Advertisement

Symmetry and Complexity

  • Mircea Vasile Diudea
Chapter
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 10)

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.

References

  1. Balasubramanian K (1994) Computer generation of automorphism graphs of weighted graphs. J Chem Inf Comput Sci 34:1146–1150CrossRefGoogle Scholar
  2. Balinski ML (1961) On the graph structure of convex polyhedra in n-space. Pac J Math 11:431–434CrossRefGoogle Scholar
  3. Blatov VA, O’Keeffe M, Proserpio DM (2010) Vertex-, face-, point-, Schläfli-, and Delaney-symbols in nets, polyhedra and tilings: recommended terminology. CrystEngComm 12:44–48CrossRefGoogle Scholar
  4. Bonnet O (1853) Note sur la therorie generale des surfaces. C R Acad Sci Paris 37:529–532Google Scholar
  5. Buekenhout F, Parker M (1998) The number of nets of the regular convex polytopes in dimension ≤4. Disc Math 186:69–94CrossRefGoogle Scholar
  6. 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 479–497Google Scholar
  7. Dehmer M, Grabner M (2013) The discrimination power of molecular identification numbers revisited. MATCH Commun Math Comput Chem 69(3):785–794Google Scholar
  8. 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):311–325CrossRefGoogle Scholar
  9. Dehmer M, Mowshowitz A (2011) Generalized graph entropies. Complexity 17(2):45–50CrossRefGoogle Scholar
  10. Dehmer M, Mowshowitz A, Emmert-Streib F (2013) Advances in network complexity. Wiley-Blackwell, WeinheimCrossRefGoogle Scholar
  11. 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):3287–3300CrossRefGoogle Scholar
  12. Diudea MV (1994) Layer matrices in molecular graphs. J Chem Inf Comput Sci 34:1064–1071CrossRefGoogle Scholar
  13. Diudea MV (2013) Quasicrystals: between spongy and full space filling. In: Diudea MV, Nagy CL (eds) Diamond and related nanostructures. Springer, Dordrecht, pp 335–385CrossRefGoogle Scholar
  14. Diudea MV, Bende A, Nagy CL (2014) Carbon multi-shell cages. Phys Chem Chem Phys 16:5260–5269CrossRefGoogle Scholar
  15. Diudea MV, Bucila VR, Proserpio DM (2013) 1-periodic nanostructures. MATCH Commun Math Comput Chem 70:545–564Google Scholar
  16. Diudea MV, Gutman I, Jäntschi L (2002) Molecular topology. NOVA, New York, NYGoogle Scholar
  17. Diudea MV, Ilić A, Varmuza K, Dehmer M (2010) Network analysis using a novel highly discriminating topological index. Complexity 16(6):32–39CrossRefGoogle Scholar
  18. Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, DordrechtCrossRefGoogle Scholar
  19. Diudea MV, Rosenfeld VR (2017) The truncation of a cage graph. J Math Chem 55:1014–1020CrossRefGoogle Scholar
  20. Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem A 42(6):1283–1294Google Scholar
  21. 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
  22. Euler L (1752–1753) Elementa doctrinae solidorum-Demonstratio nonnullarum insignium proprietatum, quibus solida hedris planis inclusa sunt praedita. Novi Comment Acad Sci I Petropolitanae 4:109–160Google Scholar
  23. Graovac A, Pisanski T (1991) On the Wiener index of a graph. J Math Chem 8:53–62CrossRefGoogle Scholar
  24. Harary F (1969) Graph theory. Addison-Wesley, Reading, MACrossRefGoogle Scholar
  25. Hargittai M, Hargittai I (2010) Symmetry through the eyes of a chemist. Springer, DordrechtGoogle Scholar
  26. Higuchi Y (2001) Combinatorial curvature for planar graphs. J Graph Theory 38:220–229CrossRefGoogle Scholar
  27. Klein DJ (2002) Topo-combinatoric categorization of quasi-local graphitic defects. Phys Chem Chem Phys 4:2099–2110CrossRefGoogle Scholar
  28. Nagy CL, Diudea MV (2009) Nano-studio. Babes-Bolyai Univ, ClujGoogle Scholar
  29. Nagy CL, Diudea MV (2017) Ring signature index. MATCH Commun Math Comput Chem 77(2):479–492Google Scholar
  30. Pirvan-Moldovan A, Diudea MV (2016) Euler characteristic of polyhedral graphs. Croat Chem Acta 89(4):471–479Google Scholar
  31. Razinger M, Balasubramanian K, Munk ME (1993) Graph automorphism perception algorithms in computer-enhanced structure elucidation. J Chem Inf Comput Sci 33:197–201CrossRefGoogle Scholar
  32. 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)Google Scholar
  33. Schulte E (1985) Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures. J Comb Theory Ser A 40(2):305–330CrossRefGoogle Scholar
  34. Schulte E (2014) Polyhedra, complexes, nets and symmetry. Acta Crystallogr A 70:203–216CrossRefGoogle Scholar
  35. Stefu M, Diudea MV (2005) CageVersatile_CVNET. Babes-Bolyai University, ClujGoogle Scholar
  36. Ștefu M, Parvan-Moldovan A, Kooperazan-Moftakhar F, Diudea MV (2015) Topological symmetry of C60-related multi-shell clusters. MATCH Commun Math Comput Chem 74:273–284Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Mircea Vasile Diudea
    • 1
  1. 1.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania

Personalised recommendations