The Spectrum of Cayley Graphs on Symmetric Group Generated by Certain Subset of r-Cycles

Let Sn be the symmetric group on [n] = {1, 2, . . . , n} and Cn(r ) be the set of all r -cycles in Sn that do not fix 1, i.e., Cn(r ) = {α ∈ Sn | α(1) = 1 and αis an r -cycle}. In this paper, we give a reduction formula of the eigenvalues of the Cayley graph (Sn, Cn(r )). Then we apply it to determine all the eigenvalues of the Cayley graph (Sn, Cn(r )) for r = 3, n − 1 and n.