Metric intersection problems in Cayley graphs and the Stirling recursion

Details Metric intersection problems in Cayley graphs and the Stirling recursion Metric intersection problems in Cayley graphs and the Stirling recursion

2012-02-20

arxiv.org/abs/1202.4493v1

Mathematics

Combinatorics

18 pages

Scientific paper

In the symmetric group Sym(n) with n at least 5 let H be a conjugacy class of elements of order 2 and let \Gamma be 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 \Gamma of radius r and centre g. We show that the intersection numbers \Phi(\Gamma; 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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