Bell and Stirling Numbers for Graphs

The Bell number B(G) of a simple graph G is the number of partitions of its vertex set whose blocks are independent sets of G. The number of these partitions with k blocks is the (graphical) Stirling number S(G, k) of G. We explore integer sequences of Bell numbers for various one-parameter families of graphs, generalizations of the relation B(Pn) = B(En−1) for path and edgeless graphs, one-parameter graph families whose Bell number sequences are quasigeometric, and relations among the polynomial A(G,α) = ∑ S(G, k)α, the chromatic polynomial and the Tutte polynomial, and some implications of these relations.