A theory and algorithm for reaction route (RR) network analysis is developed in analogy with electrical networks and is based on the combined use of RR theory, graph theory, and Kirchhoff's laws. The result is a powerful new approach of "RR graphs" that is useful in not only topological representation of complex reactions and mechanisms but, when combined with techniques of electrical network analysis, is able to provide revealing insights into the mechanism as well as the kinetics of the overall reactions involving multiple elementary reaction steps including the effect of topological constraints. Unlike existing graph theory approaches of reaction networks, the approach developed here is suitable for linear as well as nonlinear kinetic mechanisms and for single and multiple overall reactions. The theoretical approach for the case of a single overall reaction involving minimal kinetic mechanisms (unit stoichiometric numbers) is developed in Part I of this series followed by its application to examples of heterogeneous and enzyme catalytic reactions in Part II.