Fast counting of medium-sized rooted subgraphs

We prove that counting copies of any graph F in another graph G can be achieved using basic matrix operations on the adjacency matrix of G. Moreover, the resulting algorithm is competitive for medium-sized F : our algorithm recovers the best known complexity for rooted 6-clique counting and improves on the best known for 9-cycle counting. Underpinning our proofs is the new result that, for a general class of graph operators, matrix operations are homomorphisms for operations on rooted graphs.