A Sharp Forbidden Interval for the Nontrivial Adjacency Eigenvalues of Trivially Perfect Graphs

We prove a sharp forbidden interval for the nontrivial adjacency eigenvalues of trivially perfect graphs. More precisely, we show that if $G$ is a trivially perfect graph, then $\operatorname{Spec}(G)\cap [\sqrt{8}-4,0]\subseteq \{-1,0\}$. Moreover, we prove that the interval is best possible at both endpoints: there are connected trivially perfect graphs with eigenvalues arbitrarily close to $\sqrt{8}-4$ from below, and connected trivially perfect graphs with positive eigenvalues converging to $0$.