  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.
