Ela the A-like Matrices for a Hypercube

Let D denote a positive integer and let QD denote the graph of the D-dimensional hypercube. Let X denote the vertex set of QD and let A ∈ MatX(R) denote the adjacency matrix of QD. A matrix B ∈ MatX(R) is called A-like whenever both (i) BA = AB; (ii) for all x, y ∈ X that are not equal or adjacent, the (x, y)-entry of B is zero. Let L denote the subspace of MatX(R) consisting of the A-like elements. The subspace L is decomposed into the direct sum of its symmetric part and antisymmetric part. A basis for each part is given. The dimensions of the symmetric part and antisymmetric part are D + 1 and ( D 2 )