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∣s(G)=∑u∈V(G)∣d(u)-2m/n∣. We prove thats2(G)2n22m⩽μ(G)-2mn⩽s(G).In addition we derive similar inequalities for bipartite G.We also prove that the inequalityμk(G)+μn-k+2(G¯)⩾-1-22s(G)holds for every k = 2, … , n.Finally we prove that for every graph G of order n,μn(G)+μn(G¯)⩽-1-s2(G)2n3.We show that these inequalities are tight up to a constant factor.