On the smallest eigenvalues of $3$-colorable graphs

TL;DR

This paper demonstrates that the smallest eigenvalues of 3-colorable graphs are densely distributed below a specific threshold, impacting the methods used to analyze spherical two-distance sets and chromatic numbers of signed graphs.

Contribution

It establishes the density of smallest eigenvalues for 3-colorable graphs below a certain limit, revealing limitations of the forbidden-subgraph approach in related graph problems.

Findings

Smallest eigenvalues of 3-colorable graphs are dense in a specific interval.

The result constrains the refinement of forbidden-subgraph methods.

Implications for spherical two-distance sets and signed graph chromatic numbers.

Abstract

We prove that the set of the smallest eigenvalues attained by $3$-colorable graphs is dense in $(-\infty, -\lambda^*)$, where $\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$ and $\rho$ is the positive real root of $x^3 = x + 1$. As a consequence, in the context of spherical two-distance sets, our result precludes any further refinement of the forbidden-subgraph method through the chromatic number of signed graphs.