On the smallest eigenvalues of $3$-colorable graphs

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