eigenvalues of nonregular graphs

Let λ 1 be the greatest eigenvalue and λ n 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 − λ 1 > n Δ − 2 m n ( D ( n Δ − 2 m ) + 1 ) ⩾ 1 n ( D + 1 ) , where Δ is the maximum degree of G. equality improves previous bounds of Stevanović and of Zhang. It also implies that a lower bound on λ n obtained by Alon and Sudakov for (possibly regular) connected nonbipartite graphs also holds for connected nonregular graphs.