Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by { G k } the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If En is the graph on n vertices with no edges then { En k } coincides with { n k } , the ordinary Stirling number of the second kind, and so we refer to { G k } as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of En, is asymptotically normal — essentially, as n grows, the histogram of ({ En k }) k≥0 , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences (Gn)n≥0 of graphs is there asymptotic normality of ({ Gn k })