Sparsity of Graphs that Allow Two Distinct Eigenvalues

TL;DR

This paper investigates the minimum number of edges needed for a connected graph to have exactly two distinct eigenvalues, providing exact thresholds and characterizations for such graphs.

Contribution

It establishes precise edge count thresholds for graphs with two eigenvalues and characterizes graphs that meet these bounds.

Findings

Minimum edges for $q(G)=2$ are $2n-4$ (even $n$) and $2n-3$ (odd $n$).

Characterization of graphs achieving these bounds.

Provides structural insights into graphs with exactly two eigenvalues.

Abstract

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. It is shown that the minimum number of edges necessary for a connected graph $G$ to have $q(G)=2$ is $2n-4$ if $n$ is even, and $2n-3$ if $n$ is odd. In addition, a characterization of graphs for which equality is achieved in either case is given.