Eigenvalues and degree deviation in graphs

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)-2m/n . We prove that In addition we derive similar inequalities for bipartite G. We also prove that the inequality holds for every k = 2, … , n. Finally we prove that for every graph G of order n, We show that these inequalities are tight up to a constant factor.