Cayley图的度量交问题与Stirling递归
Metric intersection problems in Cayley graphs and the Stirling recursion
摘要：
在n ≥ 5的对称群Sym（n）中，设H是2阶元素的共轭类，设Γ是顶点集是由H生成的群G的Cayley图（G是Sym（n）或Alt（n）），其边集由H确定.我们感兴趣的是这个图的度量结构。特别地，对于g \in G，令B_{r}（g）是在Γ中半径为r且中心为g的公制球。我们证明了相交数Φ（r; r，g）：=|\，B_{r}（e）\，\cap \，B_{r}（g）\，|是n和r中的广义Stirling函数。这些结果的动机是错误图和相关的重建问题的研究。
英文摘要：
In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of elements of order 2 and let Γbe the Cayley graph whose vertex set is the group G generated by H (so G is Sym(n) or Alt(n)) and whose edge set is determined by H. We are interested in the metric structure of this graph. In particular, for g\in G let B_{r}(g) be the metric ball in Γof radius r and centre g. We show that the intersection numbers Φ(Γ; r, g):=|\,B_{r}(e)\,\cap\,B_{r}(g)\,| are generalized Stirling functions in n and r. The results are motivated by the study of error graphs and related reconstruction problems.
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