On $2$-integral Cayley graphs
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866916880212557824 |
|---|---|
| author | Abdollahi, Alireza Arezoomand, Majid Feng, Tao Wang, Shixin |
| author_facet | Abdollahi, Alireza Arezoomand, Majid Feng, Tao Wang, Shixin |
| contents | In this paper, we introduce the concept of $k$-integral graphs. A graph $Γ$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $Γ$ over rational field $\mathbb Q$ is equal to $k$. We prove that the set of all finite connected graphs with given algebraic degree and maximum degree is finite. $1$-integral graphs are just integral ones, graphs all of whose eigenvalues are integer. We study
$2$-integral Cayley graphs over finite groups $G$ with respect to Cayley sets which are a union of conjugacy classes of $G$. Among other general results, we completely characterize all finite abelian groups having a connected $2$-integral Cayley graph with valency $2,3,4$ and $5$.
Furthermore, we classify finite groups $G$ for which all Cayley graphs over $G$ with bounded valency are $2$-integral. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On $2$-integral Cayley graphs Abdollahi, Alireza Arezoomand, Majid Feng, Tao Wang, Shixin Combinatorics Group Theory 05C25, 05C50 In this paper, we introduce the concept of $k$-integral graphs. A graph $Γ$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $Γ$ over rational field $\mathbb Q$ is equal to $k$. We prove that the set of all finite connected graphs with given algebraic degree and maximum degree is finite. $1$-integral graphs are just integral ones, graphs all of whose eigenvalues are integer. We study $2$-integral Cayley graphs over finite groups $G$ with respect to Cayley sets which are a union of conjugacy classes of $G$. Among other general results, we completely characterize all finite abelian groups having a connected $2$-integral Cayley graph with valency $2,3,4$ and $5$. Furthermore, we classify finite groups $G$ for which all Cayley graphs over $G$ with bounded valency are $2$-integral. |
| title | On $2$-integral Cayley graphs |
| topic | Combinatorics Group Theory 05C25, 05C50 |
| url | https://arxiv.org/abs/2401.15306 |