Some graphical properties of matrices with non-negative entries

Abstract

The parameters S and M were introduced in (1), (3) and (4) in connection with the stochastic rank and the term rank of a matrix. The results of this paper involve these same parameters. Two of these results are the following. Let A be a matrix of non-negative integers with sum S and maximum row or column sum M and let a = [S/M] (the greatest integer in S/M). It is shown in theorem 1 that A is expressible as a sum of subpermutat ion matrices of rank a and a sub-permutation matrix of rank q, 0 ~< q < a. Further, let K be a bipartite graph with finite vertex sets X and Y. Let S be the total number of edges and let M be the maximum number of edges which have the same vertex. It is shown as a consequence of theorem 3 that K has a transversal G of order a = [S/M] with vertex sets U and V such that every vertex of X which has M edges is an element of U and every vertex of Y which has M edges is an element of V. Theorem 1 may be proved as a corollary of theorem 3 but the connection of theorem 1 with the doubly stochastic extension of a matrix is interesting. It is on this connection that the proof of theorem 1 in section 3 is based.