Omega Polynomial in Nanostructures

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


Omega polynomial, developed in 2006 in Cluj, Romania, counts the number of topologically parallel edges in all the opposite edge stripes of a connected graph. Definitions and relations with other polynomials and well-known topological indices are given. Within this chapter, omega polynomial is computed in several 3D nanostructures and crystal networks, and analytical formulas as well as examples are given. This polynomial is viewed as an alternative to the crystallographic description.


Bipartite Graph Plane Graph General Graph Complete Bipartite Graph Parallel Edge 
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.


  1. Ashrafi AR, Ghorbani M, Jalali M (2008a) Computing sadhana polynomial of V-phenylenic nanotubes and nanotori. Indian J Chem 47A:535–537Google Scholar
  2. Ashrafi AR, Jalali M, Ghorbani M, Diudea MV (2008b) Computing PI and omega polynomials of an infinite family of fullerenes. MATCH Commun Math Comput Chem 60:905–916Google Scholar
  3. Ashrafi AR, Manoochehrian B, Yousefi-Azari H (2006) On the PI polynomial of a graph. Util Math 71:97–108Google Scholar
  4. Aurenhammer F, Hagauer J (1995) Recognizing binary Hamming graphs in O(n2logn) time. Math Syst Theory 28:387–396CrossRefGoogle Scholar
  5. Brešar B (2001) Partial Hamming graphs and expansion procedures. Discret Math 237:13–27CrossRefGoogle Scholar
  6. Brešar B, Imrich W, Klavžar S (2003) Fast recognition algorithms for classes of partial cubes. Discret Appl Math 131:51–61CrossRefGoogle Scholar
  7. Diudea MV (2005) Nanostructures novel architecture. Nova, New YorkGoogle Scholar
  8. Diudea MV (2006) Omega polynomial. Carpath J Math 22:43–47Google Scholar
  9. Diudea MV (2010a) Counting polynomials and related indices by edge cutting procedures. MATCH Commun Math Comput Chem 64(3):569–590Google Scholar
  10. Diudea MV (2010b) Counting polynomials in partial cubes. In: Gutman I, Furtula B (eds) Novel molecular structure descriptors – theory and applications II. University of Kragujevac, Kragujevac, pp 191–215Google Scholar
  11. Diudea MV (2010c) Nanomolecules and nanostructures – polynomials and indices. MCM Ser. 10. University of Kragujevac, KragujevacGoogle Scholar
  12. Diudea MV, Klavžar S (2010) Omega polynomial revisited. Acta Chem Sloven 57:565–570Google Scholar
  13. Diudea MV, Cigher S, John PE (2008) Omega and related counting polynomials. MATCH Commun Math Comput Chem 60:237–250Google Scholar
  14. Diudea MV, Dorosti N, Iranmanesh A (2010a) Cluj CJ polynomial and indices in a dendritic molecular graph. Studia UBB Chemia LV 4:247–253Google Scholar
  15. Diudea MV, Vizitiu AE, Mirzargar M, Ashrafi AR (2010b) Sadhana polynomial in nano-dendrimers. Carpath J Math 26:59–66Google Scholar
  16. Diudea MV, Florescu MS, Khadikar PV (2006) Molecular topology and its applications. EFICON, BucharestGoogle Scholar
  17. Diudea MV, Ilić A (2009) Note on omega polynomial. Carpath J Math 25(2):177–185Google Scholar
  18. Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, DordrechtCrossRefGoogle Scholar
  19. Diudea MV, Petitjean M (2008) Symmetry in multi tori. Symmetry Culture Sci 19(4):285–305Google Scholar
  20. Diudea MV, Szefler B (2012) Omega polynomial in polybenzene multitori. Iran J Math Sci Info 7:67–74Google Scholar
  21. Djoković DŽ (1973) Distance preserving subgraphs of hypercubes. J Combin Theory Ser B 14:263–267CrossRefGoogle Scholar
  22. Eppstein D (2008) Recognizing partial cubes in quadratic time. 19th ACM-SIAM Symp. Discrete Algorithms, San Francisco, 1258–1266Google Scholar
  23. Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes of New York 27:9–15Google Scholar
  24. 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
  25. Harary F (1969) Graph theory. Addison-Wesley, ReadingGoogle Scholar
  26. Imrich W, Klavžar S (1993) A simple O(mn) algorithm for recognizing Hamming graphs. Bull Inst Comb Appl 9:45–56Google Scholar
  27. John PE, Khadikar PV, Singh J (2007a) A method of computing the PI index of benzenoid hydrocarbons using orthogonal cuts. J Math Chem 42:37–45CrossRefGoogle Scholar
  28. John PE, Vizitiu AE, Cigher S, Diudea MV (2007b) CI index in tubular nanostructures. MATCH Commun Math Comput Chem 57:479–484Google Scholar
  29. Khadikar PV (2000) On a novel structural descriptor PI. Nat Acad Sci Lett 23:113–118Google Scholar
  30. Khadikar PV, Agrawal VK, Karmarkar S (2002) Prediction of lipophilicity of polyacenes using quantitative structure-activity relationships. Bioorg Med Chem 10:3499–3507CrossRefGoogle Scholar
  31. Klavžar S (2008a) A bird’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
  32. Klavžar S (2008b) Some comments on co graphs and CI index. MATCH Commun Math Comput Chem 59:217–222Google Scholar
  33. Nagy CsL, Diudea MV (2009) Nano Studio software, “Babes-Bolyai” University, ClujGoogle Scholar
  34. O’Keeffe M, Adams GB, Sankey OF (1992) Predicted new low energy forms of carbon. Phys Rev Lett 68:2325–2328CrossRefGoogle Scholar
  35. Szefler B, Diudea MV (2012) Polybenzene revisited. Acta Chim Slov 59:795–802Google Scholar
  36. Wilkeit E (1990) Isometric embeddings in Hamming graphs. J Combin Theory Ser B 50:179–197CrossRefGoogle Scholar
  37. Winkler PM (1984) Isometric embedding in products of complete graphs. Discret Appl Math 8:209–212CrossRefGoogle 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 Physical Chemistry, Faculty of Pharmacy, Collegium MedicumNicolaus Copernicus UniversityBydgoszczPoland

Personalised recommendations