On the satisfaction frequency of spectral characterization conditions
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910079753650176 |
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| author | Lvov, Nikita Van Werde, Alexander |
| author_facet | Lvov, Nikita Van Werde, Alexander |
| contents | We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_26932 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the satisfaction frequency of spectral characterization conditions Lvov, Nikita Van Werde, Alexander Probability Combinatorics Number Theory Spectral Theory 05C50, 05C80, 60B20, 15B52 We give the first specific conjectures on how frequently graphs satisfy sufficient conditions for being uniquely characterized by spectral information. These conjectures arise from a theoretical framework that we developed based on abstract-algebraic random matrix statistics. Specifically, we rephrase conditions from the literature in terms of Z[x]-modules associated to the adjacency matrix, and study the distribution of those modules in analytically tractable profinite random matrix ensembles. We applied this new method to two distinct conditions. The first requires square-freeness of the determinant of the walk matrix, and the second uses the discriminant of the characteristic polynomial. |
| title | On the satisfaction frequency of spectral characterization conditions |
| topic | Probability Combinatorics Number Theory Spectral Theory 05C50, 05C80, 60B20, 15B52 |
| url | https://arxiv.org/abs/2603.26932 |