A spectral lower bound for the divisorial gonality of metric graphs

Abstract

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 \[γ_{div}(Γ) \geq C \frac{μ(Γ) . \ell_{\min}^{\mathrm{geo}}(Γ). λ_1(Γ)}{d_{\max}},\] 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.

arXiv comment22 pages, added new recent references, minor revision