The Schreier graph of a group G, a subgroup H, and a generating set S⊆G is denoted by Sch(G,H,S) or Sch(H\G,S). Its vertices are the right cosets Hg = {hg : h in H} for g in G, and its edges are of the form (Hg, Hgs) for g in G and s in S.
More generally, if X is a G-set, the Schreier graph of the action of G on X (with respect to S⊆G) is denoted by Sch(G,X,S) or Sch(X,S). Its vertices are the elements of X, and its edges are of the form (x,xs) for x in X and s in S. This includes the original Schreier coset graph definition, as H\G is a naturally a G-set with respect to multiplication from the right. From an algebraic-topological perspective, the graph Sch(X,S) has no distinguished vertex, whereas Sch(G,H,S) has the distinguished vertex H, and is thus a pointed graph.
The Cayley graph of the group G itself is the Schreier coset graph for H = {1G} (Gross & Tucker 1987, p. 73).
A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma (Conder 2003).
The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids. A subgroup H of a group G determines a covering morphism of groupoids and if S is a generating set for G then its inverse image under p is the Schreier graph of (G, S).