Spectral strengthening of a theorem on transversal critical graphs

A transversal set of a graph $G$ is a set of vertices incident to all edges of $G$. The transversal number of $G$, denoted by $\tau(G)$, is the minimum cardinality of a transversal set of $G$. A simple graph $G$ with no isolated vertex is called $\tau$-critical if $\tau(G-e) < \tau(G)$ for every edge $e\in E(G)$. For any $\tau$-critical graph $G$ with $\tau(G)=t$, it has been shown that $|V(G)|\le 2t$ by Erd\H{o}s and Gallai and that $|E(G)|\le {t+1\choose 2}$ by Erd\H{o}s, Hajnal and Moon. Most recently, it was extended by Gy\'arf\'as and Lehel to $|V(G)| + |E(G)|\le {t+2\choose 2}$. In this paper, we prove stronger results via spectrum. Let $G$ be a $\tau$-critical graph with $\tau(G)=t$ and $|V(G)|=n$, and let $\lambda_1$ denote the largest eigenvalue of the adjacency matrix of $G$. We show that $n + \lambda_1\le 2t+1$ with equality if and only if $G$ is $tK_2$, $K_{s+1}\cup (t-s)K_2$, or $C_{2s-1}\cup (t-s)K_2$, where $2\leq s\leq t$; and in particular, $\lambda_1(G)\le t$ with equality if and only if $G$ is $K_{t+1}$. We then apply it to show that for any nonnegative integer $r$, we have $n\left(r+ \frac{\lambda_1}{2}\right) \le {t+r+1\choose 2}$ and characterize all extremal graphs. This implies a pure combinatorial result that $r|V(G)| + |E(G)| \le {t+r+1\choose 2}$, which is stronger than Erd\H{o}s-Hajnal-Moon Theorem and Gy\'arf\'as-Lehel Theorem. We also have some other generalizations.