Dimensionality of hypercube clusters

Abstract

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 $$d>3$$ .