Multiplicities of eigenvalues of the Star graph

The Star graph $S_n$, $n\geqslant 2$, is the Cayley graph on the symmetric group $\mathrm{Sym}_n$ generated by the set of transpositions [4] $\{(1~2), (1~3), \ldots, (1~n)\}$. We consider the spectrum of the Star graph as the spectrum of its adjacency matrix. It is known that the spectrum of $S_n$ is integral. Analytic formulas for multiplicities of eigenvalues $\pm(n-k)$ for $k = 2, 3, 4, 5$ in the Star graph are given in this paper. We also prove that any fixed integer has multiplicity at least $2^{\frac{1}{2}n \log n (1-o(1))}$ as an eigenvalue of $S_n$.