Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Diameter of Cayley graphs of permutation groups generated by
  transposition trees   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.
  $$\diam(\Gamma) \le \beta \le \max_{\pi \in S_n}{c(\pi)-n+\sum_{i=1}^n
\dist_T(i,\pi(i))}.$$

20
The categories of this paper are: acc-phys
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ex
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The categories of this paper are: chem-ph