Title: Distinct eigenvalues of the Transposition graph Show Chinese title

Title: 置换图的不同特征值

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 $n\geqslant 4$. 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 $n \geqslant 19$. Show Chinese abstract

Abstract: 置换图$T_n$是在对称群上由所有置换生成的Cayley图。 已知$T_n$的所有特征值都是整数。 此外，对于任何$n\geqslant 4$，零都是其特征值。 但该图$T_n$的谱的确切分布是未知的。 在本文中，我们证明了区间$[-\frac{n-4}{2}, \frac{n-4}{2}]$中的整数在$T_n$的谱中，如果$n \geqslant 19$。