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

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 $m\le n$. We show that if $m$ is fixed, the answer is affirmative. In addition, when $m\to \infty$,weprovidepartialsolutionstotheprobleminthecasethat$d$ is fixed or that $d\to \mathrm{\infty }$ under the constraint $d=o(m^{1\mathrm{/}2})$.