Rank of the vertex-edge incidence matrix of $r$-out hypergraphs

Abstract

We consider a space of sparse Boolean matrices of size n × n, which have finite co-rank over GF (2) with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant with value about 0.2574 for large n.The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs The result that the null space is finite, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random k-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices n, [4], [6].For fields of higher order, the co-rank is typically Poisson distributed.