A simplified explicit solution of networks with two internal nodes

In an earlier paper, a general procedure was described for obtaining explicit solutions for the element values of single-element-kind networks. A matrix formulation was given for obtaining a system of simultaneous multilinear equations representing the externally observable behavior of the network. In this paper a “path” concept is introduced which leads to a systematic topological method for obtaining the same “modified” system of equations directly from the graph. This method avoids the cumbersome problem, implicit in the well-known topological formulas, of counting trees. The pertinent system determinant is expressed by compact topological representations derived from the key subgraph of the network. In addition, a greatly simplified solution procedure for the “key” subset of the system of equations applicable to such networks with two internal nodes is given. An example is included.