Eigenvalues and degree deviation in graphs

Abstract Let G be a graph with n vertices and m edges and let μ(G) = μ1(G) ⩾ ⋯ ⩾ μn(G) be the eigenvalues of its adjacency matrix. Set s ( G ) = ∑ u ∈ V ( G ) ∣ d ( u ) - 2 m / n ∣ . We prove that s 2 ( G ) 2 n 2 2 m ⩽ μ ( G ) - 2 m n ⩽ s ( G ) . In addition we derive similar inequalities for bipartite G. We also prove that the inequality μ k ( G ) + μ n - k + 2 ( G ¯ ) ⩾ - 1 - 2 2 s ( G ) holds for every k = 2, … , n. Finally we prove that for every graph G of order n, μ n ( G ) + μ n ( G ¯ ) ⩽ - 1 - s 2 ( G ) 2 n 3 . We show that these inequalities are tight up to a constant factor.