Inspired by the Li--Yau eigenvalue-diameter estimates, we investigate lower bounds for the first Dirichlet eigenvalue in terms of the diameter (or inscribed radius) of a graph. Let G = (V, E) be a graph with boundary B. Assume that the interior Ω= V B is connected. Let r be the inscribed radius of (G, B) and d be the maximum degree of G. We prove that λ₁ (G, B) d - 1r dʳ, which can be viewed as an analogue of the Lin--Yau bound and the Meng--Lin bound for normalized Dirichlet/Laplacian eigenvalues. We also derive the inequality λ₁ (G, B) 1r |Ω|. In particular, for a tree T with at least 3 vertices, we show that λ₁ (T) 4 ² π4r + 6 1 (r + 1) ². Notably, both of the two preceding bounds are sharp up to a constant factor. We additionally examine upper bounds on the first Dirichlet eigenvalue under constraints on the numbers of interior and boundary vertices.
Lin et al. (Mon,) studied this question.