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We give a self-contained introduction to the theory of directed graphs,leading up to the relationship between the Perron-Frobenius eigenvectors of agraph and its autocatalytic sets. Then we discuss a particular dynamical systemon a fixed but arbitrary graph, that describes the population dynamics ofspecies whose interactions are determined by the graph. The attractors of thisdynamical system are described as a function of graph topology. Finally weconsider a dynamical system in which the graph of interactions of the speciescoevolves with the populations of the species. We show that this systemexhibits complex dynamics including self-organization of the network byautocatalytic sets, growth of complexity and structure, and collapse of thenetwork followed by recoveries. We argue that a graph theoretic classificationof perturbations of the network is helpful in predicting the future impact of aperturbation over short and medium time scales.

Graph theory and the evolution of autocatalytic networks