A theory of even functionals and their algorithmic applications

We present a theory of even functionals of degree k. Even functional are homogeneous polynomials which are invariant with respect to permutations and reflections. These are evaluated on real symmetric matrices. Important examples of even functionals include functions for enumerating embeddings of graphs with k edges into a weighted graph with arbitrary (positive or negative) weights, and computing kth moments (expected values of kth powers) of a binary form. This theory provides a uniform approach for evaluating even functionals and links their evaluation with expressions that have matrices as operands. In particular, we show that any even functional of degree less than 7 can be computed in time sufficient to multiply two n × n matrices.