A spectral lower bound for the divisorial gonality of metric graphs [article]

Omid Amini, Janne Kool
2014 arXiv   pre-print
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) ≥ C μ(Γ) . ℓ_^geo(Γ). λ_1(Γ)/d_, where the volume μ(Γ) is the total length of the edges in Γ, ℓ_^geo is the minimum length of all the geodesic paths between points of Γ of valence different from two, and d_ is the largest valence of points of
more » ... Along the way, we also establish discrete versions of the above inequality concerning finite simple graph models of Γ and their spectral gaps.
arXiv:1407.5614v2 fatcat:t76hx7l7jvg6dkmiojgloguvga