Two-sided group digraphs and graphs

© 2015 Wiley Periodicals, Inc. We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets L,R of a group G, we define the two-sided group digraph 2S - (G;L,R) to have vertex set G, and an arc from x to y if and only if y=ℓ-1xr for some ℓ∈L and r∈R. In common with Cayley graphs and digraphs, two-sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which 2S - (G;L,R) may be viewed as a simple graph of valency |L|·|R|, and we call such graphs two-sided group graphs. We also give sufficient conditions for two-sided group digraphs 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.