  We consider the symmetric group $\mathrm{Sym}_n,\,n\geqslant 2$, generated by
the set $S$ of transpositions $(1~i),\,2 \leqslant i \leqslant n$, and the
Cayley graph $S_n=Cay(\mathrm{Sym}_n,S)$ called the Star graph. For any
positive integers $n\geqslant 3$ and $m$ with $n > 2m$, we present a family of
$PI$-eigenfunctions of $S_n$ with eigenvalue $n-m-1$. We establish a connection
of these functions with the standard basis of a Specht module. In the case of
largest non-principal eigenvalue $n-2$ we prove that any eigenfunction of $S_n$
can be reconstructed by its values on the second neighbourhood of a vertex.
