TY - JOUR

T1 - The Star graph eigenfunctions with non-zero eigenvalues

AU - Kabanov, Vladislav V.

AU - Konstantinova, Elena V.

AU - Shalaginov, Leonid

AU - Valyuzhenich, Alexandr

N1 - Publisher Copyright: © 2020 Elsevier Inc. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2021/2/1

Y1 - 2021/2/1

N2 - We consider the symmetric group SymΩ with Ω={1,…,n} for any integer n⩾2 and a set S={(1i),i∈{2,…,n}}. The Star graph Sn=Cay(SymΩ,S) is the Cayley graph over the symmetric group SymΩ with the generating set S. For n⩾3, the spectrum of the Star Sn is integral such that for each integer 1⩽k⩽n−1, the values ±(n−k) are its eigenvalues; if n⩾4, then 0 is also an eigenvalue of Sn. A family of PI-eigenfunctions of the Star graph Sn,n⩾3, has been obtained recently for eigenvalues [Formula presented]. We generalise the family of PI-eigenfunctions and present a family of eigenfunctions for all non-zero eigenvalues of this graph.

AB - We consider the symmetric group SymΩ with Ω={1,…,n} for any integer n⩾2 and a set S={(1i),i∈{2,…,n}}. The Star graph Sn=Cay(SymΩ,S) is the Cayley graph over the symmetric group SymΩ with the generating set S. For n⩾3, the spectrum of the Star Sn is integral such that for each integer 1⩽k⩽n−1, the values ±(n−k) are its eigenvalues; if n⩾4, then 0 is also an eigenvalue of Sn. A family of PI-eigenfunctions of the Star graph Sn,n⩾3, has been obtained recently for eigenvalues [Formula presented]. We generalise the family of PI-eigenfunctions and present a family of eigenfunctions for all non-zero eigenvalues of this graph.

KW - Eigenfunction

KW - Eigenvalue

KW - Star graph

KW - Symmetric group

UR - http://www.scopus.com/inward/record.url?scp=85092077395&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2020.09.042

DO - 10.1016/j.laa.2020.09.042

M3 - Article

AN - SCOPUS:85092077395

VL - 610

SP - 222

EP - 226

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -