On expansion of $G_{n, d}$ with respect to $G_{m, d}$

TL;DR

This paper explores the nonlinear expansion properties of random regular graphs when embedded into metric spaces derived from other random regular graphs, providing affirmative results for fixed parameters and partial solutions in asymptotic regimes.

Contribution

It investigates whether random regular graphs are expanders in the metric space of another random regular graph, offering new insights into nonlinear graph expansion theory.

Findings

Affirmative expansion results for fixed m.

Partial solutions when m tends to infinity.

Results depend on fixed or asymptotic degrees d.

Abstract

In several works, Mendel and Naor have introduced and developed theory surrounding a nonlinear expansion constant similar to the spectral gap for sequences of graphs, in which one considers embeddings of a graph $G$ into a metric space $X$ \cite{mendel2010towards, mendel2013nonlinear, mendel2014expanders}. Here, we investigate the open question of whether the random regular graph $G_{n, d}$ is an expander when embedded into the metric space of a random regular graph $G_{m, d}$ a.a.s., where $m\leq n$. We show that if $m$ is fixed, the answer is affirmative. In addition, when $m\to \infty$, we provide partial solutions to the problem in the case that $d$ is fixed or that $d\to \infty$ under the constraint $d=o(m^{1/2})$.