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(P_n) = B(E_n-1) for path and edgeless graphs, one-parameter graph families whose Bell number sequences are quasigeometric, and relations among the polynomial A(G,α) = Σ S(G,k) α^k, the chromatic polynomial and the Tutte polynomial, and some implications of these relations.