Dimensionality of hypercube clusters

We investigate clusters of hypercubes in d -dimensional space as a function of the number of vertices, N , and number of cluster shells, L . The number of links, vertices, and exterior vertices exhibit ‘magic number’ characteristics versus L , as the dimension of the space changes. Starting with only the spatial coordinates, we create an adjacency and distance matrix that facilitates the calculation of topological indices, including the Wiener, hyper-Wiener, reverse Wiener, Szeged, Balaban, and Kirchhoff indices. Some known topological formulas for hypercubes when L = 1 are experimentally verified. The asymptotic limits with N of the topological indices are shown to exhibit power law behavior whose exponent changes with d and type of topological index. The asymptotic graph energy is linear with N , whose slope changes with d , and in 2D agrees numerically with previous calculations. Also, the thermodynamic properties such as entropy, free energy, and enthalpy of these lattices show logarithmic behavior with increasing N . The hypercubic clusters are projected onto 3D space when the dimensionality d > 3 .