Fast counting of medium-sized rooted subgraphs

TL;DR

This paper introduces a matrix-based algorithm for counting medium-sized rooted subgraphs in graphs, achieving state-of-the-art complexity for certain subgraph types and providing a new theoretical framework for graph operators.

Contribution

It presents a novel matrix operation approach for subgraph counting, improving complexity bounds and establishing a theoretical link between matrix operations and rooted graph homomorphisms.

Findings

Achieves best known complexity for rooted 6-clique counting

Improves on existing methods for 9-cycle counting

Provides a new theoretical foundation for matrix operations on rooted graphs

Abstract

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.