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Spectral Theory and Descriptive Combinatorics for Non-Regular Borel pmp Graphs

  • Cecelia Higgins
  • Pieter Spaas
  • Sofia Torelli
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Abstract

We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.

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Cite

BibTeXRIS
BibTeX
@misc{higgins2026spectral,
  title = {{Spectral Theory and Descriptive Combinatorics for Non-Regular Borel pmp Graphs}},
  author = {Cecelia Higgins and Pieter Spaas and Sofia Torelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13551},
  eprint = {2610.13551},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13551},
}
RIS
TY  - UNPB
TI  - Spectral Theory and Descriptive Combinatorics for Non-Regular Borel pmp Graphs
AU  - Cecelia Higgins
AU  - Pieter Spaas
AU  - Sofia Torelli
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13551
UR  - https://arxiv.org/abs/2610.13551
AB  - We continue the systematic study initiated in arXiv:2602.05185 of the relationship between the spectral theory and descriptive combinatorics of bounded-degree Borel pmp graphs, particularly in the non-regular setting. We first adapt a classical result of Nikiforov to give a new spectral lower bound on the approximate measurable chromatic number. For certain non-regular graphs, this bound improves on the approximate measurable version of Hoffman's classical bound established in arXiv:2602.05185. Next, we strengthen a characterization in arXiv:2602.05185 of approximate measurable bipartiteness in terms of the spectrum of the adjacency operator by generalizing to the case of non-regular graphs. Finally, we introduce the measurable domination number and adapt several classical results of Lu, Liu, and Tian to give a bound on the spectral gap of the Laplacian operator in terms of the measurable domination number.
ER  -