Note on relative Cayley graphs of groups and its complement

The relative Cayley graph of group [Formula: see text] with respect to its proper subgroup [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] for some [Formula: see text] where [Formula: see text] is an inversed-closed subset of [Formula: see text]. In this paper, we give a necessary and sufficient condition for [Formula: see text] to be bipartite, complete bipartite, complete, connected, self-centered, star graph, and vertex pancyclic. Also, we give a necessary and sufficient condition for [Formula: see text] to be Eulerian.