  We introduce a family of graphs that generalises the class of Cayley graphs.
For non-empty subsets L, R of a group G, the two-sided Cayley graph 2SC(G;L,R)
is the directed graph with vertex set G and an arc from x to y if and only if
y=a^{-1}xb for some a in L and b in R. Thus, in common with Cayley graphs,
two-sided Cayley graphs may be useful to model networks as the same routing and
communication scheme can be implemented at each vertex. We determine when
two-sided Cayley graphs are simple undirected graphs, and give sufficient
conditions for them to be connected, vertex-transitive, or Cayley graphs.
Several open problems are posed. Many examples are given, including one on 12
vertices with connected components of sizes 4 and 8.
