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Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier

Jun-Ting Hsieh, Ting-Chun Lin, Sidhanth Mohanty, Ryan O'Donnell, Rachel Yun Zhang

Abstract

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.

Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier

Abstract

We construct the first explicit two-sided vertex expanders that bypass the spectral barrier. Previously, the strongest known explicit vertex expanders were given by -regular Ramanujan graphs, whose spectral properties imply that every small subset of vertices has at least distinct neighbors. However, it is possible to construct Ramanujan graphs containing a small set with no more than neighbors. In fact, no explicit construction was known to break the -barrier. In this work, we give an explicit construction of an infinite family of -regular graphs (for large enough ) where every small set expands by a factor of . More generally, for large enough , we give an infinite family of -biregular graphs where small sets on the left expand by a factor of , and small sets on the right expand by a factor of . In fact, our construction satisfies an even stronger property: small sets on the left and right have unique-neighbor expansion and respectively. Our construction follows the tripartite line product framework of Hsieh, McKenzie, Mohanty & Paredes, and instantiates it using the face-vertex incidence of the -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.

Paper Structure

This paper contains 20 sections, 29 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Our results
  3. Related work
  4. Technical overview
  5. Notation
  6. Explicit construction of 3/5-two-sided unique-neighbor expanders
  7. Construction
  8. Proof overview of Theorem \ref{['thm:main-UNE']}
  9. Proof of Theorem \ref{['thm:main-UNE']}
  10. Construction of the base graph
  11. Properties of the spherical building
  12. Construction of base graph from Ramanujan complex: balanced case
  13. Imbalanced case
  14. Bounding triangles
  15. Edge density of small sets in the Ramanujan complex
  16. ...and 5 more sections

Key Result

Theorem 1

For any $\varepsilon > 0$ and $\beta \in (0, 1]$, there is a large enough $d(\varepsilon,\beta)$ such that for all ${d}_L, {d}_R \geqslant d(\varepsilon,\beta)$ with $\frac{{d}_L}{{d}_R} \in [\beta, \beta+\varepsilon]$, there is an explicit infinite family of $(5{d}_L, 5{d}_R)$-biregular graphs $(Z_

Theorems & Definitions (62)

  • Theorem 1
  • Remark 1.1
  • Lemma 1.2: Triangle density bound in $4$D Ramanujan complex, informal
  • Theorem 2.1: Formal \ref{['thm:main-informal']}
  • Definition 2.2: Construction
  • Definition 2.3: Structured bipartite graph
  • Definition 2.4: Small-set triangle expansion
  • Definition 2.5: Small-set skeleton expansion
  • Lemma 2.7
  • Definition 2.8: Pseudorandom gadget
  • ...and 52 more