The graphs of active networks

The results obtained in previous papers are employed to develop the properties of the graphs of linear networks which may contain valves and transformers. A reduced graph is obtained which facilitates the setting up of the determinant of the network. This graph may be split into two separate graphs, termed the current and voltage graphs, such that there is a 1:1 correspondence between every algebraic operation on the H-matrix and every topological operation on the graphs. Thus every problem which can be solved with the one can be solved with the other. The derivative of a graph with respect to a network element corresponds to the derivative of an H-determinant with respect to a network element. With an extended definition of a tree on a network the set of trees on any network is shown to be equal to the nodal determinant of the network. Practical applications of the methods described are outlined.