What makes graph automorphisms interesting? I've completed a short course on graph theory and we never studied graph isomorphisms in depth, but I've seen at least a bit of this covered in most graph theory books I've grabbed, that grabbed my attention.
Is there any (big?) connection with another field that makes graph automorphisms interesting (besides the trivial 'automorphisms form a group under composition')? 

For a start, a number of the sporadic simple groups were first discovered as automorphism groups of graphs. The Higman-Sims group is perhaps the simplest example.
