  The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group
$Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. 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.
