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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.05185 |
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| _version_ | 1866917249180237824 |
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| author | Higgins, Cecelia Spaas, Pieter Tenenbaum, Alexander |
| author_facet | Higgins, Cecelia Spaas, Pieter Tenenbaum, Alexander |
| contents | We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matchings, we introduce a measurable version of Tutte's condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_05185 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Spectral Theory for Borel PMP Graphs Higgins, Cecelia Spaas, Pieter Tenenbaum, Alexander Logic Combinatorics Operator Algebras Spectral Theory 03E15, 47B02, 05C50 We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of approximate measurable bipartiteness. Next, we adapt classical theorems of Wilf and Hoffman to give novel upper and lower bounds on the approximate measurable chromatic number. Using similar techniques, we then show that the approximate measurable chromatic number of a pmp graph generated by $n$ bounded-to-one functions is at most $2n + 1$. Next, concerning matchings, we introduce a measurable version of Tutte's condition and show that a spectral assumption analogous to the one from a classical theorem of Brouwer and Haemers implies this measurable Tutte condition. Finally, we show that the spectrum is continuous under local-global convergence. |
| title | Spectral Theory for Borel PMP Graphs |
| topic | Logic Combinatorics Operator Algebras Spectral Theory 03E15, 47B02, 05C50 |
| url | https://arxiv.org/abs/2602.05185 |