Metric intersection problems in Cayley graphs and the Stirling recursion

TL;DR

This paper investigates the metric intersection properties of Cayley graphs generated by conjugacy classes of involutions in symmetric and alternating groups, revealing they relate to generalized Stirling functions.

Contribution

It establishes that the intersection numbers in these Cayley graphs are generalized Stirling functions, connecting algebraic graph properties with combinatorial functions.

Findings

Intersection numbers are generalized Stirling functions in n and r.

Results apply to Cayley graphs of Sym(n) and Alt(n) generated by involution classes.

Provides insights into metric structure relevant for error graphs and reconstruction problems.

Abstract

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.