A number of properties including graph energies and the number of spanning trees of nD-hypercubes have been numerically enumerated. Likewise, the graph spectra, Laplacian spectra, spectral polynomials and Laplacian polynomials of hypercubes up to dimension 23 have been numerically computed without the use of any factorings of characteristic polynomials. Moreover, we have obtained the exact analytic expressions of these indices for any nD-hypercube. We have computed the topological indices and compiled numerical tables of these indices for up to 12-cubes. The focus of the present study is to explore the topological indices of nD-hypercubes in an exhaustive manner in that we consider both distance and degree-based indices including some recently introduced indices based on both distance degrees and vertex degrees such as the Sombor index, RMS-Sombor index, geometric-mean Szeged index as well as edge, vertex, vertex-edge and total versions of these indices. Furthermore, graph spectra, matching polynomials, distance degree sequences, topological indices and other properties of graphs pertinent to chemical applications have been the subject matter of several studies. The subject matters of hypercubes, polycubes and wreath product groups have been dealt with by several researchers over the years, including techniques to obtain topological indices by cut methods which involve subgraphs of hypercubes. The symmetries of non-rigid molecules and the NMR groups of NMR graphs are all connected to symmetries of nonrigid molecules and those of hypercubes. Moreover, the observed transient chirality in water clusters is akin to the spontaneous generation of optical activity of molecules exhibiting rapid internal rotations whose rotation digraphs are finite topologies and Borel fields. Consequently, the assignments of the observed spectra, nuclear spin species, and tunneling splittings in the spectra require symmetries of nonrigid groups molecules that are represented by the automorphism groups of hypercubes or wreath product groups in the nonrigid limit. Water clusters exhibit semi-rigid to nonrigid structures owing to their potential energy surfaces that contain multiple minima divided by surmountable potential energy barriers. In the context of molecular science and drug discovery, hypercubes have been employed to partition big data sets into equivalence classes, biochemical imaging, and in the representations of symmetries of nonrigid molecules and water clusters. Hypercubes are highly symmetric structures that have been the focus of numerous studies due to their varied applications in many fields such as artificial intelligence, parallel architectures, recursive structures, the last Fermat’s theorem, Minkowski norm, neural networks, big data, genetic regulatory networks, the periodic table of elements, phylogenetic trees, moonlighting functions of intrinsically disordered proteins, water clusters, and so forth. The distance degree sequence vectors have been obtained numerically for up to 108-dimensional cubes and their frequencies are found to be in binomial distributions akin to the spectra of n-cubes. We invoke a robust dynamic programming technique to handle the computationally intensive generation of matching polynomials of hypercubes and compute all matching polynomials up to the 6-cube. These results are used to independently validate the exact analytical expressions that we have obtained for the topological indices as well as graph, Laplacian spectra and their polynomials. The symmetries of these hypercubes constitute the hyperoctahedral wreath product groups which also pave the way for the symmetry-based elegant computations. We invoke symmetry-based recursive Hadamard transforms to obtain the graph and Laplacian spectra of nD-hypercubes and the computed numerical results are constructed for up to 23-dimensional hypercubes. Moreover, computations are used to provide independent numerical values for the topological indices of the 11- and 12-cubes. We obtain a large number of degree and distance-based topological indices, graph and Laplacian spectra and the corresponding polynomials, entropies and matching polynomials of n-dimensional hypercubes through the use of Hadamard symmetry and recursive dynamic computational techniques.
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