Lower Bounds on the Homology of Vietoris–Rips Complexes of Hypercube Graphs

We provide novel lower bounds on the Betti numbers of Vietoris–Rips complexes of hypercube graphs of all dimensions and at all scales. In more detail, let Q_n be the vertex set of 2^n vertices in the n -dimensional hypercube graph, equipped with the shortest path metric. Let VR(Q_n;r) be its Vietoris–Rips complex at scale parameter r ≥ 0 , which has Q_n as its vertex set, and all subsets of diameter at most r as its simplices. For integers r更多