AI 中文总结
本文研究对称群上正规Cayley图的Aldous性质，证明了其与补图的性质互斥性，确定了大部分具有该性质的正规Cayley图，解决了2023年提出的开放问题，还完成了线图及特征值相关分类。
AI 中文摘要
Aldous谱隙猜想指出，任意连通图上的随机游走与交换过程具有相同的谱隙，等价于：对称群$S_n$关于对换集的任意连通Cayley图的第二大特征值，由$S_n$的标准表示实现。这一著名猜想于2010年被完全证明，此后学界对寻找$S_n$上其他具有该性质（现称Aldous性质）的Cayley图产生了浓厚兴趣。本文首先证明：当$n \ge 5$时，除了正规Cayley图$2K_{n!/2}$与其补图$K_{n!/2,n!/2}$这一特殊情况外，$S_n$上的一个正规Cayley图与其补图中最多仅有一个具有Aldous性质。随后，我们确定了对于足够大的$n$，所有具有Aldous性质的正规Cayley图$\mathrm{Cay}(S_n, S)$，仅排除$S$同时包含支撑大小在$\{2,3,\dots,n-2\}$的置换和支撑大小在$\{n-1,n\}$的置换，但$S$未包含所有支撑大小为$n$的置换这一情况。特别地，我们证明：若正规Cayley图$\mathrm{Cay}(S_n, S)$非完全，且$S$中所有置换的支撑大小均为$n-1$或$n$，或$S$包含所有支撑大小为$n$的置换，则该图不具有Aldous性质，从而解决了Li、Xia和Zhou于2023年提出的一个开放问题。在此过程中，我们确定了$S_n$上所有为线图的正规Cayley图，并分类了$S_n$上所有严格第二大特征值不超过1的正规Cayley图。
英文摘要
Aldous' spectral gap conjecture states that the random walk and the interchange process on any connected graph have the same spectral gap, or, equivalently, the second largest eigenvalue of any connected Cayley graph on the symmetric group $S_n$ with respect to a set of transpositions is achieved by the standard representation of $S_n$. This celebrated conjecture, proved in its general form in 2010, has inspired much interest in searching for other Cayley graphs on $S_n$ possessing this property, now known as the Aldous property. In this paper, we first prove that for $n \ge 5$ at most one of a normal Cayley graph on $S_n$ and its complement can possess the Aldous property except when these two graphs are $2K_{n!/2}$ and $K_{n!/2,n!/2}$ respectively. We then determine, for sufficiently large $n$, all normal Cayley graphs $\mathrm{Cay}(S_n, S)$ that have the Aldous property, except for the case when $S$ contains a permutation with support size in $\{2, 3, \dots, n-2\}$ and a permutation with support size in $\{n-1, n\}$, but not all permutations with support size $n$ are contained in $S$. In particular, we show that a non-complete normal Cayley graph $\mathrm{Cay}(S_n, S)$ does not have the Aldous property if all permutations in $S$ have support size $n-1$ or $n$, or all permutations with support size $n$ are contained in $S$, thereby solving an open problem posed by Li, Xia and Zhou in 2023. Along the way we determine all normal Cayley graphs on $S_n$ that are line graphs, and classify all normal Cayley graphs on $S_n$ with the strictly second largest eigenvalue at most $1$.
