In this talk, we study the possible bifurcations of periodic orbits by reducing them to graphs. The aforementioned allows to study the genericity of routes to chaos, as well as to analyze their possible complexity by applying graph-theoretic tools. In particular, we will prove there is no upper bound on the possible complexity of routes to chaos, which correlates well with several numerical studies. Time permitting, we will also discuss how our results suggest general fermionic description of bifurcations via the virial expansion, and a universal description via the Rado graph.