Abstract
Hypercube Q n is an n-dimensional analogue of the Cube (n = 3); it is a regular graph of degree n and can be obtained by the Cartesian product of P 2 graph or can be drawn as a Hӓsse diagram. Hypercube is a regular polytope in the space of any number of dimensions; its Schläfli symbols is {4,3n−2} and has as a dual the n-orthoplex {3n−2,4}. The number of k-cubes contained in an n-cube Q n (k) comes from the coefficients of (2k + 1)n. A “spongy hypercube” G(d, v, Q n + 1) = G(d, v) □n P 2 is defined in this chapter; on each edge of the original polyhedral graph, a local hypercube Q n is evolved; these hypercubes are incident in a hypervertex, according to the original degree, d. It means that, in a spongy hypercube, the original 2-faces are not counted.
The k-faces of a spongy hypercube are combinatorially counted from the previous rank faces; their alternating summation accounts for the genus of the embedded surface. Tubular and toroidal hypercubes were also designed. Analytical formulas for counting Omega and Cluj polynomials, respectively, in hypercubes were derived.
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Diudea, M.V. (2018). Spongy Hypercubes. In: Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-64123-2_11
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