TITLE: A graph similar to the Bruhat graph, what is it called?
QUESTION [5 upvotes]: The weak Bruhat graph (or 1-skeleton of the permutohedron) $B_n$ can be constructed as follows:

the vertices of $B_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they can be transformed into each other by a transposition of two neighboring numbers, e.g.

$$(5,3,\underline{\smash{4,1}},2) \quad\mapsto \quad (5,3,\underline{\smash{1,4}},2).$$
Now, consider the graph $X_n$, basically defined in the same way, but also allowing the "cyclic transposition":

the vertices of $X_n$ are the permutations of the tuple $(1,...,n)$, two are joined by an edge, if they can be transformed into each other by a transposition of two neighboring numbers, or the first and the last number, i.e.

$$(\underline 5,3,4,1,\underline 2) \quad\mapsto \quad (\underline 2,3,4,1,\underline 5).$$

Question: Is there a name for the graph $X_n$ in the literature, and what is it used for?

$B_n$ is a spanning subgraph of $X_n$.
In fact, $B_n$ can be obtained from $X_n$ by deleting a matching.
In contrast to the Bruhat graph, $X_n$ is always arc-transitive.
It has $n!$ vertices and degree $n$ (if $n\ge 3$). For example, $X_3=K_{3,3}$.

REPLY [6 votes]: A Markov chain on the symmetric group with this transition graph (but with directed edges and weights) was investigated by Lam and Williams.  This has since received considerable attention, and has been connected to "TASEP on a ring" if you are looking for search words (it doesn't appear that the graph itself has a name in this context).
I should point out that one should not give this graph the tempting names "circular Bruhat graph" or "cyclic Bruhat graph" as both of these have been used to refer to the graph underlying Postnikov's circular Bruhat order, which is different.