Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2310.15378 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909494520315904 |
|---|---|
| author | Podestá, Ricardo A. Videla, Denis E. |
| author_facet | Podestá, Ricardo A. Videla, Denis E. |
| contents | We study the spectrum of generalized Paley graphs $Γ(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of $Γ(k,q)$ are given by the Gaussian periods $η_{i}^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $Γ(k,q)$ with $1\le k \le 4$ and of $Γ(5,q)$ for $p\equiv 1\pmod 5$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $Γ(k,q)$ is integral if and only if $p$ divides $(q-1)/(p-1)$. Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs $Γ(k,q)$ with $1\le k \le 4$ or where $(k,q)$ is a semiprimitive pair. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15378 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral properties of generalized Paley graphs Podestá, Ricardo A. Videla, Denis E. Combinatorics We study the spectrum of generalized Paley graphs $Γ(k,q)=Cay(\mathbb{F}_q,R_k)$, undirected or not, with $R_k=\{x^k:x\in \mathbb{F}_q^*\}$ where $q=p^m$ with $p$ prime and $k\mid q-1$. We first show that the eigenvalues of $Γ(k,q)$ are given by the Gaussian periods $η_{i}^{(k,q)}$ with $0\le i\le k-1$. Then, we explicitly compute the spectrum of $Γ(k,q)$ with $1\le k \le 4$ and of $Γ(5,q)$ for $p\equiv 1\pmod 5$ and $5\mid m$. Also, we characterize those GP-graphs having integral spectrum, showing that $Γ(k,q)$ is integral if and only if $p$ divides $(q-1)/(p-1)$. Next, we focus on the family of semiprimitive GP-graphs. We show that they are integral strongly regular graphs (of pseudo-Latin square type). Finally, we characterize all integral Ramanujan graphs $Γ(k,q)$ with $1\le k \le 4$ or where $(k,q)$ is a semiprimitive pair. |
| title | Spectral properties of generalized Paley graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.15378 |