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?

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    $\begingroup$ 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. $\endgroup$ Apr 14, 2021 at 10:02
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    $\begingroup$ 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 $\endgroup$ Apr 14, 2021 at 10:44
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    $\begingroup$ 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. $\endgroup$ Apr 14, 2021 at 20:59

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We are working on these questions currently with Ignacio Garcia-Marco. As a postive answer I can tell you that the multiplication table of S codes a semigroup. If you take elements 1,6 as connection set, then the right Cayley graph is Cay(S,{1,6}). Its underlying undirected simple graphs is the Petersen graph. From the negative side we can also show, that any directed Cayley graph (of a semigroup) whose underlying undirected simple graph is the Petersen graph has loops.


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