Abstract
Classical geometric symmetry refers to some operations acting on geometric properties of a polyhedron, that leave the object invariant; it is reflected in several molecular properties, such as dipole moments, IR vibrations, 13C-NMR signals etc. Topological symmetry, defined in terms of connectivity, is addressed to constitutive aspects of a molecule and it is involved in synthesis and/or structure elucidation. Complexity refers to the state or quality of being complex/intricate/complicated, or being the union of some interacting (by some rules) parts. Structural complexity is addressed to the organization of matter. It is studied by the aid of graphs associated to molecules/ions/crystals, on which basis several descriptors are calculated. Topological symmetry speaks about structural complexity by considering the type of atoms/vertices and their reciprocal distribution. Genus and rank (or space dimension) of a structure are parameters of complexity acting by means of Euler characteristic of the embedding surface. The fourth chapter introduces to: Euler characteristic, topological symmetry, indices of centrality, ring signature index and Euler characteristic, as reflected in pairs of map operations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Balasubramanian K (1994) Computer generation of automorphism graphs of weighted graphs. J Chem Inf Comput Sci 34:1146–1150
Balinski ML (1961) On the graph structure of convex polyhedra in n-space. Pac J Math 11:431–434
Blatov VA, O’Keeffe M, Proserpio DM (2010) Vertex-, face-, point-, Schläfli-, and Delaney-symbols in nets, polyhedra and tilings: recommended terminology. CrystEngComm 12:44–48
Bonnet O (1853) Note sur la therorie generale des surfaces. C R Acad Sci Paris 37:529–532
Buekenhout F, Parker M (1998) The number of nets of the regular convex polytopes in dimension ≤4. Disc Math 186:69–94
Dehmer M, Emmert-Streib F, Tsoy RY, Varmuza K (2011) Quantifying structural complexity of graphs: information measures in mathematical chemistry. In: Putz M (ed) Quantum frontiers of atoms and molecules. Nova Publishing House, New York, NY, pp 479–497
Dehmer M, Grabner M (2013) The discrimination power of molecular identification numbers revisited. MATCH Commun Math Comput Chem 69(3):785–794
Dehmer M, Grabner M, Mowshowitz A, Emmert-Streib F (2013) An efficient heuristic approach to detecting graph isomorphism based on combinations of highly discriminating invariants. Adv Comput Math 39(2):311–325
Dehmer M, Mowshowitz A (2011) Generalized graph entropies. Complexity 17(2):45–50
Dehmer M, Mowshowitz A, Emmert-Streib F (2013) Advances in network complexity. Wiley-Blackwell, Weinheim
Devos M, Mohar B (2007) An analogue of the Descartes-Euler formula for infinite graphs and Higuchi’s conjecture. Trans Am Math Soc 359(7):3287–3300
Diudea MV (1994) Layer matrices in molecular graphs. J Chem Inf Comput Sci 34:1064–1071
Diudea MV (2013) Quasicrystals: between spongy and full space filling. In: Diudea MV, Nagy CL (eds) Diamond and related nanostructures. Springer, Dordrecht, pp 335–385
Diudea MV, Bende A, Nagy CL (2014) Carbon multi-shell cages. Phys Chem Chem Phys 16:5260–5269
Diudea MV, Bucila VR, Proserpio DM (2013) 1-periodic nanostructures. MATCH Commun Math Comput Chem 70:545–564
Diudea MV, Gutman I, Jäntschi L (2002) Molecular topology. NOVA, New York, NY
Diudea MV, Ilić A, Varmuza K, Dehmer M (2010) Network analysis using a novel highly discriminating topological index. Complexity 16(6):32–39
Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, Dordrecht
Diudea MV, Rosenfeld VR (2017) The truncation of a cage graph. J Math Chem 55:1014–1020
Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem A 42(6):1283–1294
Epstein D (2016) Euler’s formula references (The geometry Junkyard, Theory Group, ICS, UC Irvine). https://www.ics.uci.edu/~eppstein/junkyard/euler/refs.html
Euler L (1752–1753) Elementa doctrinae solidorum-Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita. Novi Comment Acad Sci I Petropolitanae 4:109–160
Graovac A, Pisanski T (1991) On the Wiener index of a graph. J Math Chem 8:53–62
Harary F (1969) Graph theory. Addison-Wesley, Reading, MA
Hargittai M, Hargittai I (2010) Symmetry through the eyes of a chemist. Springer, Dordrecht
Higuchi Y (2001) Combinatorial curvature for planar graphs. J Graph Theory 38:220–229
Klein DJ (2002) Topo-combinatoric categorization of quasi-local graphitic defects. Phys Chem Chem Phys 4:2099–2110
Nagy CL, Diudea MV (2009) Nano-studio. Babes-Bolyai Univ, Cluj
Nagy CL, Diudea MV (2017) Ring signature index. MATCH Commun Math Comput Chem 77(2):479–492
Pirvan-Moldovan A, Diudea MV (2016) Euler characteristic of polyhedral graphs. Croat Chem Acta 89(4):471–479
Razinger M, Balasubramanian K, Munk ME (1993) Graph automorphism perception algorithms in computer-enhanced structure elucidation. J Chem Inf Comput Sci 33:197–201
Schläfli L (1901) Theorie der vielfachen Kontinuität Zürcher und Furrer, Zürich (Reprinted in: Ludwig Schläfli, 1814–1895, Gesammelte Mathematische Abhandlungen, Band 1, 167–387, Verlag Birkhäuser, Basel, 1950)
Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Comb Theory Ser A 40(2):305–330
Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Crystallogr A 70:203–216
Stefu M, Diudea MV (2005) CageVersatile_CVNET. Babes-Bolyai University, Cluj
Ștefu M, Parvan-Moldovan A, Kooperazan-Moftakhar F, Diudea MV (2015) Topological symmetry of C60-related multi-shell clusters. MATCH Commun Math Comput Chem 74:273–284
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Diudea, M.V. (2018). Symmetry and Complexity. In: Multi-shell Polyhedral Clusters. Carbon Materials: Chemistry and Physics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-64123-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-64123-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64121-8
Online ISBN: 978-3-319-64123-2
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)