Diameter of Cayley graphs of permutation groups generated by transposition trees

TL;DR

This paper evaluates the accuracy of a known upper bound for the diameter of Cayley graphs generated by transposition trees, introduces an efficient algorithm for estimating the diameter, and analyzes its performance across different tree structures.

Contribution

It assesses the sharpness of the existing diameter bound, introduces a polynomial-time algorithm for diameter estimation, and demonstrates its effectiveness on various tree families.

Findings

The upper bound is sharp for trees of maximum and minimum diameter.

The algorithm provides a lower-complexity estimate of the diameter.

For all tested trees, the algorithm's estimate is an upper bound on the true diameter.

Abstract

Let $\Gamma$ 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 $\Gamma$ is bounded as $$\diam(\Gamma) \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))},$$ where the maximization is over all permutations $\pi$, $c(\pi)$ denotes the number of cycles in $\pi$, 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 $\beta$ computed by our algorithm happens to also be an upper bound on the diameter, i.e. $$\diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n \dist_T(i,\pi(i))}.$$