LAUSR

Abstract We consider the symmetric group Sym Ω with Ω = { 1 , … , n } for any integer n ⩾ 2 and a set S = { ( 1 i ) , i ∈ { 2 , … , n } } . The Star graph S n = Cay ( Sym Ω , S ) is the Cayley graph over the symmetric group Sym Ω with the generating set S. For n ⩾ 3 , the spectrum of the Star S n 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 S n . A family of PI-eigenfunctions of the Star graph S n , n ⩾ 3 , has been obtained recently for eigenvalues n − 2 , … , n − ⌊ n 2 ⌋ − 1 . We generalise the family of PI-eigenfunctions and present a family of eigenfunctions for all non-zero eigenvalues of this graph.