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arXiv · 2610.30662

Aleshin graphs are expanders

Olga Kharlampovich·Alina Vdovina
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Abstract

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 $n\geq10$.

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Olga Kharlampovich, Alina Vdovina. 2026-10-22. Aleshin graphs are expanders. https://arxiv.org/abs/2610.30662

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