The network flows approach for matrices with given row and column sums

Abstract

We study the set of integral matrices with given row and column sums where each position has a bound on the size of the entry in it. Such matrices correspond to maximum integral flows in certain networks. The well known existence theorem follows from the max flow-min cut theorem. A general network flow result, specialized to our setting, yields a useful interchange theorem which has a number of corollaries. Prompted by another network flow result, new results are obtained for invariant positions i.e. entries which are the same regardless of which matrix is chosen. This leads to a classification of invariant edges in graphs of a given degree sequence, important in the study of split graphs and threshold graphs.