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

TL;DR

This paper introduces a conjecture relating the minimum number of distinct eigenvalues of a graph and its complement, verifying it for several classes of graphs including trees and small graphs.

Contribution

It proposes a Nordhaus-Gaddum type conjecture for the minimum number of eigenvalues and verifies it for multiple graph classes and small graph sizes.

Findings

Conjecture holds for trees, unicyclic graphs, and graphs with q(G) ≤ 4.

Verified the conjecture for all graphs with up to 7 vertices.

Computed q(G^c) for all trees and graphs with q(G) = |G| - 1.

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 $q(G)+q(G^c)\le |G|+2$, where $G^c$ is the complement of $G$. We compute $q(G^c)$ 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 $|G|\le 7$.