Aleshin graphs are expanders

We prove that the Aleshin level graphs are two-sided expanders. We establish linear diameter bounds for Belaterra and Aleshin graph families. At level $n$, the diameters of the Bellaterra and undirected Aleshin graphs are a $|V_n|=2^n$. This improves the quadratic diameter bounds of Malyshev and Pak to the correct order of growth. For the Aleshin graphs we obtain a normalized spectral gap of at least $1-2\sqrt2/3$ and an edge-expansion constant of at least $3-2\sqrt2$, thereby establishing a Malyshev–Pak's Conjecture with explicit constants. An exact calculation at level ten also proves that the undirected Aleshin graphs are not Ramanujan for any $n\geq10$.