Non-isomorphic $d$-integral circulant graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909702136266752 |
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| author | Poddar, Sauvik Das, Angsuman |
| author_facet | Poddar, Sauvik Das, Angsuman |
| contents | The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with algebraic degree $d$. Let $C(d)$ be the least positive integer such that there exists a circulant graph of order $C(d)$ having algebraic degree $d$. A graph $G$ is called $d$-integral if $Deg(G)=d$. We call a $d$-integral circulant graph \textit{minimal} if order of that graph equals $C(d)$. Let $\mathcal{F}_{n,d}$ denote the collection of isomorphism classes of connected, $d$-integral circulant graphs of some given possible order $n$. In this paper we compute the exact value of $C(d)$ and provide some bounds on $|\mathcal{F}_{n,d}|$, thereby showing that the minimal $d$-integral circulant graph is not unique. Moreover, we find the exact value of $|\mathcal{F}_{p,d}|$ where both $p$ and $d$ are prime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-isomorphic $d$-integral circulant graphs Poddar, Sauvik Das, Angsuman Combinatorics 05E18, 05C25, 05C30 The algebraic degree $Deg(G)$ of a graph $G$ is the dimension of the splitting field of the adjacency polynomial of $G$ over the field $\mathbb{Q}$. It can be shown that for every positive integer $d$, there exists a circulant graph with algebraic degree $d$. Let $C(d)$ be the least positive integer such that there exists a circulant graph of order $C(d)$ having algebraic degree $d$. A graph $G$ is called $d$-integral if $Deg(G)=d$. We call a $d$-integral circulant graph \textit{minimal} if order of that graph equals $C(d)$. Let $\mathcal{F}_{n,d}$ denote the collection of isomorphism classes of connected, $d$-integral circulant graphs of some given possible order $n$. In this paper we compute the exact value of $C(d)$ and provide some bounds on $|\mathcal{F}_{n,d}|$, thereby showing that the minimal $d$-integral circulant graph is not unique. Moreover, we find the exact value of $|\mathcal{F}_{p,d}|$ where both $p$ and $d$ are prime. |
| title | Non-isomorphic $d$-integral circulant graphs |
| topic | Combinatorics 05E18, 05C25, 05C30 |
| url | https://arxiv.org/abs/2507.17407 |