Minimum supports of eigenfunctions with the second largest eigenvalue of the Star graph

The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sy{m}_n$ generated by the set of transpositions $\{(12),(13),\dots ,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$ corresponding to the second largest eigenvalue and obtain a characterization of eigenfunctions with the minimum cardinality of the support.