I don't know how to type subscripts here, so K(m,n) denotes the complete bipartite graph on parts of cardinality m and n.
My question is; How many nonisomorphic spanning subgraphs are there of of K(m,n)? This is such an obvious question, it has probably been answered. I just don't know whereto look. There is an obvious, but complex to use, recursion for the constructions. Given the set of nonisomorphic subgraphs of K(m-1,n) -- or of Km,n-1) -- appending the n-1 edges from the missing vertex in the first case or m-1 in the second edges in all inequivalent ways will generate the set for K(m,n). But this is not a numerical problem so no simple recursion seems possible -- yet it may have well been solved using Polya's counting theorem. Do any of you know the answer, or where it can be found?