We consider the symmetric group Symn, n⩾2, generated by the set S of transpositions (1i),2⩽i⩽n, and the Cayley graph Sn=Cay(Symn,S) called the Star graph. For any positive integers n⩾3 and m with n>2m, we present a family of eigenfunctions of Sn with eigenvalue n−m−1 and call them PI-eigenfunctions. We then establish a connection of these functions with the standard basis of a Specht module. Finally, in the case of the largest non-principal eigenvalue n−2 we prove that any eigenfunction of Sn can be reconstructed by its values on the second neighbourhood of a vertex.
Предметные области OECD FOS+WOS
- 1.01 МАТЕМАТИКА