Applications of dimensionality reduction and exponential sums to graph automorphism

Given a connected undirected graph, we associate a simplex with it such that two graphs are isomorphic if and only if their corresponding simplices are congruent under an isometric map. In the first part of the paper, we study the effectiveness of a dimensionality reduction approach to Graph Automorphism. More precisely, we show that orthogonal projections of the simplex onto a lower dimensional space preserves an automorphism if and only if the space is an invariant subspace of the automorphism. This insight motivates the study of invariant subspaces of an automorphism. We show the existence of some interesting (possibly lower dimensional) invariant subspaces of an automorphism. As an application of the correspondence between a graph and its simplex, we show that there are roughly a quadratic number of invariants that uniquely characterize a connected undirected graph up to isomorphism. In the second part, we present an exponential sum formula for counting the number of automorphisms of a graph and study the computation of this formula. As an application, we show that for a fixed prime p and any graph G, we can count, modulo p, the number of permutations that violate a multiple of p edges in G in polynomial time.