High-girth near-Ramanujan graphs with lossy vertex expansion

TL;DR

This paper constructs high-girth near-Ramanujan graphs with bounded vertex expansion and analyzes vertex expansion properties in Ramanujan graphs with large girth, using advanced spectral graph theory tools.

Contribution

It provides explicit constructions of high-girth near-Ramanujan graphs with controlled vertex expansion and analyzes vertex expansion in Ramanujan graphs with large girth.

Findings

Constructed $d$-regular graphs with girth $ o ext{logarithmic in } n$ and bounded vertex expansion.

Showed that Ramanujan graphs with girth proportional to $ ext{log } n$ have high vertex expansion for small sets.

Used spectral tools like the nonbacktracking operator and Ihara--Bass formula in analysis.

Abstract

Kahale proved that linear sized sets in $d$-regular Ramanujan graphs have vertex expansion $\sim\frac{d}{2}$ and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than $\frac{d}{2}$. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether high-girth Ramanujan graphs have improved vertex expansion. Our results are two-fold: 1. For every $d = p+1$ for prime $p$ and infinitely many $n$, we exhibit an $n$-vertex $d$-regular graph with girth $\Omega(\log_{d-1} n)$ and vertex expansion of sublinear sized sets bounded by $\frac{d+1}{2}$ whose nontrivial eigenvalues are bounded in magnitude by $2\sqrt{d-1}+O\left(\frac{1}{\log n}\right)$. 2. In any Ramanujan graph with girth $C\log n$, all sets of size bounded by $n^{0.99C/4}$ have vertex expansion $(1-o_d(1))d$. The tools in analyzing our construction include the nonbacktracking operator of an infinite graph, the Ihara--Bass formula, a trace moment method inspired by Bordenave's proof of Friedman's theorem, and a method of Kahale to study dispersion of eigenvalues of perturbed graphs.