Tight lower bound for the spectral radius of connected graphs with given matching number

Let $\mathscr{G}_{n,k}$ denote the family of all connected graphs of order $n$ with matching number $k$. Liu, Lou, and Trevisan~(Linear Algebra Appl., 2026) posed the following problem: Determine the spectrally minimal graphs in $\mathscr{G}_{n,k}$. In this paper we prove that for every graph $G \in \mathscr{G}_{n,k}$, $\rho(G) \ge \sqrt{\frac{n + 2k - 3}{k}},$ and we completely characterize the extremal graphs when $k \mid (n-3)$. As applications, we establish $\rho(G) + k \ge 3\sqrt[3]{n/4}$ for $k \ge 2$, settling the asymptotic order of $\rho + k$ as $\Theta(n^{1/3})$ -- strictly smaller than the $\Theta(\sqrt{n})$ order suggested by the disproved Aouchiche--Hansen conjecture.