On the smallest eigenvalues of $3$-colorable graphs

Abstract

We prove that the set of the smallest eigenvalues attained by $3$-colorable graphs is dense in $(-\infty ,-\hat{I}{»}^{\ast })$, where $\hat{I}{»}^{\ast }=\ddot{I}{}^{1/2}+\ddot{I}{}^{-1/2}\approx 2.01980$ and $\ddot{I}$ 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.

13 pages, 3 figures; full proofs of Lemmas 11 and 12 added in Appendix A