Commutativity of the adjacency matrices of graphs

We say that two graphs G 1 and G 2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph K n , n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n − 1 linearly independent commuting adjacency matrices of size n ; and if this bound occurs, then there exists a Hadamard matrix of order n . Finally, we determine the centralizers of some families of graphs.