Spectral bounds of multi-way Cheeger constants via cyclomatic number
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913513841098752 |
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| author | Ge, Chuanyuan |
| author_facet | Ge, Chuanyuan |
| contents | As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant in forms of $C(k)\sqrt{λ_k(G)}$, where $λ_k(G)$ is the $k$-th eigenvalue of the graph Laplacian and $C(k)$ is a constant depending only on $k$. In this article, we prove some new bounds for multi-way Cheeger constants. By shifting the index of the eigenvalue via cyclomatic number, we establish upper bound estimates with an absolute constant instead of $C(k)$. This, in particular, gives a more direct proof of Miclo's higher order Cheeger inequalities on trees. We also show a new lower bound of the multi-way Cheeger constants in terms of the spectral radius of the graph. The proofs involve the concept of discrete nodal domains and a probability argument showing generic properties of eigenfunctions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07097 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral bounds of multi-way Cheeger constants via cyclomatic number Ge, Chuanyuan Combinatorics Spectral Theory As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant in forms of $C(k)\sqrt{λ_k(G)}$, where $λ_k(G)$ is the $k$-th eigenvalue of the graph Laplacian and $C(k)$ is a constant depending only on $k$. In this article, we prove some new bounds for multi-way Cheeger constants. By shifting the index of the eigenvalue via cyclomatic number, we establish upper bound estimates with an absolute constant instead of $C(k)$. This, in particular, gives a more direct proof of Miclo's higher order Cheeger inequalities on trees. We also show a new lower bound of the multi-way Cheeger constants in terms of the spectral radius of the graph. The proofs involve the concept of discrete nodal domains and a probability argument showing generic properties of eigenfunctions. |
| title | Spectral bounds of multi-way Cheeger constants via cyclomatic number |
| topic | Combinatorics Spectral Theory |
| url | https://arxiv.org/abs/2409.07097 |