A graph-theoretic approach to developing reaction route (RR) networks is described that is a powerful new methodology for investigating the integrated architecture and dynamics of complex chemical and biological reaction systems. A distinct feature of our RR graphs is that while the edges represent reaction steps comprising the mechanism, the nodes simply represent their interconnectivity (not necessarily individual species, as has been the convention so far). As a result, all pathways can be traced simply as trails or walks on the RR graphs. Further, these networks are directly converted into wiring diagrams, allowing the use of electrical circuit theory, including Kirchhoff's current and voltage laws. Thus, they not only provide the reaction network topology but can be used to rigorously analyze network dynamics and the dominant pathways and bottleneck steps. We highlight the approach with three illustrations: (1) an enzymatic reaction system, (2) a heterogeneous catalytic process, and (3) an atmospheric chemistry example. The last example involves 41 reaction steps and provides the most comprehensive chart of atmospheric ozone chemistry to date, while the catalysis illustration with 17 steps provides the most complete view of water-gas shift chemistry.