  Evolutionary games on graphs play an important role in the study of evolution
of cooperation in applied biology. Using rigorous mathematical concepts from a
dynamical systems and graph theoretical point of view, we formalize the notions
of attractor, update rules and update orders. We prove results on attractors
for different utility functions and update orders. For complete graphs we
characterize attractors for synchronous and sequential update rules. In other
cases (for $k$-regular graphs or for different update orders) we provide
sufficient conditions for attractivity of full cooperation and full defection.
We construct examples to show that these conditions are not necessary. Finally,
by formulating a list of open questions we emphasize the advantages of our
rigorous approach.
