Title

Eigenvalue problems of Nordhaus-Gaddum type

Abstract

Let G be a graph with n vertices and m edges and let μ1 (G) ≥ ⋯ ≥ μn (G) be the eigenvalues of its adjacency matrix. We discuss the following general problem. For k fixed and n large, find or estimatefk (n) = under(max, v (G) = n) | μk (G) | + | μk (over(G, -)) | .In particular, we prove thatfrac(4, 3) n - 2 ≤ f1 (n) < (sqrt(2) - c) nfor some c > 10- 7 independent of n. We also show that frac(sqrt(2), 2) n - 3 < f2 (n) < frac(sqrt(2), 2) n, frac(sqrt(2), 2) n - 3 < fn (n) ≤ frac(sqrt(3), 2) n . © 2006 Elsevier B.V. All rights reserved.

Publication Title

Discrete Mathematics

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