A biclique in a graph Γ is a complete bipartite subgraph of Γ. We give bounds for the maximum number of edges in a biclique in terms of the eigenvalues of matrix representations of Γ. These bounds show a strong similarity with Lovász’s bound θ(Γ) for the Shannon capacity of Γ. Motivated by this similarity we investigate bicliques and the bounds in certain product graphs.

A class of graphs is said to be biclique-free if there is an integer t such that no graph in the class contains Kt,t as a subgraph. Large families of graph classes, such as any nowhere dense class of graphs or d-degenerate graphs, are biclique-free. We show that various domination problems are fixed-parameter tractable on biclique-free classes of graphs, when parameterizing by both solution siz...

Bipartite graphs have been proven useful in modeling a wide range of relationship networks. Finding the maximum edge biclique within a bipartite graph is a well-known problem in graph theory and data mining, with numerous realworld applications across different domains. We propose a probabilistic algorithm for finding the maximum edge biclique using a Monte Carlo subspace clustering approach. E...

A biclique is a maximal bipartite complete induced subgraph of G. The biclique graph of a graph G, denoted by KB(G), is the intersection graph of the family of all bicliques of G. In this work we give a natural definition of the distance between bicliques in a graph. We give a useful formula that relates the distance between bicliques in a graph G and the distance between their respectives vert...

A technique for clustering data by common attribute values involves grouping rows and columns of a binary matrix to make the minimum number of submatrices all 1’s. As binary matrices can be viewed as adjacency matrices of bipartite graphs, the problem is equivalent to partitioning a bipartite graph into the smallest number of complete bipartite sub-graphs (commonly called “bicliques”). We show ...

We propose a new formulation of the clustering problem that differs from previous work in several aspects. First, the goal is to explicitly output a collection of simple and meaningful conjunctive descriptions of the clusters. Second, the clusters might overlap, i.e., a point can belong to multiple clusters. Third, the clusters might not cover all points, i.e., not every point is clustered. Fin...

We introduce a new method for identifying optimal incomplete data sets from large sequence databases based on the graph theoretic concept of alpha-quasi-bicliques. The quasi-biclique method searches large sequence databases to identify useful phylogenetic data sets with a specified amount of missing data while maintaining the necessary amount of overlap among genes and taxa. The utility of the ...

We present polynomial time algorithms for induced biclique optimization problems in the following families of graphs: polygon-circle graphs, 4-hole-free graphs, complements of intervalfilament graphs and complements of subtree-filament graphs. Such problems are to find maximum: induced bicliques, induced balanced bicliques and induced edge bicliques. These problems have applications for bicliqu...