Distinct eigenvalues of the Transposition graph

Abstract

Transposition graph Tn is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of Tn are integers. Moreover, zero is its eigenvalue for any n⩾4. But the exact distribution of the spectrum of the graph Tn is unknown. In this paper we prove that integers from the interval [−n−42,n−42] lie in the spectrum of Tn for any n⩾19.