The Topological Conditions: The Properties of the Pair of Conjugate Tress

Graph Theory is used for the study of real-world Systems possessing a binary relation between elements of a certain set within the system description. Among other discipline (Circuit Theory) has received outstanding contributions from the study of graphs. Some of the contributions may be found in the solution to specific problems related to electrical network analysis, nonlinear circuit theory, circuit diagnosis and circuit synthesis [1,2]. A line of research developed in recent years has attempted to determine the relationships between the topology of a circuit and its functionality, which has derived in a deeper knowledge in the general problem of nonlinear circuits [3,4]. One important work has been reported in [5] and [6], where a topological criterion for the existence and uniqueness of the solution of linear circuits has been proposed. This criterion is based on two definitions of Graph Theory: the pair of conjugate trees and the uniform partial orientation of the resistors. In this paper, the attention is focused on several properties of the pair of conjugate trees; these are highlighted by looking at the resulting loop and cutset matrices.