Given a symmetric group $S_n $ and the generator set : $S = \{(12),(23), . . . ,(n − 1 n) \}$ is there any closed form expression for the diameter of the Cayley graph generated by this set of generator ? Is there any upper bound on that diameter ?
For the given generator set: $diameter(S_2) = 1$, $diameter(S_3) = 3$, $diameter(S_4) = 6$ so does the following hold ?
$diameter(S_n) = nC_2$ [w.r.t the given generator set]
It will be great to have the tightest bound on the diameter w.r.t the given generator set.