Turing-complete computation models on graphs
As in my comment, the configuration graph (Chapter 4.1.1. of Arora-Barak) is a simple way to view a Nondeterministic Turing Machine (NTM) as a graph. The vertices of the graph are labeled by configurations: the internal state of the NTM, the contents of the tape, and head positions. There is a  directed edge from vertex $c_1$ to vertex $c_2$ if the NTM can move from $c_1$ to $c_2$ in one step. 

This is especially useful for arguing about space-bounded computation. If the space used by the NTM is bounded by $s(n)$, then on input size $n$ you only need to consider a configuration graph with $O(2^{s(n)})$ vertices. Then a lot of complexity results follow from algorithms for directed $s$-$t$ connectivity: Savitch's theorem (NPSPACE = PSPACE) follows from an algorithm that uses $O(\log^2 |V|)$ space, PSPACE in EXP (or, scaling down, L in P) follows from DFS,  etc. Similarly, you can show that directed connectivity is complete for NL under L reductions, so NL is equal to L if and only if there is a logspace algorithm for directed $s$-$t$ connectivity.