标题： 关于$3$-可着色图的最小特征值 显示英文标题

标题： On the smallest eigenvalues of $3$-colorable graphs

摘要： 我们证明了由$3$-可着色图获得的最小特征值的集合在$(-\infty, -\lambda^*)$中是稠密的，其中$\lambda^* = \rho^{1/2} + \rho^{-1/2} \approx 2.01980$和$\rho$是$x^3 = x + 1$的正实根。 作为推论，在球面二距离集的背景下，我们的结果排除了通过有号图的色数对禁止子图方法进行进一步改进的可能。 显示英文摘要

摘要： 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.