I'm looking for a source from which to learn about Schreier coset graphs. Especially, examples in which combinatorial properties (specifically, diameter) of Schreier graphs are calculated.
Also, is there an english translation of Schreier's work: "Die Untergruppen der freien Gruppen"?
Thank you!
The paper (Annexstein, Baumslag and Rosenberg, "Group action graphs and parallel architectures", SIAM Journal on Computing, pp. 544-569, 1990) studies various properties of coset graphs - including diameter - that arise in network contexts. In that paper, Schreier coset graphs are called [Cayley] coset graphs. The coset graph of $G$ with respect to a subgroup $H$ and a subset $S$ is denoted $Cos(G,H,S)$.
Not every vertex-transitive graph can be realized as a Cayley graph (the Petersen graph is the usual counterexample). But every vertex-transitive graph can be realized as a coset graph. This was proved by (Sabidussi, "Vertex-transitive graphs", 1964).
It is mentioned in (Heydemann, "Cayley graphs and interconnection networks", in Graph Symmetry: Algebraic Methods and Applications, Eds. Hahn and Sabidussi, p. 189) that coset graphs have been used in constructing graphs for the degree/diameter problem.