Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier
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2024
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| author | Hsieh, Jun-Ting Lin, Ting-Chun Mohanty, Sidhanth O'Donnell, Ryan Zhang, Rachel Yun |
| author_facet | Hsieh, Jun-Ting Lin, Ting-Chun Mohanty, Sidhanth O'Donnell, Ryan Zhang, Rachel Yun |
| contents | We construct the first explicit two-sided vertex expanders that bypass the spectral barrier.
Previously, the strongest known explicit vertex expanders were given by $d$-regular Ramanujan graphs, whose spectral properties imply that every small subset of vertices $S$ has at least $0.5d|S|$ distinct neighbors. However, it is possible to construct Ramanujan graphs containing a small set $S$ with no more than $0.5d|S|$ neighbors. In fact, no explicit construction was known to break the $0.5 d$-barrier.
In this work, we give an explicit construction of an infinite family of $d$-regular graphs (for large enough $d$) where every small set expands by a factor of $\approx 0.6d$. More generally, for large enough $d_1,d_2$, we give an infinite family of $(d_1,d_2)$-biregular graphs where small sets on the left expand by a factor of $\approx 0.6d_1$, and small sets on the right expand by a factor of $\approx 0.6d_2$. In fact, our construction satisfies an even stronger property: small sets on the left and right have unique-neighbor expansion $0.6d_1$ and $0.6d_2$ respectively.
Our construction follows the tripartite line product framework of Hsieh, McKenzie, Mohanty & Paredes, and instantiates it using the face-vertex incidence of the $4$-dimensional Ramanujan clique complex as its base component. As a key part of our analysis, we derive new bounds on the triangle density of small sets in the Ramanujan clique complex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11627 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier Hsieh, Jun-Ting Lin, Ting-Chun Mohanty, Sidhanth O'Donnell, Ryan Zhang, Rachel Yun Combinatorics Computational Complexity Discrete Mathematics Data Structures and Algorithms We construct the first explicit two-sided vertex expanders that bypass the spectral barrier. Previously, the strongest known explicit vertex expanders were given by $d$-regular Ramanujan graphs, whose spectral properties imply that every small subset of vertices $S$ has at least $0.5d|S|$ distinct neighbors. However, it is possible to construct Ramanujan graphs containing a small set $S$ with no more than $0.5d|S|$ neighbors. In fact, no explicit construction was known to break the $0.5 d$-barrier. In this work, we give an explicit construction of an infinite family of $d$-regular graphs (for large enough $d$) where every small set expands by a factor of $\approx 0.6d$. More generally, for large enough $d_1,d_2$, we give an infinite family of $(d_1,d_2)$-biregular graphs where small sets on the left expand by a factor of $\approx 0.6d_1$, and small sets on the right expand by a factor of $\approx 0.6d_2$. In fact, our construction satisfies an even stronger property: small sets on the left and right have unique-neighbor expansion $0.6d_1$ and $0.6d_2$ respectively. Our construction follows the tripartite line product framework of Hsieh, McKenzie, Mohanty & Paredes, and instantiates it using the face-vertex incidence of the $4$-dimensional Ramanujan clique complex as its base component. As a key part of our analysis, we derive new bounds on the triangle density of small sets in the Ramanujan clique complex. |
| title | Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier |
| topic | Combinatorics Computational Complexity Discrete Mathematics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2411.11627 |