Graphs with Bipartite Complement that Admit Two Distinct Eigenvalues

TL;DR

This paper investigates the spectral properties of graphs with bipartite complements, establishing conditions under which such graphs have exactly two distinct eigenvalues, and characterizes cases where this does not hold.

Contribution

It proves that graphs with complements having few edges have two eigenvalues and characterizes bipartite complement cases where this property fails.

Findings

Graphs with complement edges ≤ floor(n/2)-1 have q(G)=2

Conjecture that graphs with complement edges ≤ n-3 have q(G)=2

Characterization of bipartite complement graphs with n-2 edges where q(G)>2

Abstract

The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor -1$ have $q(G)=2$. We conjecture that any $G$ with $e(\overline{G}) \leq n-3$ satisfies $q(G) = 2$. We show that this conjecture is true if $\overline{G}$ is bipartite and in other sporadic cases. Furthermore, we characterize $G$ with $\overline{G}$ bipartite and $e(\overline{G}) = n-2$ for which $q(G) > 2$.