A Correspondence Between Maximal Complete Bipartite Subgraphs and Closed Patterns

For an undirected graph without self-loop, we prove: (i) that the number of closed patterns in the adjacency matrix of is even; (ii) that the number of the closed patterns is precisely double the number of maximal complete bipartite subgraphs of ; (iii) that for every maximal complete bipartite subgraph, there always exists a unique and distinct pair of closed patterns that matches the two vertex sets of the subgraph. Therefore, we can efficiently enumerate all maximal complete bipartite subgraphs by using algorithms for mining closed patterns which have been extensively studied in the data mining field.