Infinite families of regular expanders of arbitrary constant degree obtained via the modified zig-zag product

TL;DR

This paper extends the zig-zag product to generate infinite families of regular expanders with any constant degree, demonstrating that the modified product preserves expansion properties, thus broadening the construction of expanders.

Contribution

It introduces a generalized zig-zag product that produces infinite families of regular expanders of arbitrary constant degree, expanding the toolkit for constructing expanders.

Findings

The modified zig-zag product maintains good expansion properties.

The second largest eigenvalue is effectively controlled in the new construction.

The resulting graphs are regular expanders, though not Ramanujan.

Abstract

We generalize the zig-zag product construction to produce infinite families of regular graphs of any constant degree. We analyze the second largest eigenvalue of this new zig-zag product to show that the modified zig-zag product of good expanders is again a good expander (yet not Ramanujan).