Generalized networks: Networks embedded on a matroid, part I

AbstractThe definition of ordinary electric networks is based on graphs. Since such a formulation is inadequate in many respects, we define a new network concept, namely, a generalized network, which is a network based on a matroid. To make the paper self‐contained an introduction to basic matroid theory is given in the first part of the paper. We then define a generalized network and formulate the network analysis and network synthesis problems for generalized networks. A number of new results are obtained for both analysis and synthesis and some old results for networks on graphs are generalized to networks on matroids. It is shown that the principle of duality, which does not hold for networks on graphs but is valid for generalized networks, adds power and insight for solving analysis and synthesis problems. The analysis problem, though formulated in terms of resistance networks, holds for networks containing all types of elements. The synthesis problem that is treated is the crucial one of the realization of a p‐port resistance network.