The Extremal Graphs for (Sum-) Balaban Index of Spiro and Polyphenyl Hexagonal Chains

Document Type: Research Paper

Authors

College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, P. R. China

Abstract

As highly discriminant distance-based topological indices, the Balaban index and the sum-Balaban index of a graph $G$ are defined as
$J(G)=\frac{m}{\mu+1}\sum\limits_{uv\in E} \frac{1}{\sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=\frac{m}{\mu+1}\sum\limits_{uv\in E} \frac{1}{\sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=\sum\limits_{v\in V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the number of edges and $\mu$ is the cyclomatic number of $G$. They are useful distance-based descriptor in chemometrics. In this paper, we focus on the extremal graphs of spiro and polyphenyl hexagonal chains with respect to the Balaban index and the sum-Balaban index.

Keywords

Main Subjects


1. V. Andová, M. Knor, R. Škrekovski, Distance based indices in nanotubical graphs:part 2, J. Math. Chem. DOI: 10.1007/s10910-018-0933-2.
2. Y. Bai, B. Zhao, P. Zhao, Extremal Merrifild-Simmons index and Hosoya index of polyphenyl chains, MATCH Commun. Math. Comput. Chem. 62 (2009) 649–656.
3. A. T. Balaban, Highly discriminating distance-based topological index, Chem. Phys. Lett. 89 (1982) 399–404.
4. A. T. Balaban, Topological index based on topological distance in molecular graphs, Pure Appl. Chem. 55 (1983) 199–206.
5. A. T. Balaban, P. V. Khadikar, S. Aziz, Comparison of topological index based on iterated ’sum’ versus ’product’ operations, Iranian J. Math. Chem. 1 (2010) 43–67.
6. A. T. Balaban, L. B. Kier, N. Joshi, Structure-property analysis of octane numbers for hydrocarbons (alkanes, cycloalkanes, alkenes), MATCH Commun. Math. Comput. Chem. 28 (1992) 13–27.
7. D. Bonchev, E. J. Markel, A. H. Dekmezian, Long chain branch polymer chain dimensions: application of topology to the Zimm-Stockmayer model, Polymer 43 (2002) 203–222.

8. M. Bureš, V. Pekárek, T. Ocelka, Thermochemical properties and relative stability of polychlorinated biphenyls, Environ. Tox. Pharm. 25 (2008) 2610–2617.
9. X. Chen, B. Zhao, P. Zhao, Six-membered ring spiro chains with extremal Merrifild-Simmons index and Hosoya index, MATCH Commun. Math. Comput. Chem. 62 (2009) 657–665.
10. H. Deng, On the Balaban index of trees, MATCH Commun. Math. Comput. Chem. 66 (2011) 253–260.
11. H. Deng, On the Sum-Balaban index, MATCH Commun. Math. Comput. Chem. 66 (2011) 273–284.
12. H. Deng, Wiener indices of spiro and polyphenyl hexagonal chains, Math. Comput. Modelling 55 (2012) 634–644.
13. H. Deng, Z. Tang, Kirchhoff indices of spiro and polyphenyl hexagonal chains, Util. Math. 95 (2014) 113–128.
14. H. Dong, X. Guo, Character of graphs with extremal Balaban index, MATCH Commun. Math. ComputChem. 63 (2010) 799–812.
15. H. Dong, X. Guo, Character of trees with extremal Balaban index, MATCH Commun. Math. Comput. Chem. 66 (2011) 261–272.
16. T. Došlić, M. Litz, Matchings and independent sets in polyphenylene chains, MATCH Commun. Math. Comput. Chem. 67 (2012) 313–330.
17. T. Došlić, F. Måløy, Chain hexagonal cacti: Matchings and independent sets, Discrete Math. 310 (2010) 1676–1690.
18. W. Fang, Y. Gao, Y. Shao, W. Gao, G. Jing, Z. Li, Maximum Balaban index and Sum-Balaban index of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 75 (2016) 129–156.
19. W. Fang, H. Yu, Y. Gao, X. Li, G. Jing, Z. Li, Maximum Balaban index and Sum-Balaban index of tricyclic graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 717–742.
20. D. R. Flower, On the properties of bit string-based measures of chemical similarity, J. Chem. Inf. Comput. Sci. 38 (1998) 379–386.
21. A. Graja, Low-Dimensional Organic Conductors, World Scientific, Singapore, 1992.
22. G. Grassy, B. Calas, A. Yasri, R. Lahana, J. Woo, S. Iyer, M. Kaczorek, F. Floch, R. Buelow, Computer-assisted rational design of immunosuppressive compounds, Nat. Biotechnol. 16 (1998) 748–752.
23. G. Huang, M. Kuang, H. Deng, The extremal graph with respect to the matching energy for a random polyphenyl chain, MATCH Commun. Math. Comput. Chem. 73 (2015) 121–131.

24. G. Huang, M. Kuang, H. Deng, The expected values of Hosoya index and Merrifield-Simmons index in a random polyphenylence chain, J. Combin. Opt. 32 (2) (2016) 550–562.
25. M. Knor, R. Škrekovski, A. Tepeh, Balaban index of cubic graphs, MATCH Commun. Math. Comput. Chem. 73 (2015) 519–528.
26. Q. R. Li, Q. Yang, H. Yin, S. Yang, Analysis of by-products from improved Ullmann reaction using TOFMS and GCTOFMS, J. Univ. Sci. Technol. China 34 (2004) 335–341.
27. S. Li, B. Zhou, On the Balaban index of trees, Ars Combin. 101 (2011) 503–512.
28. G. Luthe, J. A. Jacobus, L. W. Robertson, Receptor interactions by polybrominated diphenyl ethers versus polychlorinated biphenyls: A theoretical structure-activity assessment, Environ. Tox. Pharm. 25 (2008) 202–210.
29. L. Sun, Bounds on the Balaban index of trees, MATCH Commun. Math. Comput. Chem. 63 (2010) 813–818.
30. S. Tepavčević, A. T. Wroble, M. Bissen, D. J. Wallace, Y. Choi, L. Hanley, Photoemission studies of polythiophene and polyphenyl films produced via surface polymerization by ion-assisted deposition, J. Phys. Chem. B 109 (2005) 7134–7140.
31. R. Xing, B. Zhou, A. Graovac, On Sum-Balaban index, Ars Combin. 104 (2012) 211–223.
32. W. Yang, F. Zhang, Wiener index in random polyphenyl chains, MATCH Commun. Math. Comput. Chem. 68 (2012) 371–376.
33. L. You, X. Dong, The maximum Balaban index (Sum-Balaban index) of unicyclic graphs, J. Math. Res. Appl. 34 (2014) 292–402.
34. L. You, H. Han, The maximum Balaban index (Sum-Balaban index) of trees with given diameter, Ars Combin. 112 (2013) 115–128.
35. P. Zhao, B. Zhao, X. Chen, Y. Bai, Two classes of chains with maximal and minmal total 􀟨-electron energy, MATCH Commun. Math. Comput. Chem. 62 (2009) 525–536.