Catalogue of the Star graph eigenvalue multiplicities

The Star graph $$S_n$$Sn, $$n\geqslant 2$$n⩾2, is the Cayley graph over the symmetric group $$\mathrm {Sym}_n$$Symn generated by transpositions $$(1~i),\,2\leqslant i \leqslant n$$(1i),2⩽i⩽n. This set of transpositions plays an important role in the representation theory of the symmetric group. The spectrum of $$S_n$$Sn contains all integers from $$-(n-1)$$-(n-1) to $$n-1$$n-1, and also zero for $$n\geqslant 4$$n⩾4. In this paper we observe methods for getting explicit formulas of eigenvalue multiplicities in the Star graphs $$S_n$$Sn, present such formulas for the eigenvalues $$\pm (n-k)$$±(n-k), where $$2\leqslant k \leqslant 12$$2⩽k⩽12, and finally collect computational results of all eigenvalue multiplicities for $$n\leqslant 50$$n⩽50 in the catalogue.