On the p-ranks of the adjacency matrices of distance-regular graphs

Let be a distance regular graph with adjacency matrix A. Let I be the identity matrix and J the all-1 matrix. Let p be a prime. We study the p-rank of the matrices A + bJ - cI for integral b; c.Graphs;mathematics

The Maximum Order of Adjacency Matrices With a Given Rank

AMS Subject Classification: 05B20, 05C50.Graph;Adjacency matrix

The Maximum Order of Reduced Square(0, 1)-Matrices with a Given Rank

We look for the maximum order of a square (0, 1)-matrix A with a fixed rank r, provided A has no repeated rows or columns. If A is the adjacency matrix of a graph, Kotlov and Lovász [J. Graph Theory 23, 1996] proved that the maximum order equals Θ(2r/2). In this note we show that this result remains correct if A is symmetric, but becomes false if symmetry is not required.(0;1)-matrix;rank;graph

The Maximum Edge Biclique Problem is NP-Complete

We prove that the maximum edge biclique problem in bipartite graphs is NP-complete.A biclique in a bipartite graph is a vertex induced subgraph which is complete.The problem of finding a biclique with a maximum number of vertices is known to be solvable in polynomial time but the complexity of finding a biclique with a maximum number of edges was still undecided.