The path matrix and switching functions

Abstract

Part I introduces a new matrix, the path matrix, in the theory of linear graph. The matrix is defined and its properties are given in a number of lemmas and theorems. They include (1) a relation between the path matrix and the incidence matrix of a connected graph (Theorem 1) and (2) the rank of the path matrix, which is shown to be e − v + 2 − S (Theorem 3), where e is the number of edges, v the number of vertices and S the number of independent circuits contained in the “path-isolated subgraphs” of a connected graph. It is clear that there is a one-to-one correspondence between the union of all paths between two vertices and a two-terminal switching function. An explicit formula is obtained which relates a switching function to the topology of the graph.The formula shows that a switching function is expressible in terms of the path matrix in much the same manner as the determinant of the admittance matrix is expressed in terms of the vertex matrix. Part II demonstrates a synthesis procedure which obtains a graph from a given path matrix without first converting it into a circuit matrix by means of the “free element” as did Okada, Seshu and Gould.The procedure is based on a number of relations between paths and cut sets, which are derived in the paper. As there is a one-one correspondence between a path matrix and a switching function of n variables, the procedure is directly applicable without modification to the synthesis of a switching function of n variables using n contacts.