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 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(Ḡ) ≥-1-22s(G)holds for every k = 2, ... , n. Finally we prove that for every graph G of order n,μn(G)+μn(Ḡ)≤-1-s2(G)2n3.We show that these inequalities are tight up to a constant factor. © 2005 Elsevier Inc. All rights reserved.
Linear Algebra and Its Applications
Nikiforov, V. (2006). Eigenvalues and degree deviation in graphs. Linear Algebra and Its Applications, 414 (1), 347-360. https://doi.org/10.1016/j.laa.2005.10.011