Bipartite unique-neighbour expanders via Ramanujan graphs

TL;DR

This paper presents a simpler construction of bipartite unique-neighbour expanders using Ramanujan graphs, with potential practical advantages despite being weaker than previous lossless expanders.

Contribution

It introduces a novel, simpler method to construct bipartite unique-neighbour expanders by composing Ramanujan graphs with fixed-size gadgets, extending previous work.

Findings

Constructed an infinite family of bipartite unique-neighbour expanders.

Proved a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs.

Bound relies on exact Ramanujan-ness, not nearly-Ramanujan graphs.

Abstract

We construct an infinite family of bounded-degree bipartite unique-neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to be implementable in practice due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalizes the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction we prove a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs. Our bound generalizes Kahale's average degree bound to bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs.