Estimates of the first Dirichlet eigenvalue of graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916991051235328 |
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| author | Lin, Huiqiu Liu, Lianping You, Zhe Zhao, Da |
| author_facet | Lin, Huiqiu Liu, Lianping You, Zhe Zhao, Da |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_04557 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimates of the first Dirichlet eigenvalue of graphs Lin, Huiqiu Liu, Lianping You, Zhe Zhao, Da Combinatorics Analysis of PDEs 05C50, 35R02, 47A75, 49J40, 49R05 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. |
| title | Estimates of the first Dirichlet eigenvalue of graphs |
| topic | Combinatorics Analysis of PDEs 05C50, 35R02, 47A75, 49J40, 49R05 |
| url | https://arxiv.org/abs/2510.04557 |