A spectral lower bound for the divisorial gonality of metric graphs

TL;DR

This paper establishes a spectral lower bound for the divisorial gonality of compact metric graphs, linking geometric, spectral, and combinatorial properties with a universal constant.

Contribution

It introduces a Yang-Li-Yau type inequality relating divisorial gonality to spectral and geometric graph invariants, including discrete analogues.

Findings

Derived a universal constant lower bound for divisorial gonality.

Connected spectral gap with geometric and combinatorial graph parameters.

Established discrete inequalities for finite graph models.

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 \[\gamma_{div}(\Gamma) \geq C \frac{\mu(\Gamma) . \ell_{\min}^{\mathrm{geo}}(\Gamma). \lambda_1(\Gamma)}{d_{\max}},\] 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.