换位树生成的置换群的Cayley图的直径
Diameter of Cayley graphs of permutation groups generated by transposition trees
摘要：
设$Γ$是由$n$顶点上的换位树$T$生成的置换群的Cayley图。在一个经常被引用的文件\cite{Akers:Krishnamurthy:1989}（也见\cite{Hahn:Sabidussi:1997}），它表明，直径的凯莱图$Γ$是有界的$$\diam(Γ) \le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))},$$，其中最大化是在所有的排列$π$，$c(π)$表示的循环数在$π$，和$\dist_T$是距离函数在$T$。在这项工作中，我们首先评估的性能（清晰度和严格性）的上限。我们表明，上界是尖锐的最大直径的所有树木，也为所有树木的最小直径，我们展示了一些家庭的树木的界限是严格的。然后，我们表明，对于每一个$n$，存在一棵树上的$n$顶点，使上限和真实直径值之间的差异至少是$n-4$。观察到，评估该上限需要$n!$（乘以多项式）计算的数量级。我们提供了一个算法，获得的估计直径，但它只需要的顺序（多项式）$n$计算;此外，通过我们的算法获得的值小于或等于先前已知的直径上限。这个结果是可能的，因为我们的算法直接使用$n$顶点上的转置树，并且不需要检查任何排列（只有证明需要检查排列）。对于到目前为止检查的所有树族，由我们的算法计算的值$β$恰好也是直径的上界，即$$\diam(Γ) \le β\le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))}.$$
英文摘要：
Let $Γ$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the Cayley graph $Γ$ is bounded as $$\diam(Γ) \le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))},$$ where the maximization is over all permutations $π$, $c(π)$ denotes the number of cycles in $π$, and $\dist_T$ is the distance function in $T$. In this work, we first assess the performance (the sharpness and strictness) of this upper bound. We show that the upper bound is sharp for all trees of maximum diameter and also for all trees of minimum diameter, and we exhibit some families of trees for which the bound is strict. We then show that for every $n$, there exists a tree on $n$ vertices, such that the difference between the upper bound and the true diameter value is at least $n-4$. Observe that evaluating this upper bound requires on the order of $n!$ (times a polynomial) computations. We provide an algorithm that obtains an estimate of the diameter, but which requires only on the order of (polynomial in) $n$ computations; furthermore, the value obtained by our algorithm is less than or equal to the previously known diameter upper bound. This result is possible because our algorithm works directly with the transposition tree on $n$ vertices and does not require examining any of the permutations (only the proof requires examining the permutations). For all families of trees examined so far, the value $β$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. $$\diam(Γ) \le β\le \max_{π\in S_n}{c(π)-n+\sum_{i=1}^n \dist_T(i,π(i))}.$$
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