论文标题
换位图的光谱
Spectrum of the Transposition graph
论文作者
论文摘要
换位图$ t_n $定义为所有转置生成的对称组上的Cayley图。众所周知，$ t_n $的所有特征值都是整数。但是，对频谱的明确描述是未知的。在本文中，我们证明，对于任何整数$ k \ geqslant 0 $，存在$ n_0 $，因此对于任何$ n \ geqslant n_0 $和任何$ m \ in \ {0，\ dots，k \} $，$ m $，$ m $是$ t_n $的eigenvalue。特别是，事实证明，对于任何$ n \ neq2 $，零是$ t_n $的特征值，而对于任何奇数$ n \ geqslant 7 $，对于任何$ n \ n \ geqslant 14 $而言，一个是$ t_n $的特征值。我们还提供了$ t_n $的第三和第四大特征值的精确值。
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. However, an explicit description of the spectrum is unknown. In this paper we prove that for any integer $k\geqslant 0$ there exists $n_0$ such that for any $n\geqslant n_0$ and any $m \in \{0, \dots, k\}$, $m$ is an eigenvalue of $T_n$. In particular, it is proved that zero is an eigenvalue of $T_n$ for any $n\neq2$, and one is an eigenvalue of $T_n$ for any odd $n\geqslant 7$ and for any even $n \geqslant 14$. We also present exact values of the third and the fourth largest eigenvalues of $T_n$ with their multiplicities.
