On the gcd graphs over polynomial rings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909824343605248 |
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| author | Mináč, Ján Nguyen, Tung T. Tân, Nguyen Duy |
| author_facet | Mináč, Ján Nguyen, Tung T. Tân, Nguyen Duy |
| contents | Gcd-graphs over the ring of integers modulo $n$ are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over $\mathbb{Z}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the gcd graphs over polynomial rings Mináč, Ján Nguyen, Tung T. Tân, Nguyen Duy Number Theory Commutative Algebra Combinatorics 5C25, 05C50, 05C51 Gcd-graphs over the ring of integers modulo $n$ are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over $\mathbb{Z}.$ |
| title | On the gcd graphs over polynomial rings |
| topic | Number Theory Commutative Algebra Combinatorics 5C25, 05C50, 05C51 |
| url | https://arxiv.org/abs/2409.01929 |