Estimates of the first Dirichlet eigenvalue of graphs

TL;DR

This work develops Li–Yau–type lower and upper bounds for the first Dirichlet eigenvalue on graphs with boundary, relating λ1 to inscribed radius r, diameter D, and maximum degree d. The authors introduce volume- and degree-based lower bounds, with λ1(G,B) ≥ 1/(r|Ω|) and λ1(G,B) ≥ (d−1)/(r d^r), and establish a sharp general upper bound λ1(G,B) ≤ |E(Ω,B)|/|Ω|, along with a precise tree theory: λ1(T) ≤ |B|/|Ω| and a detailed extremal-tree classification for several leaf-count regimes using star-like path trees with explicit eigenvalues σ1, σ2, σ3, and σ4. The results provide sharp bounds (up to constants) and complete extremal characterizations in many tree cases, contributing to discrete Faber–Krahn-type questions and broadening the understanding of spectral graph theory under Dirichlet boundary conditions. The methods combine Rayleigh quotient analysis with combinatorial path-packings and explicit constructions of extremal trees, highlighting intrinsic links between geometry (radius/diameter) and spectra in graphs.

Abstract

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 $Ω= V \setminus B$ is connected. Let $r$ be the inscribed radius of $(G, B)$ and $d$ be the maximum degree of $G$. We prove that $$λ_1(G, B) \geq \frac{d - 1}{r d^r},$$ 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 $$λ_1(G, B) \geq \frac{1}{r |Ω|}.$$ In particular, for a tree $T$ with at least $3$ vertices, we show that $$λ_1(T) \geq 4 \sin^2 \fracπ{4r + 6} \geq \frac{1}{(r + 1)^2}.$$ 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.

Figures (2)

• Figure 1: The path-cliques graph $PC(\ell, \alpha)$
• Figure 2: a star-like path tree $SLP(p,q;c;d,e)$

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Lemma 2.1
• proof
• Lemma 2.2
• proof