Distance property of chemical graphs

Abstract

We have developed a rigorous computational technique to compute exact analytic expressions for a number of distance-based topological indices of chemical graphs. There are two main advantages of our technique over existing techniques of similar nature: first, our technique is significantly diverse as it also covers the Wiener index and eccentricity-based topological indices besides Szeged-like indices, and secondly we have considerably reduced the algorithmic and computational complexity in comparison to previous techniques. Our proposed technique generates certain vertex and edge partitions of a graph which are essential in computing the exact analytical formulas of distance-based and eccentricity-based indices. To ensure the applicability of our technique,we have computed various distance-based and eccentricity-based topological indices for certain infinite families of polyomino chain system. Moreover, we find analytical exact expressions of certain degree-based topological indices for these polyomino chains. These topological indices can be obtained as a by-product of our technique.

Keywords

• Combinatorial algorithms,Distance-based topological indices,Eccentricity-based topological indices,Degree-based topological indices,Polyomino chain system

References

1. Ashrafi, A.R. Doslic, T. and Saheli, M. The eccentric connectivity index of $TUC_4C_8(R)$ nanotubes, MATCH Commun. Math. Comput. Chem. 65, 221-230, 2011.
2. Ashrafi, A.R. Ghorbani, M. and Jalali, M. The $PI$ and edge Szeged polynomials of an infinitefamily of fullerenes, Fullerenes, Nanotubes and Carbon Nanostructures 18 (3), 107-116,2010.
3. Aouchiche, M. and Hansen, P. On a conjecture about the Szeged index, European J. Combin. 31, 1662-1666, 2010.
4. Arockiaraj M. Kavithah, S.R.J. and Balasubramanian, K. Vertex-cut methods for distance- based topological indices and its application to inorganic networks, J. Math. Chem. 54, 1728-1747, 2016.
5. Baca, M. Horváthová, J. Mokrisová, M. and Suhányiová, A. On topological indices of fullerenes, Appl. Math. Comput. 251, 154-161, 2015.
6. Imran, M. and Hayat, S. On counting polynomials of certain polyomino chains, Bulg. Chem. Commun. 48, 332-337, 2016.
7. Diudea, M.V. Nanomolecules and nanostructures: polynomials and indices, University of Kragujevac, Kragujevac, 2010.
8. Diudea, M.V. Ursu O. and Nagy, Cs.L. TOPOCLUJ, Babes-Bolyai University, Cluj, 2002.

9. Dureja, H. and Madan, A.K. Superaugmented eccentric connectivity indices: new-generationhighly discriminating topological descriptors for $QSAR/QSPR$ modeling, Med. Chem. Res.16, 331-341, 2007.
10. Eliasi, M. and Taeri, B. Szeged index of armchair polyhex nanotubes, MATCH Commun. Math. Comput. Chem., 59 437-450, 2008.
11. Estrada, E. Torres, L. Rodríguez, L. and Gutman, I. An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem., 37A, 849-855, 1998.
12. The GAP Team, GAP, Groups, Algorithms and Programming, Lehrstuhl De fur Mathematik, RWTH, Aachen, 1992.
13. Golomb, S.W. Polyominoes, Princeton University Press, Princeton, New Jersey, 1994.
14. Graovac A. Ghorbani, M. and Hosseinzadeh, M.A. Computing fifth geometric-arithmetic index for nanostar dendrimers, J. Math. Nanosci. 1, 33-42, 2011.
15. Graovac, A. and Hosseinzadeh, M.A. Computing $ABC_4$ index of nanostar dendrimers, Optoelectron.Adv. Mater. Rapid Commun. 4, 1419-1422, 2010.
16. Graovac, A. Ori, O. Faghani, M. and Ashrafi, A.R. Distance property of fullerenes, Iranian J. Math. Chem. 1, 5-15, 2010.
17. Gutman, I. A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York, 27, 9-15, 1994.
18. Gutman, I. and Dobrynin, A.A. The Szeged index - a success story, Graph Theory Notes New York 34, 37-44, 1998.
19. Hayat, S. and Imran, M. Computation of topological indices of certain networks, Appl. Math. Comput. 240, 213-228, 2014.
20. HyperChem package Release 7.5 for Windows, Hypercube Inc., 1115 NW 4th Street, Gainesville, Florida 32601, USA, 2002.
21. Imran, M. Hayat, S. and Malik, M.Y.H. On topological indices of certain interconnection networks, Appl. Math. Comput. 244, 936-951, 2014.
22. Karelson, M. Molecular descriptors in $QSAR/QSPR$, Wiley, New York, 2000.
23. Khadikar, P.V. Karmarkar, S. and Agrawal, V.K. A novel $PI$ index and its applications to $QSPR/QSAR$ studies, J. Chem. Inf. Comput. Sci., 41, 934-949, 2001.
24. Klarner, D.A. Polyominoes, In: J. E. Goodman, J. O'Rourke, (eds.), Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, (1997), 225-242, Chapter 12.
25. Liu, J.-B. and Cao, J. Applications of Laplacian spectra for n-prism networks, Neurocomputing, 198, 69-73, 2016.
26. Liu, J.-B. Pan, X.-F. Yu, L. and Li, D. Complete characterization of bicyclic graphs withminimal Kirchhoff index, Discrete Appl. Math. 200, 95-107, 2016.
27. Liu, J.-B. and Pan, X.-F. Minimizing Kirchhoff index among graphs with a given vertex bipartiteness, Appl. Math. Comput. 291, 84-88, 2016.
28. Liu, J.-B. Wang, W.R. Zhang, Y.M. and Pan, X.-F. On degree resistance distance of cacti, Discrete Appl. Math. 203, 217-225, 2016.
29. Liu, J.-B. Wang, C. Wang, S. and Wei, B. Zagreb indices and multiplicative Zagreb indices of Eulerian graphs, Bull. Malays. Math. Sci. Soc. DOI: 10.1007/s40840-017-0463-2.
30. MATLAB and Statistics Toolbox Release 2012b, The MathWorks, Inc., Natick, Massachusetts, United States.
31. Mehranian, Z. Mottaghi, A. and Ashrafi, A.R. The topological study of $IPR$ fullerenes by studying their Szeged and revised Szeged indices, J. Theor. Comput. Chem. 11 (3), 547-559,2012.
32. Mottaghi, A. and Ashrafi, A.R. Topological edge properties of $C_{12n+60} fullerenes, BeilsteinJ. Nanotechnol., 4, 400-405, 2013.
33. Pisanski, T. and Randi¢, M. Use of the Szeged index and the revised Szeged index for meauring network bipartivity, Discrete Appl. Math. 158, 1936-1944, 2010.
34. Randi¢, M. On characterization of molecular branching, J. Amer. Chem. Soc. 97 (23), 6609-6615, 1975.
35. Randi¢, M. On generalization of Wiener index for cyclic structures, Acta Chim. Slov. 49, 483-496, 2002.
36. Sharma, V. Goswami, R. and Madan, A.K. Eccentric connectivity index: a novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci. 37, 273-282, 1997.
37. Siddiqui, M.K. Imran M. and Ahmad A., On Zagreb indices, Zagreb polynomials of some nanostar dendrimers, Appl. Math. Comput. 280, 132-139, 2016.
38. Vukisevi¢, D. and Furtula, B. Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46, 1369-1376, 2009.
39. Wang, C. Wang, S. and Wei, B. Cacti with extremal PI index, Trans. Comb. 5, 1-8, 2016.
40. Wang, S. and Wei, B. Multiplicative Zagreb indices of k-trees, Discrete Appl. Math. 180, 168-175, 2015.
41. Wiener, H. Structural determination of the paraffin boiling points, J. Am. Chem. Soc. 69,17-20, 1947.
42. Xing, R. and Zhou, B. On the revised Szeged index, Discrete Appl. Math. 159, 69-78, 2011.
43. Yarahmadia, Z. Ashrafi A.R. and Moradic, S. Extremal polyomino chains with respect to Zagreb indices, App. Math. Lett. 25, 166-171, 2012.
44. Zhou, B. and Du, Z. On eccentric connectivity index, MATCH Commun. Math. Comput. Chem. 63, 181-198, 2010.