A Spectral Lower Bound for the Divisorial Gonality of Metric Graphs

Omid Amini, Janne Kool
2015 International mathematics research notices  
Let Γ be a compact metric graph, and denote by ∆ the Laplace operator on Γ with the first non-trivial eigenvalue λ1. We prove the following Yang-Li-Yau type inequality on divisorial gonality γ div of Γ. There is a universal (explicit) constant C such that where the volume µ(Γ) is the total length of the edges in Γ, ℓ geo min is the non-zero minimum length of all the geodesic paths between points of Γ of valence different from two, and dmax is the largest valence of points of Γ. Along the way,
more » ... also establish discrete versions of the above inequality concerning finite simple graph models of Γ and their spectral gaps.
doi:10.1093/imrn/rnv213 fatcat:7ip2ejlia5etrf63yog5vyd6l4