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)-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.
Publication Title
Linear Algebra and Its Applications
Recommended Citation
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
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