Eigenvector-based identification of bipartite subgraphs

Abstract

We report our experiments on identifying large bipartite subgraphs of simple connected graphs which are based on the sign pattern of eigenvectors belonging to the extremal eigenvalues of different graph matrices: adjacency, signless Laplacian, Laplacian, and normalized Laplacian matrix. We compare these methods to a ‘local switching’ algorithm based on the proof of the Erdös’ bound that each graph contains a bipartite subgraph with at least half of its edges. Experiments with one scale-free and three random graph models, which cover a wide range of real-world networks, show that the methods based on the eigenvectors of the normalized Laplacian and the adjacency matrix, while yielding comparable results to the local switching algorithm, are still outperformed by it. We also formulate two edge bipartivity indices based on the former eigenvectors, and observe that the method of iterative removal of edges with maximum bipartivity index until one obtains a bipartite subgraph, also yields comparable results to the local switching algorithm.