In this work, we consider the class of Cayley graphs known as generalized Paley graphs (GP-graphs for short) given by \(\Gamma (k,q) = \textrm{Cay}({\mathbb {F}}_q, \{x^k: x\in {\mathbb {F}}_q^* \})\) , 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 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 \(K_2\) .
