Distinct eigenvalues of the Transposition graph

Elena V. Konstantinova; Artem Kravchuk

paper

Distinct eigenvalues of the Transposition graph

arXiv:2306.01627

Abstract

Transposition graph $T_{n}$ is defined as a Cayley graph over the symmetric group generated by all transpositions. It is known that all eigenvalues of $T_{n}$ are integers. Moreover, zero is its eigenvalue for any $n ⩾ 4$ . But the exact distribution of the spectrum of the graph $T_{n}$ is unknown. In this paper we prove that integers from the interval $[- \frac{n - 4}{2}, \frac{n - 4}{2}]$ lie in the spectrum of $T_{n}$ if $n ⩾ 19$ .

11 pages. arXiv admin note: text overlap with arXiv:2204.03153