Enumeration of Bipartite Graphs and Bipartite Blocks

We use the theory of combinatorial species to count unlabeled bipartite graphs and bipartite blocks (nonseparable or 2-connected graphs). We start with bicolored graphs, which are bipartite graphs that are properly colored in two colors. The two-element group S2 acts on these graphs by switching the colors, and connected bipartite graphs are orbits of connected bicolored graphs under this action. From first principles we compute the S2-cycle index for bicolored graphs, an extension of the ordinary cycle index, introduced by Henderson, that incorporates the S2-action. From this we can compute the S2-cycle index for connected bicolored graphs, and then the ordinary cycle index for connected bipartite graphs. The cycle index for connected bipartite graphs allows us, by standard techniques, to count unlabeled bipartite graphs and unlabeled blocks.