The semibasis in network analysis and graph theoretical degrees of freedom

Abstract

A characterization of a linear lumped constant network analysis is made by the concept of semibasis. The semibasis <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> , over a coefficient set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> (of rational functions of edge immittances over the real field) is a set of variables such that any variable of the network can be represented by or whole the analysis equations can be written by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</tex> , over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . Graph theoretical conditions of the semibasis over certain familiar coefficient sets such as the field of rational functions, the ring of polynomials, and the sets of quadratic or linear polynomials are given. Considerations on the minimum semibasis, the cardinality of which is called the graph theoretical degree of freedom, over the linear or quadratic polynomial coefficient sets are made. Some graph theoretical unsolved problems related to the principal partition of a graph and the central tree are shown to be essential.