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A Nordhaus-Gaddum conjecture for the minimum number of distinct eigenvalues of a graph (1807.06436v1)

Published 17 Jul 2018 in math.SP and math.CO

Abstract: We propose a Nordhaus-Gaddum conjecture for $q(G)$, the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph $G$: for every graph $G$ excluding four exceptions, we conjecture that $q(G)+q(Gc)\le |G|+2$, where $Gc$ is the complement of $G$. We compute $q(Gc)$ for all trees and all graphs $G$ with $q(G)=|G|-1$, and hence we verify the conjecture for trees, unicyclic graphs, graphs with $q(G)\le 4$, and for graphs with $|G|\le 7$.

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