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 $K_2$.
Categories: math.CO