Matroids, generalized networks, and electric network synthesis

Matroid theory has been applied to solve problems in generalized assignment, operations research, control theory, network theory, flow theory, generalized flow theory or linear programming, coding theory, and telecommunication network design. The operations of matroid union, matroid partitioning, matroid intersection, and the theorem on the greedy algorithm, Rado’s theorem, and Brualdi’s symmetric version of Rado’s theorem have been important for some of these applications. In this paper we consider the application of matroids to solve problems in network synthesis. Previously Bruno and Weinberg defined a generalized network, which is a network based on a matroid rather than a graph; for a generalized network the duality principle holds whereas it does not hold for a network based on a graph. We use the concept of the generalized network to formulate a solution to the following problem: What are the necessary and sufficient conditions for a singular matrix of real numbers, of order p and rank s, to be realizable as the open-circuit resistance matrix of a resistance p-port network. A simple algorithm is given for carriyng out the synthesis. We then present a number of unsolved problems, included among which is what could be called the four-color problem of network synthesis, namely, the resistance n-port problem.