A Nordhaus–Gaddum conjecture for the minimum number of distinct eigenvalues of a graph

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)≤|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)≤4, and for graphs with |G|≤7.