Connected components and non-bipartiteness of generalized Paley graphs
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| Format: | Preprint |
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2024
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| author | Podestá, Ricardo A. Videla, Denis E. |
| author_facet | Podestá, Ricardo A. Videla, Denis E. |
| contents | In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by $Γ(k,q) = Cay(\mathbb{F}_q, \{x^k : x\in \mathbb{F}_q^* \})$, where $\mathbb{F}_q$ is a finite field with $q$ elements, both in the directed and undirected case. Hence $q=p^m$ with $p$ prime, $m\in \mathbb{N}$ and one can assume that $k\mid q-1$. We first give the connected components of an arbitrary GP-graph. We show that these components are smaller GP-graphs all isomorphic to each other (generalizing a Lim and Praeger's result from 2009 to the directed case). We then characterize those GP-graphs which are disjoint unions of odd cycles. Finally, we show that $Γ(k,q)$ is non-bipartite except for the graphs $Γ(2^m-1,2^m)$, $m \in \mathbb{N}$, which are isomorphic to $K_2 \sqcup \cdots \sqcup K_2$, the disjoint union of $2^{m-1}$ copies of $K_2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Connected components and non-bipartiteness of generalized Paley graphs Podestá, Ricardo A. Videla, Denis E. Combinatorics Primary 05C25, Secondary 05C50, 05C75, 05C20, 05C40 In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by $Γ(k,q) = Cay(\mathbb{F}_q, \{x^k : x\in \mathbb{F}_q^* \})$, where $\mathbb{F}_q$ is a finite field with $q$ elements, both in the directed and undirected case. Hence $q=p^m$ with $p$ prime, $m\in \mathbb{N}$ and one can assume that $k\mid q-1$. We first give the connected components of an arbitrary GP-graph. We show that these components are smaller GP-graphs all isomorphic to each other (generalizing a Lim and Praeger's result from 2009 to the directed case). We then characterize those GP-graphs which are disjoint unions of odd cycles. Finally, we show that $Γ(k,q)$ is non-bipartite except for the graphs $Γ(2^m-1,2^m)$, $m \in \mathbb{N}$, which are isomorphic to $K_2 \sqcup \cdots \sqcup K_2$, the disjoint union of $2^{m-1}$ copies of $K_2$. |
| title | Connected components and non-bipartiteness of generalized Paley graphs |
| topic | Combinatorics Primary 05C25, Secondary 05C50, 05C75, 05C20, 05C40 |
| url | https://arxiv.org/abs/2410.00281 |