ABSTRACT

Electrical circuit theory is one of the earliest applications of graph theory to a problem in physical science. The dynamic behavior of an electrical circuit is governed by three laws: Kirchhoff’s voltage law, Kirchhoff’s current law, and Ohm’s law. Each element in a circuit is associated with two variables, namely, the current variable and the voltage variable. Kirchhoff’s voltage law requires that the algebraic sum of the voltages around a circuit be zero, and Kirchhoff’s current law requires that the algebraic sum of the currents across a cut be zero. Thus, circuits and cuts define a linear relationship among the voltage variables and a linear relationship among the current variables, respectively. It is for this reason that circuits, cuts, and the vector spaces associated with them have played a major role in the discovery of several fundamental properties of electrical circuits arising from the structure or the interconnection of the circuit elements. Several graph theorists and circuit theorists have immensely contributed to the development of what we may now call the structural theory of electrical circuits. The significance of the results to be presented in this section goes well beyond their application to circuit theory. They will bring out the fundamental duality that exists between circuits and cuts and the influence of this duality on the structural theory of graphs. Most of the results in this section are also relevant to the development of combinatorial optimization theory as well as matroid theory.