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.

Characteristic polynomials, spectral-based Riemann-Zeta functions and entropy indices of n-dimensional hypercubes

“…The computed graph energies and spectral-based graph entropies of n-cubes are shown in Table 7 . Graph energies have been the topic of considerable interest over the years [ 39 , 40 ] as they seem to be one of the important spectra-based quantifies of graphs. Kaatz and Bultheel [ 40 ] have shown how graph entropies and graph energies can be used together to compute graph-based free energies and other properties to predict the properties of nanoclusters and hypercube-based clusters.…”

“…The computations of graph-theoretically derived properties of nanomaterials and hypercube clusters such as topological indices, graph spectra and perfect matchings have received considerable attention over the years [34][35][36][37][38][39][40][41][42]. A number of topological indices of the nD-hypercubes have been recently considered [22].…”

Magic Mathematical Relationships for Nanoclusters

“…The simple cubic lattice structure has previously been analyzed by others [4], although without the detail we provide. We have previously studied the d -dimensional hypercube forms [14]. Polonium is the only element which takes the simple cubic structure.…”

“…Here we see that the tabulated data may be succinctly summarized as magic formulas. Also the degree of the polynomial of the index follows the rules from 3D space [14]. Some topological indices for the Platonic solids have previously been published [36].…”

“…An efficient algorithm for the calculation of the distance matrix from the adjacency matrix exists [13]. Using these definitions, we can calculate the Wiener index, W ( G ), the hyper-Wiener index, W W ( G ), the reverse Wiener index rW ( G ), and the Szeged index, Sz ( G ), as previously detailed [14]. These calculations use the the same algorithm that we have previously used for topological indices and nanoclusters [12].…”