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 γdiv of Γ. There is a universal (explicit) constant C such that γdiv(Γ) ≥ C μ(Γ).` min(Γ).λ1(Γ) dmax , where the volume μ(Γ) is the total length of the edges in Γ, ` min is the minimum length of edges in the minimal model of Γ, and dmax is the largest valency 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.