Spectrum 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. However, an explicit description of the spectrum is unknown. In this paper we prove that for any integer k⩾0 there exists n(k) such that for any n⩾n(k) and any m∈{0,…,k}, m is an eigenvalue of Tn. In particular, it is proved that zero is an eigenvalue of Tn for any n≠2, and the integer 1 is an eigenvalue of Tn for any odd n⩾7 and for any even n⩾14. We also present exact values of the third and the fourth largest eigenvalues of Tn with their multiplicities.