Enumeration of connected invariant graphs

Let h be a finite group acting on unlabeled graphs which does not change connectivity. Examples include edge reversal in directed graphs and permutations of colors in edge and/or vertex colored graphs. The generating functions of h-invariant (directed) graphs and h-invariant (weakly) connected (directed) graphs are discussed. This leads to a recursive formula for calculating the number of connected graphs when the total number of graphs is known. This is then applied to self-dual signed graphs, self-converse digraphs, and color cyclic graphs. Asymptotic expansions are also obtained. As expected, almost all of the above h-invariant graphs are connected and the asymptotic number of disconnected graphs has a simple interpretation.