PreprintCC BY 4.0arXiv
Aleshin graphs are expanders
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 |Vn|=2n. 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√2/3 and an edge-expansion constant of at least 3−2√2, 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≥10.
Related
BibTeX
@misc{kharlampovich2026aleshin,
title = {{Aleshin graphs are expanders}},
author = {Olga Kharlampovich and Alina Vdovina},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30662},
eprint = {2610.30662},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30662},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Aleshin graphs are expanders
AU - Olga Kharlampovich
AU - Alina Vdovina
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30662
UR - https://arxiv.org/abs/2610.30662
AB - 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√2/3 and an edge-expansion constant of at least 3−2√2, 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≥10.
ER -