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A spectral lower bound for the divisorial gonality of metric graphs

Let $Γ$ be a compact metric graph, and denote by $Δ$ the Laplace operator on $Γ$ with the first non-trivial eigenvalue $λ_1$. We prove the following Yang-Li-Yau type inequality on divisorial gonality $γ_{div}$ of $Γ$. 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 $μ(Γ)$ is the total length of the edges in $Γ$, $\ell_{\min}^{\mathrm{geo}}$ is the minimum length of all the geodesic paths between points of $Γ$ of valence different from two, and $d_{\max}$ is the largest valence of points of $Γ$. Along the way, we also establish discrete versions of the above inequality concerning finite simple graph models of $Γ$ and their spectral gaps.