Explicit Two-Sided Vertex Expanders Beyond the Spectral Barrier

TL;DR

This paper presents the first explicit construction of two-sided vertex expanders that surpass the spectral barrier, achieving higher expansion factors for small sets than previously possible with Ramanujan graphs.

Contribution

The authors construct explicit infinite families of bipartite and biregular graphs with small sets expanding by approximately 0.6 times their degree, breaking the 0.5 barrier of prior explicit constructions.

Findings

Constructed infinite families of graphs with >0.5d expansion for small sets.

Achieved unique-neighbor expansion of 0.6d for small sets.

Derived new bounds on triangle density in Ramanujan clique complexes.

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.