Spectrum of Cayley graphs on the symmetric group generated by transpositions

Abstract

For an integer n ≥ 2, let X n be the Cayley graph on the symmetric group S n generated by the set of transpositions {(1 2), (1 3), . . . , (1 n)}. It is shown that the spectrum of X n contains all integers from −(n − 1) to n − 1 (except 0 if n = 2 or n = 3).

On the eigenvalues of certain Cayley graphs and arrangement graphs

“…It was conjectured by Abdollahi and Vatandoost [1] that the eigenvalues of Γ(S n , Cy(2)) are integers, and contains all integers in the range from −(n − 1) to n − 1 (with the sole exception that when n = 2 or 3, zero is not an eigenvalue of Γ(S n , Cy(2)). The second part of the conjecture was proved by Krakovski and Mohar [17]. In fact, they showed that for n ≥ 2 and each integer 1 ≤ l ≤ n − 1, ±(n − l) are eigenvalues of Γ(S n , Cy(2)) with multiplicity at least n−2 l−1 .…”

“…, n} and S ⊆ S n be closed under conjugation. Since In general, if S is not closed under conjugation, then the eigenvalues of Γ(S n , S) may not be integers [13] (see also [1,17,20] for related results on the eigenvalues of certain Cayley graphs). Problem 1.3.…”

“…In particular, A(n, k, 1) is a k(n − k)-regular graph. We note here that A(n, k, 1) is called partial permutation graph in [17].…”

Catalogue of the Star graph eigenvalue multiplicities

“…They verified this conjecture numerically using GAP for n 6. In 2012, Krakovski and Mohar [14] proved the second part of the conjecture. More precisely, they proved that for n 2 and for each integer 1 k n − 1, the values ±(n − k) are eigenvalues of S n with multiplicity at least n−2 k−1 .…”

The smallest eigenvalues of the 1-point fixing graph

“…We note here that A(n, k, 1) was first introduced in [4] as an interconnection network model for parallel computation. Furthermore, A(n, k, 1) is called the partial permutation graph by Krakovski and Mohar in [13]. The eigenvalues of the arrangement graphs A(n, k, 1) were first studied in [2] by using a method developed by Godsil and McKay [10].…”