# Is the Petersen graph a “Cayley graph” of some more general group-like structure?

The Petersen graph is the smallest vertex-transitive graph which is not a Cayley graph. Is it the "Cayley graph" of some slightly more general group-like structure?

• I don't know if you're interested, but it is the Schützenberger graph of an inverse semigroup. However, that might be overkill, as every undirected graph appears as a Schützenberger graph of an inverse semigroup. – Diego Martínez Apr 14 at 10:02
• Perhaps you are looking for the definition of a Schreier coset graph. These are like Cayley graphs but they are defined for quotients $G/H$. Every vertex-transitive graph can be written as a coset graph. en.wikipedia.org/wiki/Schreier_coset_graph – Gjergji Zaimi Apr 14 at 10:44
• It depends, of course, on what one means by "group-like". I want to echo the comment by Gjergjii Zaimi. The symmetric group $S_5$ is vertex transitive on the vertices of the Cayley graph and so we may create a graph with vertices defined not be single elements of the group but by cosets. Pick the stabilizer of a vertex and create a graph on those 10 cosets ... and one can recover the Petersen graph as a vertex of cosets, instead of singletons. – Ken W. Smith Apr 14 at 20:59