Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ashrafi AR, Ghorbani M, Jalali M (2008a) Computing sadhana polynomial of V-phenylenic nanotubes and nanotori. Indian J Chem 47A:535–537
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 60:905–916
Ashrafi AR, Manoochehrian B, Yousefi-Azari H (2006) On the PI polynomial of a graph. Util Math 71:97–108
Aurenhammer F, Hagauer J (1995) Recognizing binary Hamming graphs in O(n2logn) time. Math Syst Theory 28:387–396
Brešar B (2001) Partial Hamming graphs and expansion procedures. Discret Math 237:13–27
Brešar B, Imrich W, Klavžar S (2003) Fast recognition algorithms for classes of partial cubes. Discret Appl Math 131:51–61
Diudea MV (2005) Nanostructures novel architecture. Nova, New York
Diudea MV (2006) Omega polynomial. Carpath J Math 22:43–47
Diudea MV (2010a) Counting polynomials and related indices by edge cutting procedures. MATCH Commun Math Comput Chem 64(3):569–590
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 191–215
Diudea MV (2010c) Nanomolecules and nanostructures – polynomials and indices. MCM Ser. 10. University of Kragujevac, Kragujevac
Diudea MV, Klavžar S (2010) Omega polynomial revisited. Acta Chem Sloven 57:565–570
Diudea MV, Cigher S, John PE (2008) Omega and related counting polynomials. MATCH Commun Math Comput Chem 60:237–250
Diudea MV, Dorosti N, Iranmanesh A (2010a) Cluj CJ polynomial and indices in a dendritic molecular graph. Studia UBB Chemia LV 4:247–253
Diudea MV, Vizitiu AE, Mirzargar M, Ashrafi AR (2010b) Sadhana polynomial in nano-dendrimers. Carpath J Math 26:59–66
Diudea MV, Florescu MS, Khadikar PV (2006) Molecular topology and its applications. EFICON, Bucharest
Diudea MV, Ilić A (2009) Note on omega polynomial. Carpath J Math 25(2):177–185
Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, Dordrecht
Diudea MV, Petitjean M (2008) Symmetry in multi tori. Symmetry Culture Sci 19(4):285–305
Diudea MV, Szefler B (2012) Omega polynomial in polybenzene multitori. Iran J Math Sci Info 7:67–74
Djoković DŽ (1973) Distance preserving subgraphs of hypercubes. J Combin Theory Ser B 14:263–267
Eppstein D (2008) Recognizing partial cubes in quadratic time. 19th ACM-SIAM Symp. Discrete Algorithms, San Francisco, 1258–1266
Gutman I (1994) A formula for the Wiener number of trees and its extension to graphs containing cycles. Graph Theory Notes of New York 27:9–15
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
Imrich W, Klavžar S (1993) A simple O(mn) algorithm for recognizing Hamming graphs. Bull Inst Comb Appl 9:45–56
John PE, Khadikar PV, Singh J (2007a) A method of computing the PI index of benzenoid hydrocarbons using orthogonal cuts. J Math Chem 42:37–45
John PE, Vizitiu AE, Cigher S, Diudea MV (2007b) CI index in tubular nanostructures. MATCH Commun Math Comput Chem 57:479–484
Khadikar PV (2000) On a novel structural descriptor PI. Nat Acad Sci Lett 23:113–118
Khadikar PV, Agrawal VK, Karmarkar S (2002) Prediction of lipophilicity of polyacenes using quantitative structure-activity relationships. Bioorg Med Chem 10:3499–3507
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 60:255–274
Klavžar S (2008b) Some comments on co graphs and CI index. MATCH Commun Math Comput Chem 59:217–222
Nagy CsL, Diudea MV (2009) Nano Studio software, “Babes-Bolyai” University, Cluj
O’Keeffe M, Adams GB, Sankey OF (1992) Predicted new low energy forms of carbon. Phys Rev Lett 68:2325–2328
Szefler B, Diudea MV (2012) Polybenzene revisited. Acta Chim Slov 59:795–802
Wilkeit E (1990) Isometric embeddings in Hamming graphs. J Combin Theory Ser B 50:179–197
Winkler PM (1984) Isometric embedding in products of complete graphs. Discret Appl Math 8:209–212
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Diudea, M.V., Szefler, B. (2016). Omega Polynomial in Nanostructures. In: Ashrafi, A., Diudea, M. (eds) Distance, Symmetry, and Topology in Carbon Nanomaterials. Carbon Materials: Chemistry and Physics, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-31584-3_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-31584-3_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-31582-9
Online ISBN: 978-3-319-31584-3
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)