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We prove that counting copies of any graph $F$ in another graph $G$ can beachieved using basic matrix operations on the adjacency matrix of $G$.Moreover, the resulting algorithm is competitive for medium-sized $F$: ouralgorithm recovers the best known complexity for rooted 6-clique counting andimproves on the best known for 9-cycle counting. Underpinning our proofs is thenew result that, for a general class of graph operators, matrix operations arehomomorphisms for operations on rooted graphs.

Fast counting of medium-sized rooted subgraphs