Catalogue of the Star graph eigenvalue multiplicities

Abstract

The Star graph S n , n 2, is the Cayley graph over the symmetric group Sym n generated by transpositions (1 i), 2 i n. This set of transpositions plays an important role in the representation theory of the symmetric group. The spectrum of S n contains all integers from −(n − 1) to n − 1, and also zero for n 4. In this paper we observe methods for getting explicit formulas of eigenvalue multiplicities in the Star graphs S n , present such formulas for the eigenvalues ±(n − k), where 2 k 12, and finally collect computational results of all eigenvalue multiplicities for n 50 in the catalogue.