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We use the theory of combinatorial species to count unlabelled 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 \$\backslash mathfrak\S\_2\$ 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 \$\backslash mathfrak\S\_2\$-cycle index for bicolored graphs, an extension of the ordinary cycle index, introduced by Henderson, that incorporates the \$\backslash mathfrak\S\_2\$-action. From this we can compute the \$\backslash mathfrak\S\_2\$-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 bipartite blocks.

@article{ Gainer-Dewar2014a, abstract = {We use the theory of combinatorial species to count unlabelled 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 \$\backslash mathfrak\{S\}_2\$ 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 \$\backslash mathfrak\{S\}_2\$-cycle index for bicolored graphs, an extension of the ordinary cycle index, introduced by Henderson, that incorporates the \$\backslash mathfrak\{S\}_2\$-action. From this we can compute the \$\backslash mathfrak\{S\}_2\$-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 bipartite blocks.}, author = {Gainer-Dewar, Andrew and Gessel, Ira M.}, file = {:Users/KunihiroWASA/Dropbox/paper/2013/Gainer-Dewar, Gessel, Enumeration of bipartite graphs and bipartite blocks, 2013.pdf:pdf}, issn = {1077-8926}, journal = {The Electronic Journal of Combinatorics}, keywords = {combinatorial species,cycle index,unlabeled bipartite graphs and blocks}, number = {2}, pages = {P2.40}, title = {{Enumeration of Bipartite Graphs and Bipartite Blocks}}, url = {http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i2p40}, volume = {21}, year = {2014} }

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