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Elsevier, Linear Algebra and its Applications, (493), p. 484-493

DOI: 10.1016/j.laa.2015.12.012

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Eigenvalue multiplicity in triangle-free graphs

Journal article published in 2016 by Peter Rowlinson
This paper was not found in any repository, but could be made available legally by the author.
This paper was not found in any repository, but could be made available legally by the author.

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Abstract

Let G be a connected triangle-free graph of order n>5 with μ∉{−1,0} as an eigenvalue of multiplicity k>1. We show that if d is the maximum degree in G then k≤n−d−1; moreover, if k=n−d−1 then either (a) G is non-bipartite and k≤(μ2+3μ+1)(μ2+2μ−1), with equality only if G is strongly regular, or (b) G is bipartite and k≤d−1, with equality only if G is a bipolar cone. In each case we discuss the extremal graphs that arise.