TITLE: Graphs with commuting adjacency matrices
QUESTION [10 upvotes]: Let A and B be adjacency matrix of two undirected simple graphs. Can we assign some combinatorial interpretations to this pair of graphs if A and B  commute?

REPLY [3 votes]: $AB=BA$ says that for vertices $u,v$ there are just as many paths $u-w-v$ with $uw\in A$ and $vw\in B$ as with $uv\in B$ and $vw\in A$.