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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References
Ashrafi AR, Jalali M, Ghorbani M, Diudea MV (2008) Computing PI and Omega polynomials of an infinite family of fullerenes. MATCH Commun Math Comput Chem 60:905–916
Baker KA, Fishburn P, Roberts FS (1971) Partial orders of dimension 2. Networks 2(1):11–28
Balinski ML (1961) On the graph structure of convex polyhedra in n-space. Pac J Math 11(2):431–434
Coxeter HSM (1973) Regular polytopes, 3rd edn. Dover, New York
Coxeter HSM (1974) Regular complex polytopes. Cambridge University Press, Cambridge
Coxeter HSM (1982) Ten toroids and fifty-seven hemi-dodecahedra. Geom Dedicata 13:87–99
Diudea MV (1997) Cluj matrix invariants. J Chem Inf Comput Sci 37:300–305
Diudea MV (2002) Graphenes from 4-Valent tori. Bull Chem Soc Jpn 75:487–492
Diudea MV (2006) Omega polynomial. Carpath J Math 22:43–47
Diudea MV (2009) Cluj polynomials. J Math Chem 45:295–308
Diudea MV (2010a) Nanomolecules and nanostructures – polynomials and indices. In: MCM Ser. 10. University of Kragujevac, Kragujevac
Diudea MV (2010b) Counting polynomials and related indices by edge cutting procedures. MATCH Commun Math Comput Chem 64(3):569–590
Diudea MV, Klavžar S (2010) Omega polynomial revisited. Acta Chem Sloven 57:565–570
Diudea MV, Parv B, Ursu O (2003) TORUS software. Babes-Bolyai University, Cluj
Diudea MV, Cigher S, John PE (2008) Omega and related counting polynomials. MATCH Commun Math Comput Chem 60:237–250
Djoković DŽ (1973) Distance preserving subgraphs of hypercubes. J Combin Theory Ser B 14:263–267
Euler L (1752–1753) Elementa doctrinae solidorum. Novi Comm Acad Sci Imp Petrop 4:109–160
Gutman I (1994) A formula for the wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY 27:9–15
Gutman I, Klavžar S (1995) An algorithm for the calculation of the szeged index of benzenoid hydrocarbons. J Chem Inf Comput Sci 35:1011–1014
Harary F (1969) Graph theory. Addison-Wesley, Reading
Hillis WD (1982) New computer architectures and their relationship to physics or why computer science is no good. Int J Theor Phys 21(3/4):255–262
John PE, Vizitiu AE, Cigher S, Diudea MV (2007) CI Index in tubular nanostructures. MATCH Commun Math Comput Chem 57:479–484
Khalifeh MH, Yousefi-Azari H, Ashrafi AR (2008) A matrix method for computing szeged and vertex PI indices of join and composition of graphs. Linear Algebra Appl 429:2702–2709
Klavžar S (2008a) Abrid’s eye view of the cut method and a survey of its applications in chemical graph theory. MATCH Commun Math Comput Chem 60:255–274
Klavžar S (2008b) Some comments on co graphs and CI index. MATCH Commun Math Comput Chem 59:217–222
Lijnen E, Ceulemans A (2005) The symmetry of the Dyck graph: group structure and molecular realization (Chap. 14). In: Diudea MV (ed) Nanostructures: novel architecture. Nova, New York, pp 299–309
Mansour T, Schork M (2009) The vertex PI index and szeged index of bridge graphs. Discr Appl Math 157:1600–1606
McMullen P, Schulte P (2002) Abstract Regular Polytopes, 1st edn. Cambridge University Press, Cambridge
Nagy CsL, Diudea MV (2009) Nano studio software. “Babes-Bolyai” University, Cluj
Parvan-Moldovan A, Diudea MV (2015) Hyper-tubes of hyper-cubes. Iran J Math Chem 6:163–168
Pirvan-Moldovan A, Diudea MV (2016) Euler characteristic of polyhedral graphs. Croat Chem Acta (accepted)
Schläfli L (1901) Theorie der vielfachen Kontinuität Zürcher und Furrer, Zürich (Reprinted in: Ludwig Schläfli, 1814–1895, Gesammelte Mathematische Abhandlungen, Band 1, 167–387, VerlagBirkhäuser, Basel, 1950)
Schulte E (1985) Regular incidence-polytopes with euclidean or toroidal faces and vertex-figures. J Combin Theory Ser A 40(2):305–330
Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Cryst A70:203–216
Szymanski TH (1989) On the permutation capability of a circuit-switched hypercube. In: Proceedings of the international conference on parallel processing 1, IEEE Computer Society Press, Silver Spring, MD, pp 103–110
Winkler PM (1984) Isometric embedding in products of complete graphs. Discret Appl Math 8:209–212
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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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