Proving a conjecture concerning chromatic number, size and least eigenvalue

Abstract

Let $G$ be a simple nonempty graph with size $m$, chromatic number $\chi$, and least eigenvalue $\lambda$. We prove that \[ \chi(\chi-1) \le (m+1-\lambda^2)+\sqrt{(m+1-\lambda^2)^2-4(\lambda^2-1)(\lambda^2-m)} \] with equality if and only if $G$ is either a complete graph or a complete bipartite graph, with possibly isolated vertices. The inequality was conjectured recently by Tang and Elphick in [Electron. J. Combin. 33 (2026), \#P2.65].