Automorphism capture a natural notion of symmetry of graphs. As a result, they can be used to speed up algorithms that would otherwise run slowly by chopping down the search space.
For example, integer programming is usually solved via branch-and-bound. However, if an equation is degenerate this can take far longer than necessary to run, because it has to keep checking symmetric parts of the tree. We can use graph automorphisms to compute the orbits of variables in the linear programming problem, and then treat parts with the same orbit as identical. A recent application of such techniques to MILP can be read here.