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In several works, Mendel and Naor have introduced and developed theorysurrounding a nonlinear expansion constant similar to the spectral gap forsequences of graphs, in which one considers embeddings of a graph $G$ into ametric space $X$ \cite{mendel2010towards, mendel2013nonlinear,mendel2014expanders}. Here, we investigate the open question of whether therandom regular graph $G_{n, d}$ is an expander when embedded into the metricspace of a random regular graph $G_{m, d}$ a.a.s., where $m\leq n$. We showthat 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 $d$ isfixed or that $d\to \infty$ under the constraint $d=o(m^{1/2})$.

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