The number of labeled connected graphs modulo prime powers

Abstract

Let cn denote the number of vertex-labeled connected graphs on n vertices. Using group actions and elementary number theory, we show that the infinite sequence, cn : n ≥ 1, is ultimately periodic modulo every positive integer. We state and prove our results for sequences defined by a weighted generalization of cn and conjecture that these results are suggestive of similar periodic behavior of the Tutte polynomial evaluations of the complete graph Kn at integer points.