Estimates of the first Dirichlet eigenvalue of graphs

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 - 1/r 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) ≥ 1/r |Ω|. In particular, for a tree T with at least $3$ vertices, we show that λ₁(T) ≥ 4 sin² 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.