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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.10348 |
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| _version_ | 1866912537411321856 |
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| author | Nguyen, Tung T. Tân, Nguyen Duy |
| author_facet | Nguyen, Tung T. Tân, Nguyen Duy |
| contents | We define super-Cayley graphs over a finite abelian group $G$. Using the theory of supercharacters on $G$, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we show that a super-Cayley graph is determined by its spectrum once an indexing on the underlying group $G$ is fixed. This generalizes a theorem by Sander-Sander, which investigates the case where $G$ is a cyclic group. We then use our theory to define and study the concept of a $U$-unitary Cayley graph over a finite commutative ring $R$, where $U$ is a subgroup of the unit group of $R$. Furthermore, when the underlying ring is a Frobenius ring, we show that there is a natural supercharacter theory associated with $U$. By applying the general theory of super-Cayley graphs developed in the first part, we explore various spectral properties of these $U$-unitary Cayley graphs, including their rationality and connections to various arithmetical sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10348 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs Nguyen, Tung T. Tân, Nguyen Duy Number Theory 11L03, 11T24, 05C25 We define super-Cayley graphs over a finite abelian group $G$. Using the theory of supercharacters on $G$, we explain how their spectra can be realized as a super-Fourier transform of a superclass characteristic function. Consequently, we show that a super-Cayley graph is determined by its spectrum once an indexing on the underlying group $G$ is fixed. This generalizes a theorem by Sander-Sander, which investigates the case where $G$ is a cyclic group. We then use our theory to define and study the concept of a $U$-unitary Cayley graph over a finite commutative ring $R$, where $U$ is a subgroup of the unit group of $R$. Furthermore, when the underlying ring is a Frobenius ring, we show that there is a natural supercharacter theory associated with $U$. By applying the general theory of super-Cayley graphs developed in the first part, we explore various spectral properties of these $U$-unitary Cayley graphs, including their rationality and connections to various arithmetical sums. |
| title | Supercharacters of finite abelian groups and applications to spectra of $U$-unitary Cayley graphs |
| topic | Number Theory 11L03, 11T24, 05C25 |
| url | https://arxiv.org/abs/2508.10348 |