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Cluj Polynomial in Nanostructures

  • Mircea V. DiudeaEmail author
  • Mahboubeh Saheli
Chapter
Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 9)

Abstract

Cluj polynomial, developed in 2009–2010 in Cluj, Romania, counts the vertex proximities in a connected graph. Definitions and relations with other polynomials and topological indices are given. Within this chapter, Cluj and related polynomials are computed in several 3D nanostructures and crystal networks and analytical formulas as well as examples are given.

Keywords

Bipartite Graph Topological Index Graphite Sheet Related Polynomial Pairwise Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Adachi M, Murata Y, Okada I, Yoshikawa S (2003) Formation of Titania Nanotubes and Applications for Dye-Sensitized Solar Cells. J Electrochem Soc 150:488–493CrossRefGoogle Scholar
  2. Alipour MA, Ashrafi AR (2009) A numerical method for computing the Wiener index of one-heptagonal carbon nanocone. J Comput Theor Nanosci 6:1204–1207CrossRefGoogle Scholar
  3. Ashrafi AR, Ghorbani M, Jalali M (2008) The vertex PI and Szeged indices of an infinite family of fullerenes. J Theor Comput Chem 7:221–231CrossRefGoogle Scholar
  4. Diudea MV (1997a) Cluj matrix CJu: source of various graph descriptors. MATCH Commun Math Comput Chem 35:169–183Google Scholar
  5. Diudea MV (1997b) Cluj matrix invariants. J Chem Inf Comput Sci 37:300–305CrossRefGoogle Scholar
  6. Diudea MV (1999) Valencies of property. Croat Chem Acta 72:835–851Google Scholar
  7. Diudea MV (ed) (2005) Nanostructures, Novel Architecture. Nova, New YorkGoogle Scholar
  8. Diudea MV (2009) Cluj polynomials. J Math Chem 45:295–308CrossRefGoogle Scholar
  9. Diudea MV (2010a) Counting polynomials in partial cubes. In: Gutman I, Furtula B (eds) Novel molecular structure descriptors-theory and applications I. Univ Kragujevac, Kragujevac, pp 191–215Google Scholar
  10. Diudea MV (2010b) Counting polynomials and related indices by edge cutting procedures. In: Gutman I, Furtula B (eds) Novel molecular structure descriptors-theory and applications II. Univ Kragujevac, Kragujevac, pp 57–78Google Scholar
  11. Diudea MV, Katona G (1999) Molecular topology of dendrimers, in: G A Newkome Ed. Adv Dendritic Macromol 4:135–201CrossRefGoogle Scholar
  12. Diudea MV, Klavžar S (2010) Omega polynomial revisited. Acta Chem Sloven 57:565–570Google Scholar
  13. Diudea MV, Nagy CL (2007) Periodic Nanostructures. Springer, DordrechtCrossRefGoogle Scholar
  14. Diudea MV, Ursu O (2003) Layer matrices and distance property descriptors. Indian J Chem 42A:1283–1294Google Scholar
  15. Diudea MV, Parv B, Gutman I (1997) Detour-Cluj matrix and derived invariants. J Chem Inf Comput Sci 37:1101–1108CrossRefGoogle Scholar
  16. Diudea MV, Gutman I, Jäntschi L (2002) Molecular Topology. Nova, New YorkGoogle Scholar
  17. Diudea MV, Florescu MS, Khadikar PV (2006) Molecular Topology and Its Applications. Eficon, BucharestGoogle Scholar
  18. Diudea MV, Vizitiu AE, Janežič D (2007) Cluj and related polynomials applied in correlating studies. J Chem Inf Model 47:864–874CrossRefGoogle Scholar
  19. Diudea MV, Cigher S, John PE (2008) Omega and related counting polynomials. MATCH Commun Math Comput Chem 60:237–250Google Scholar
  20. Diudea MV, Ilić A, Ghorbani M, Ashrafi AR (2010a) Cluj and PIv polynomials. Croat Chem Acta 83:283–289Google Scholar
  21. Diudea MV, Dorosti N, Iranmanesh A (2010b) Cluj CJ polynomial and indices in a dendritic molecular graph. Carpath J Math 4:247–253Google Scholar
  22. Dorosti N, Iranmanesh A, Diudea MV (2009) Computing the Cluj index of dendrimer nanostars. MATCH Commun Math Comput Chem 62:389–395Google Scholar
  23. Du GH, Chen Q, Che RC, Yuan ZY, Peng LM (2001) Preparation and Structure Analysis of Titanium Oxide Nanotubes. Appl Phys Lett 79:3702–3704CrossRefGoogle Scholar
  24. Ebbesen TW (1998) Cones and tubes: geometry in the chemistry of carbon. Acc Chem Res 31:558–566CrossRefGoogle Scholar
  25. Enyashin AN, Seifert G (2005) Structure stability and electronic properties of TiO2 nanostructures. Phys Stat Sol 242:1361–1370CrossRefGoogle Scholar
  26. Gong D, Grimes CA, Varghese OK, Hu W, Singh RS, Chen Z, Dickey EC (2001) Titanium oxide nanotube arrays prepared by anodic oxidation. J Mater Res 16:3331–3334CrossRefGoogle Scholar
  27. Grimes CA, Ong KG, Varghese OK, Yang X, Mor G, Paulose M, Dickey EC, Ruan C, Pishko MV, Kendig JW, Mason AJ (2003) A Sentinel Sensor Network for Hydrogen Sensing. Sensors 3:69–82CrossRefGoogle Scholar
  28. Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes NY 27:9–15Google Scholar
  29. Gutman I, Klavžar S (1995) An algorithm for the calculation of the Szeged index of benzenoid hydrocarbons. J Chem Inf Comput Sci 35:1011–1014CrossRefGoogle Scholar
  30. Harary F (1969) Graph theory. Addison-Wesley, ReadingGoogle Scholar
  31. Hecht S, Frechet JMJ (2001) Dendritic encapsulation of function: applying nature’s site isolation principle from biomimetics to materials science. Angew Chem Int Ed 40:74–91CrossRefGoogle Scholar
  32. Hoyer P (1996) Formation of a titanium dioxide nanotube array. Langmuir 12:1411–1413CrossRefGoogle Scholar
  33. Ilić A (2010) On the extremal graphs with respect to the vertex PI index. Appl Math Lett 23:1213–1217CrossRefGoogle Scholar
  34. Imai H, Takei Y, Shimizu K, Matsuda M, Hirashima H (1999) Direct preparation of anatase TiO2 nanotubes in porous alumina membranes. J Mater Chem 9:2971–2972CrossRefGoogle Scholar
  35. Imai H, Matsuta M, Shimizu K, Hirashima N, Negishi N (2002) Morphology transcription with TiO2 using chemical solution growth and its application for photocatalysts. Solid State Ion 151:183–187CrossRefGoogle Scholar
  36. Ivanovskaya VV, Enyashin AN, Ivanovskii AL (2003) Electronic structure of single-walled TiO2 and VO2 nanotubes. Mendeleev Comm 13:5–7CrossRefGoogle Scholar
  37. Ivanovskaya VV, Enyashin AN, Ivanovskii AL (2004) Nanotubes and fullerene-like molecules based on TiO2 and ZrS2: Electronic structure and chemical bond Russ. J Inorg Chem 49:244–251Google Scholar
  38. Kasuga T, Hiramatsu M, Hoson A, Sekino T, Niihara K (1998) Formation of titanium oxide nanotube. Langmuir 14:3160–3163CrossRefGoogle Scholar
  39. Kasuga T, Hiramatsu M, Hoson A, Sekino T, Niihara K (1999) Titania nanotubes prepared by chemical processing. Adv Mater 11:1307–1311CrossRefGoogle Scholar
  40. Khadikar PV (2000) On a novel structural descriptor PI. Nat Acad Sci Lett 23:113–118Google Scholar
  41. Khalifeh MH, Yousefi-Azari H, Ashrafi AR (2008a) Vertex and edge PI indices of Cartesian product graphs. Discret Appl Math 156:1780–1789CrossRefGoogle Scholar
  42. Khalifeh MH, Yousefi-Azari H, Ashrafi AR (2008b) A matrix method for computing Szeged and vertex PI indices of join and composition of graphs. Linear Algebra Appl 429:2702–2709CrossRefGoogle Scholar
  43. Klavžar S (2008) A brid’s eye view of the cut method and a survey of its applications in chemical graph theory. MATCH Commun Math Comput Chem 60:255–274Google Scholar
  44. Kobayashi S, Hanabusa K, Hamasaki N, Kimura M, Shirai H (2000) Preparation of TiO2 hollow-fibers using supramolecular assemblies. Chem Mater 12:1523–1525CrossRefGoogle Scholar
  45. Krishnan A, Dujardin E, Treacy MMJ, Hugdahl J, Lynum S, Ebbesen TW (1997) Graphitic cones and the nucleation of curved carbon surfaces. Nature 388:451–454CrossRefGoogle Scholar
  46. Lakshmi BB, Dorhout PK, Martin CR (1997) Sol–gel template synthesis of semiconductor nanostructures. Chem Mater 9:857–872CrossRefGoogle Scholar
  47. Li XH, Liu WM, Li HL (2003) Template synthesis of well-aligned titanium dioxide nanotubes. Appl Phys A 80:317–320CrossRefGoogle Scholar
  48. Lin CH, Chien SH, Chao JH, Sheu CY, Cheng YC, Huang YJ, Tsai CH (2002) The synthesis of sulfated titanium oxide nanotubes. Catal Lett 80:153–159CrossRefGoogle Scholar
  49. Liu SM, Gan LM, Liu LH, Zhang WD, Zeng HC (2002) Synthesis of single-crystalline TiO2 nanotubes. Chem Mater 14:1391–1397CrossRefGoogle Scholar
  50. Mansour T, Schork M (2009) The vertex PI index and Szeged index of bridge graphs. Discr Appl Math 157:1600–1606CrossRefGoogle Scholar
  51. Mor GK, Varghese OK, Paulose M, Mukherjee N, Grimes CA (2003) Fabrication of tapered, conical-shaped titania nanotubes. J Mater Res 18:2588–2593CrossRefGoogle Scholar
  52. Mor GK, Carvalho MA, Varghese OK, Pishko MV, Grimes CA (2004) A room − temperature TiO2 − nanotube hydrogen sensor able to self-clean photoactively from environmental contamination. J Mater Res 19:628–634CrossRefGoogle Scholar
  53. Patzke GR, Krumeich F, Nesper R (2002) Oxidic nanotubes and nanorods − anisotropic modules for a future nanotechnology. Angew Chem Int Ed 41:2446–2461CrossRefGoogle Scholar
  54. Peng T, Yang H, Chang G, Dai K, Hirao K (2004) Synthesis of bamboo-shaped TiO2 nanotubes in nanochannels of porous aluminum oxide membrane. Chem Lett 33:336–337CrossRefGoogle Scholar
  55. Rao CNR, Nath M (2003) Inorganic nanotubes. Dalton Trans 1:1–24CrossRefGoogle Scholar
  56. Seo DS, Lee JK, Kim H (2001) Preparation of nanotube − shaped TiO2 powder. J Cryst Growth 229:428–432CrossRefGoogle Scholar
  57. Shi YL, Zhang XG, Li HL (2002) Liquid phase deposition templates synthesis of nanostructures of anatase titania. Mater Sci Engin A 333:239–242CrossRefGoogle Scholar
  58. Sun J, Gao L, Zhang Q (2003) TiO2 tubes synthesized by using ammonium sulfate and carbon nanotubes as templates. J Mater Sci Lett 22:339–341CrossRefGoogle Scholar
  59. Tenne R (2002) Inorganic Nanotubes and Fullerene-Like Materials. Chem Eur J 8:5296–5304CrossRefGoogle Scholar
  60. Ursu O, Diudea MV (2005) TOPOCLUJ software program. Babes-Bolyai University, ClujGoogle Scholar
  61. Varghese OK, Gong D, Paulose M, Grimes CA, Dickey EC (2003a) Crystallization and high − temperature structural stability of titanium oxide nanotube arrays. J Mater Res 18:156–165CrossRefGoogle Scholar
  62. Varghese OK, Gong D, Paulose M, Ong KG, Dickey EC, Grimes CA (2003b) Extreme changes in the electrical resistance of titania nanotubes with hydrogen exposure. Adv Mater 15:624–627CrossRefGoogle Scholar
  63. Varghese OK, Gong D, Paulose M, Ong KG, Grimes CA (2003c) Hydrogen sensing using titania nanotubes. Sens Actuators B 93:338–344CrossRefGoogle Scholar
  64. Vizitiu AE, Diudea MV (2006) Conetori of high genera. Studia Univ Babes-Bolyai 51(1):39–48Google Scholar
  65. Vizitiu AE, Diudea MV (2008) Omega and Theta polynomials in conical nanostructures. MATCH Commun Math Comput Chem 60:927–933Google Scholar
  66. Vizitiu AE, Diudea MV (2009) Cluj polynomial description of TiO2 nanostructures. Studia Univ Babes Bolyai 54(1):173–180Google Scholar
  67. Wang YQ, Hu GQ, Duan XF, Sun HL, Xue QK (2002) Microstructure and formation mechanism of titanium dioxide nanotubes. Chem Phys Lett 365:427–431CrossRefGoogle Scholar
  68. Wang W, Varghese OK, Paulose M, Grimes CA (2003) Synthesis of CuO and Cu2O crystalline nanowires using Cu(OH)2 nanowire templates. J Mater Res 18:2756–2759CrossRefGoogle Scholar
  69. Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20CrossRefGoogle Scholar
  70. Yao BD, Chan YF, Zhang XY, Zhang WF, Yang ZY, Wang N (2003) Formation mechanism of TiO2 nanotubes. Appl Phys Lett 82:281–283CrossRefGoogle Scholar
  71. Zhang S, Zhou J, Zhang Z, Du Z, Vorontsov AV, Jin Z (2000) Morphological structure and physicochemical properties of nanotube TiO2. Chin Sci Bull 45:1533–1536CrossRefGoogle Scholar
  72. Zhang M, Bando Y, Wada K (2001) Sol–gel template preparation of TiO2 nanotubes and nanorods. J Mater Sci Lett 20:167–170CrossRefGoogle Scholar
  73. Zhou Y, Li H, Koltypin Y, Hacohen YR, Gedanken A (2001) Sonochemical synthesis of titania whiskers and nanotubes. Chem Commun 24:2616–2617CrossRefGoogle Scholar
  74. Zhou Y, Cao L, Zhang F, He B, Li H (2003) Lithium insertion into TiO2 nanotube prepared by the hydrothermal process. J Electrochem Soc 150A:1246–1249CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Chemistry, Faculty of Chemistry and Chemical EngineeringBabes-Bolyai UniversityCluj-NapocaRomania
  2. 2.Department of Pure MathematicsUniversity of KashanKashanIran

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