More eigenvalue problems of Nordhaus-Gaddum type


Let G be a graph of order n and let μ1(G) ≥μ≥μn(G) be the eigenvalues of its adjacency matrix. This note studies eigenvalue problems of Nordhaus-Gaddum type. Let Ḡ be the complement of a graph G. It is shown that if s≥2 and n≥15(s-1), then| μs(G)|+|μs(Ḡ)|≤n/2(s-1)-1. Also if s≥1 and n≥4s, then|μn-s+1(G)|+|μn- s+1(Ḡ)|≤n/2s+1. If s=2k+1 for some integer k, these bounds are asymptotically tight. These results settle infinitely many cases of a general open problem. © 2014 Published by Elsevier Inc.

Publication Title

Linear Algebra and Its Applications