In the area of mathematics called combinatorial group theory, the Schreier coset graph is a graph associated with a group G, a generating set of G, and a subgroup ofG.
The Schreier graph encodes the abstract structure of the group modulo an equivalence relation formed by the cosets of the subgroup.
[1] An equivalent definition was made in an early paper of Todd and Coxeter.
[2] Given a group G, a subgroup H ≤ G, and a generating set S = {si : i in I} of G, the Schreier graph Sch(G, H, S) is a graph whose vertices are the right cosets Hg = {hg : h in H} for g in G and whose edges are of the form (Hg, Hgs) for g in G and s in S. More generally, if X is any G-set, one can define a Schreier graph Sch(G, X, S) of the action of G on X (with respect to the generating set 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.
Stallings' core graphs[3] are retracts of Schreier graphs of free groups, and are an essential tool for computing with subgroups of a free group.