The spectral radius of irregular graphs

Let λ1λ1 be the largest eigenvalue and λnλn the least eigenvalue of the adjacency matrix of a connected graph GG of order nn. We prove that if GG is irregular with diameter DD, maximum degree ΔΔ, minimum degree δδ and average degree dd, thenΔ-λ1>(n-δ)D+1Δ-d-D2-1.The inequality improves previous bounds of various authors and implies two lower bounds on λnλn which improve previous bounds of Nikiforov. It also gives some fine tuning of a result of Alon and Sudakov. A similar inequality is also obtained for the Laplacian spectral radius of a connected irregular graph.