Catalogue of the Star graph eigenvalue multiplicities

The Star graph S_n, n⩾ 2 , is the Cayley graph over the symmetric group Sym _n generated by transpositions (1i),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.