  We consider graph Turing machines, a model of parallel computation on a
graph, in which each vertex is only capable of performing one of a finite
number of operations. This model of computation is a natural generalization of
several well-studied notions of computation, including ordinary Turing
machines, cellular automata, and parallel graph dynamical systems. We analyze
the power of computations that can take place in this model, both in terms of
the degrees of computability of the functions that can be computed, and the
time and space resources needed to carry out these computations. We further
show that properties of the underlying graph have significant consequences for
the power of computation thereby obtained. In particular, we show that every
arithmetically definable set can be computed by a graph Turing machine in
constant time, and that every computably enumerable Turing degree can be
computed in constant time and linear space by a graph Turing machine whose
underlying graph has finite degree.
