Spectrum of the Transposition graph

Transposition graph Tn is defined as a Cayley over the symmetric group generated by all transpositions. It known that eigenvalues of are integers. However, an explicit description spectrum unknown. In this paper we prove for any integer k⩾0 there exists n(k) such n⩾n(k) and m∈{0,…,k}, m eigenvalue Tn. particular, it proved zero n≠2, 1 odd n⩾7 even n⩾14. We also present exact values third fourth largest with their multiplicities.