A spectral lower bound for the divisorial gonality of metric graphs

Let $\Gamma$ be a compact metric graph, and denote by $\Delta$ the Laplace operator on $\Gamma$ with the first non-trivial eigenvalue $\lambda_1$. We prove the following Yang-Li-Yau type inequality on divisorial gonality $\gamma_{div}$ of $\Gamma$. There is a universal constant $C$ such that $$\gamma_{div}(\Gamma) \geq C \frac{\mu(\Gamma) . \ell_{\min}^{\mathrm{geo}}(\Gamma). \lambda_1(\Gamma)}{d_{\max}},$$ where the volume $\mu(\Gamma)$ is the total length of the edges in $\Gamma$, $\ell_{\min}^{\mathrm{geo}}$ is the minimum length of all the geodesic paths between points of $\Gamma$ of valence different from two, and $d_{\max}$ is the largest valence of points of $\Gamma$. Along the way, we also establish discrete versions of the above inequality concerning finite simple graph models of $\Gamma$ and their spectral gaps.