Cayley graphs and coset diagrams

Let G be a finite group, and X a subset of G. The Cayley graph of G with respect to X , written Cay(G,X) has two different definitions in the literature. The vertex set of this graph is the group G. In one definition, there is an arc from g to xg for all g ∈ G and x ∈ X ; in the other definition, for the same pairs (g,x), there is an arc from g to gx. Cayley graphs are generalised by coset graphs. For these we take a subgroup H of G as part of the data. Now the vertices of the graph may be either the left cosets or the right cosets of H in G; and an arc may join the coset containing g to the coset containing xg, or to the coset containing gx. Thus, there are four possible types of coset graphs. The two types of Cayley graph are in a certain sense equivalent, while the four types of coset graph fall into two distinct types: one type encompasses all vertex-transitive graphs, while the other is seldom vertex-transitive but has more specialised uses in the theory of regular maps. The purpose of this essay is to explain where the definitions come from and what purposes they serve.