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 discre