On Vietoris–Rips complexes of hypercube graphs

We describe the homotopy types of Vietoris–Rips complexes hypercube graphs at small scale parameters. In more detail, let $$Q_n$$ be vertex set graph with $$2^n$$ vertices, equipped shortest path metric. Equivalently, is all binary strings length n, Hamming distance. The complex parameter zero points, and one graph, which equivalent to a wedge sum circles. show that two 3-spheres, furthermore we provide formula for number 3-spheres. Many questions about larger parameters remain open.