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 $\Omega = V \setminus B$ is connected. Let $r$ be the inscribed radius of $(G, B)$ and $d$ be the maximum degree of $G$. We prove that $$\lambda_1(G, B) \geq \frac{d - 1}{r d^r},$$ 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 $$\lambda_1(G, B) \geq \frac{1}{r |\Omega|}.$$ In particular, for a tree $T$ with at least $3$ vertices, we show that $$\lambda_1(T) \geq 4 \sin^2 \frac{\pi}{4r + 6} \geq \frac{1}{(r + 1)^2}.$$ 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.