Connected components and non-bipartiteness of generalized Paley graphs

TL;DR

This paper analyzes the structure of generalized Paley graphs, determining their connected components, conditions for disjoint unions of odd cycles, and identifying when they are non-bipartite, extending previous results to directed graphs.

Contribution

It characterizes the connected components and bipartiteness of generalized Paley graphs, generalizing prior work to include directed cases and providing new structural insights.

Findings

Connected components are smaller isomorphic GP-graphs.

GP-graphs are non-bipartite except for specific cases.

Identifies when GP-graphs are disjoint unions of odd cycles.

Abstract

In this work we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by $\Gamma(k,q) = Cay(\mathbb{F}_q, \{x^k : x\in \mathbb{F}_q^* \})$, 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 $\Gamma(k,q)$ is non-bipartite except for the graphs $\Gamma(2^m-1,2^m)$, $m \in \mathbb{N}$, which are isomorphic to $K_2 \sqcup \cdots \sqcup K_2$, the disjoint union of $2^{m-1}$ copies of $K_2$.