With regard to the question from Joanna A Ellis-Monaghan, I don't know if it is quite what was wanted, but John Shrimpton's thesis here was first on the cartesian closed category (actually, topos) of directed graphs, but then applying this to say that the automorphism structure for a (directed ) graph is an internal group in directed graphs and so is a graph-group. This structure is also known as a `pre-cat^1 group' (Brown-Loday, Topology, 26 (1987)), which is equivalent to a pre-crossed module. This has an associated crossed module. So the automorphism structure of a directed graph is a crossed module. This leads to the notion of inner automorphism , and centre of a graph. The paper on this is to appear in the JPAA, and a preprint is available from Bangor (if we print some more!). The aim is to relate the inner autos and centre of a graph to properties of the graph. E.g., in the finite case, the centre is a Z_2 vector space, and its dimension is related to properties of the graph. Ronnie Brown