I need some "well-known" non-regular finite graphs (at least two vertices have different valency) whose automorphism groups contain a non-trivial subgroup that acts on the vertices semi-regularly (i.e all point-stabilizers are the trivial subgroup). It is well-known that semi-Cayley graphs, bi-Cayley graphs, bicirculants and tricirculants are examples of graphs having a semiregular subgroup and there exist many papers concerned with graphs, but to my knowledge, in all of these papers, the graph considered are regular. Also it is easy to construct several graphs such that their automorphism groups contains a semiregular subgroup and the graph is non-regular. I am seeking some well-known such graphs. The $n$-sunlet graph is one example. Please, let me know if there exists any other famous graph with this property.
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1$\begingroup$ To construct examples, just start with the semiregular permutation group you want, then add a few edges, and consider their orbits. For example, you can take a semiregular cyclic group with two orbits, then add one edge between the two orbits, and one edge inside one orbit to get the sunlet graph. Is the sunlet graph really famous? In any case, just starting from that page, the sun graphs also have this property, and so on. This question seems a little broad. $\endgroup$– verretCommented Sep 29, 2017 at 19:43
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