The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908945967218688 |
|---|---|
| author | Podestá, Ricardo A. Videla, Denis E. |
| author_facet | Podestá, Ricardo A. Videla, Denis E. |
| contents | We consider the family of generalized Paley graphs (GP-graphs for short) $Γ(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(Γ(k,q)) \subset \mathbb{R}$ if and only if $Γ(k,q)$ is undirected. We then study conditions for integrality in the spectrum and give a general method to produce integral GP-graphs through cyclotomic polynomials. Using this, we construct several infinite families of integral GP-graphs. Next, we focus on directed GP-graphs (GP-digraphs). We show that GP-digraphs always have three or more eigenvalues, and then we prove that there is only one kind of GP-digraphs having three different eigenvalues: the oriented Paley graphs $\vec{\mathcal{P}}_q$ or disjoint unions of copies of them, $\vec{\mathcal{P}}_q \cup \cdots \cup \vec{\mathcal{P}}_q$. Then, we show that generically the GP-digraphs have period 1 (equivalently index of imprimitivity 1) except for $Γ(q-1,q)$ with $q$ odd, which is the disjoint union of oriented $p$-cycles, having period $p$. Finally, as an application, we study weak Waring numbers over finite fields through GP-graphs. In particular, we reduce the computation of the weak Waring numbers over finite fields to the computation of classic Waring numbers over finite fields, a result previously obtained by Cochrane and Cipra in 2012 by other means. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06513 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields Podestá, Ricardo A. Videla, Denis E. Combinatorics Primary 05C25, Secondary 05C50, 05C75 We consider the family of generalized Paley graphs (GP-graphs for short) $Γ(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(Γ(k,q)) \subset \mathbb{R}$ if and only if $Γ(k,q)$ is undirected. We then study conditions for integrality in the spectrum and give a general method to produce integral GP-graphs through cyclotomic polynomials. Using this, we construct several infinite families of integral GP-graphs. Next, we focus on directed GP-graphs (GP-digraphs). We show that GP-digraphs always have three or more eigenvalues, and then we prove that there is only one kind of GP-digraphs having three different eigenvalues: the oriented Paley graphs $\vec{\mathcal{P}}_q$ or disjoint unions of copies of them, $\vec{\mathcal{P}}_q \cup \cdots \cup \vec{\mathcal{P}}_q$. Then, we show that generically the GP-digraphs have period 1 (equivalently index of imprimitivity 1) except for $Γ(q-1,q)$ with $q$ odd, which is the disjoint union of oriented $p$-cycles, having period $p$. Finally, as an application, we study weak Waring numbers over finite fields through GP-graphs. In particular, we reduce the computation of the weak Waring numbers over finite fields to the computation of classic Waring numbers over finite fields, a result previously obtained by Cochrane and Cipra in 2012 by other means. |
| title | The nature of the spectrum of generalized Paley graphs and weak Waring numbers over finite fields |
| topic | Combinatorics Primary 05C25, Secondary 05C50, 05C75 |
| url | https://arxiv.org/abs/2604.06513 |