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Explicit two-sided unique-neighbor expanders

Jun-Ting Hsieh, Theo McKenzie, Sidhanth Mohanty, Pedro Paredes

TL;DR

The paper delivers explicit constructions of two-sided unique-neighbor expanders (UNE) with algebraic structure and unequal bipartition sizes, addressing a major gap between random constructions and explicit, analyzable graphs. It introduces the tripartite line product, combining a Ramanujan base with a compact gadget to achieve strong UNE on both sides and, in variants, lossless small-set expansion up to sizes subexponential in the graph size. The analysis hinges on sharp subgraph spectral-radius bounds for non-backtracking matrices, a refined Moore-type bound, and the use of tree extensions and folding to relate local subgraph behavior to global expansion. The results yield infinite families of explicit biregular graphs with robust UNE properties and provide a pathway to two-sided algebraic UNE expanders, with corollaries including one-sided lossless expanders and potential quantum code applications via Lin & Hsieh's framework. Open questions remain regarding the full realization of algebraic two-sided lossless expanders and their quantum-code implications, as well as extensions to broader-partite settings and group-action symmetries.

Abstract

We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of $(d_1,d_2)$-biregular graphs $(G_n)_{n\ge 1}$ (for large enough $d_1$ and $d_2$) where all sets $S$ with fewer than a small constant fraction of vertices have $Ω(d_1\cdot |S|)$ unique-neighbors (assuming $d_1 \leq d_2$). Additionally, we can also guarantee that subsets of vertices of size up to $\exp(Ω(\sqrt{\log |V(G_n)|}))$ expand losslessly.

Explicit two-sided unique-neighbor expanders

TL;DR

The paper delivers explicit constructions of two-sided unique-neighbor expanders (UNE) with algebraic structure and unequal bipartition sizes, addressing a major gap between random constructions and explicit, analyzable graphs. It introduces the tripartite line product, combining a Ramanujan base with a compact gadget to achieve strong UNE on both sides and, in variants, lossless small-set expansion up to sizes subexponential in the graph size. The analysis hinges on sharp subgraph spectral-radius bounds for non-backtracking matrices, a refined Moore-type bound, and the use of tree extensions and folding to relate local subgraph behavior to global expansion. The results yield infinite families of explicit biregular graphs with robust UNE properties and provide a pathway to two-sided algebraic UNE expanders, with corollaries including one-sided lossless expanders and potential quantum code applications via Lin & Hsieh's framework. Open questions remain regarding the full realization of algebraic two-sided lossless expanders and their quantum-code implications, as well as extensions to broader-partite settings and group-action symmetries.

Abstract

We study the problem of constructing explicit sparse graphs that exhibit strong vertex expansion. Our main result is the first two-sided construction of imbalanced unique-neighbor expanders, meaning bipartite graphs where small sets contained in both the left and right bipartitions exhibit unique-neighbor expansion, along with algebraic properties relevant to constructing quantum codes. Our constructions are obtained from instantiations of the tripartite line product of a large tripartite spectral expander and a sufficiently good constant-sized unique-neighbor expander, a new graph product we defined that generalizes the line product in the work of Alon and Capalbo and the routed product in the work of Asherov and Dinur. To analyze the vertex expansion of graphs arising from the tripartite line product, we develop a sharp characterization of subgraphs that can arise in bipartite spectral expanders, generalizing results of Kahale, which may be of independent interest. By picking appropriate graphs to apply our product to, we give a strongly explicit construction of an infinite family of -biregular graphs (for large enough and ) where all sets with fewer than a small constant fraction of vertices have unique-neighbors (assuming ). Additionally, we can also guarantee that subsets of vertices of size up to expand losslessly.
Paper Structure (31 sections, 37 theorems, 138 equations, 2 figures, 1 table)

This paper contains 31 sections, 37 theorems, 138 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our results
  3. Context and related work
  4. Technical overview
  5. Open questions
  6. Organization
  7. Preliminaries
  8. Notation.
  9. Random bipartite graph models.
  10. Graph expansion
  11. Non-backtracking matrix
  12. Notation.
  13. The line product of graphs
  14. Algebraic unique-neighbor expanders
  15. Non-backtracking matrix of subgraphs
  16. ...and 16 more sections

Key Result

Theorem 1.2

For every $\beta\in(0,1/2]$ there is a constant $\gamma > 0$ such that for all large enough $d_1,d_2$ with $1\geqslant\frac{d_1}{d_2} \geqslant \frac{\beta}{1-\beta}$, there is an explicit infinite family of $(2d_1, 2d_2)$-biregular graphs $(Z_n)_{n\geqslant 1}$ where every $Z_n$ is a $\gamma$-two-s

Figures (2)

  • Figure 1: An example of the line product.
  • Figure 2: An example of the tripartite line product. The gadget placed on the first vertex in $M$ is highlighted in $G \diamond H$.

Theorems & Definitions (88)

  • Definition 1.1: Two-sided (algebraic) unique-neighbor expander
  • Theorem 1.2: Two-sided algebraic unique-neighbor expanders.
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: Two-sided unique-neighbor expanders with small-set lossless expansion; see \ref{['thm:2-sided-un-sse']}
  • Remark 1.6: Bicycle-free Ramanujan graph construction
  • Remark 1.7: One-sided lossless expanders
  • Theorem 1.8: See \ref{['thm:biregular-subgraph']}
  • Remark 1.9
  • Remark 1.10: Sharpness of \ref{['thm:subgraph-spectral-radius-main']}
  • ...and 78 more