Schreier graph

Let $G$ be a group, $H\le G$ a subgroup and $S$ a generating set. The Schreier graph has left cosets $G/H$ as vertices and a directed edge $gH\to sgH$ labeled $s$ for each $s\in S$. Left multiplication makes this independent of the chosen representative of $gH$.

When $S$ is symmetric, the radius-$m$ graph ball about $H$ consists ofThese are cosets reached by words of length at most $m$. The single exact power $S^m$ gives the same set when $1\in S$; without this padding convention it may miss vertices reached at shorter lengths. For nonsymmetric $S$, the directed reachability version uses the displayed words, while the underlying undirected graph uses $S\cup S^{-1}$.

For $H=\{1\}$, this recovers the left-multiplication Cayley graph. More generally the Schreier graph records the group action on the set of cosets, including the stabilizer $H$.