TY - JOUR

T1 - PI-eigenfunctions of the Star graphs

AU - Goryainov, Sergey

AU - Kabanov, Vladislav

AU - Konstantinova, Elena

AU - Shalaginov, Leonid

AU - Valyuzhenich, Alexandr

PY - 2020/2/1

Y1 - 2020/2/1

N2 - 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.

AB - 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.

KW - Eigenfunction

KW - Jucys-Murphy element

KW - Polytabloid

KW - Reconstruction

KW - Specht module

KW - Star graph

KW - RECONSTRUCTION

KW - CAYLEY-GRAPHS

UR - http://www.scopus.com/inward/record.url?scp=85073606839&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2019.10.018

DO - 10.1016/j.laa.2019.10.018

M3 - Article

AN - SCOPUS:85073606839

VL - 586

SP - 7

EP - 27

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -