On the Bicyclic Graphs with Minimum Reduced Reciprocal Randic Index

Document Type : Research Paper

Authors

1 University of Management & Technology Sialkot, Pakistan

2 Velammal Engineering College, Surapet, Chennai-66 Tamil Nadu, India

3 Department of Mathematics, Savannah State University Savannah, GA 31404, USA

4 Hochschule fur Technik und Wirtschaft, Berlin, Germany and Faculty of Information Studies, Novo Mesto, Slovenia

Abstract

The reduced reciprocal Randić (RRR) index is a molecular structure descriptor (or more precisely, a topological index), which is useful for predicting the standard enthalpy of formation and normal boiling point of isomeric octanes. In this paper, a mathematical aspect of RRR index is explored, or more specifically, the graph(s) having minimum RRR index is/are identified from the collection of all n–vertex connected bicyclic graphs for n≥5. As a consequence, the best possible lower bound on the RRR index, for n–vertex connected bicyclic graphs is obtained when n≥5.

Keywords

Main Subjects


1. N. Trinajstić, Chemical Graph Theory, 2nd ed., CRC Press, Boca Raton,
Florida, 1992.
2. E. Estrada, D. Bonchev, Section 13.1. Chemical Graph Theory, in
Handbook of Graph Theory, 2nd ed., Gross, Yellen and Zhang, Eds., CRC
Press, Boca Raton, FL, 2013, pp. 1538–1558.
3. J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, Berlin, 2008.
4. F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.
5. A. T. Balaban, Chemical graph theory and the Sherlock Holmes principle,
HYLE: Int. J. Phl. Chem. 19 (2013) 107–134.
6. H. Wiener, Structual determination of paraffin boiling points, J. Amer.
Chem. Soc.69 (1947) 17–20.
7. M. Randić, On characterization of molecular branching, J. Am. Chem. Soc.
97 (1975) 6609−6615.
8. B. Bollobás, P. Erdős, Graphs of extremal weights, Ars Combin. 50 (1998)
225–233.
9. J. Gao, M. Lu, On the Randić index of unicyclic graphs, MATCH Commun.
Math. Comput. Chem. 53 (2005) 377–384.
10. Y. Zhu, G. Liu, J. Wang, in: Recent Results in the Theory of Randić Index;
I. Gutman, B. Furtula (Eds.), Univ. Kragujevac, Kragujevac, 2008, pp.
119–132.
11. I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013)
351–361.
12. X. Li, Y. Shi, A survey on the Randić index, MATCH Commun. Math.
Comput. Chem.59 (2008) 127–156.
13. Q. Cui, L. Zhong, The general Randić index of trees with given number of
pendent vertices, Appl. Math. Comput. 302 (2017) 111–121.
14. T. Divnić, L. Pavlović, B. Liu, Extremal graphs for the Randić index when
minimum, maximum degrees and order of graphs are odd, Optimization 64
(2015) 2021–2038.
15. M. Knor, B. Lužar, R. Škrekovski, Sandwiching the (generalized) Randić
index, Discrete Appl. Math. 181 (2015) 160–166.
16. F. Li, Q. Ye, The general connectivity indices of fluoranthene-type
benzenoid systems, Appl. Math. Comput. 273 (2016) 897–911.
17. T. Mansour, M.A. Rostami, S. Elumalai, B.A. Xavier, Correcting a paper
on the Randić and geometric-arithmetic indices, Turk. J. Math. 41 (2017)
27–32.
18. Y. Shi, Note on two generalizations of the Randić index, Appl. Math.
Comput. 265 (2015) 1019–1025.
19. T. Dehghan-Zadeh, A.R. Ashrafi, N. Habibi, Maximum and second
maximum of Randić index in the class of tricyclic graphs, MATCH
Commun. Math. Comput. Chem.74 (2015) 137–144.
20. A. Ali, Z. Du, On the difference between atom-bond connectivity index and
Randić index of binary and chemical trees, Int. J. Quantum Chem.117
(2017) e25446.
21. F. C. G. Manso, H. S. Júnior, R. E. Bruns, A. F. Rubira, E. C. Muniz,
Development of a new topological index for the prediction of normal
boiling point temperatures of hydrocarbons: The Fi index, J. Mol. Liquids
165 (2012) 125–132.
22. I. Gutman, B. Furtula, C. Elphick, Three new/old vertex-degree-based
topological indices, MATCH Commun. Math. Comput. Chem. 72 (2014)
617–632.
23. X. Ren, X. Hu, B. Zhao, Proving a conjecture concerning trees with
maximal reduced reciprocal Randić index, MATCH Commun. Math.
Comput. Chem. 76 (2016) 171–184.
24. A. Ali, A. A. Bhatti, A note on the minimum reduced reciprocal Randić
index of 􀝊-vertex unicyclic graphs, Kuwait J. Sci. 44 (2) (2017) 27–33.
25. S. Li, H. Zhou, On the maximum and minimum Zagreb indices of graphs
with connectivity at most 􀝇, Appl. Math. Lett. 23 (2010) 128–132.
26. Q. Zhao, S. Li, On the maximum Zagreb indices of graphs with 􀝇 cut
vertices, Acta Appl. Math.111 (2010) 93–0106.
27. B. Borovićanin, B. Furtula, On extremal Zagreb indices of trees with given
domination number, Appl. Math. Comput. 279 (2016) 208–218.
28. J. Ma, Y. Shi, Z. Wang, J. Yue, On Wiener polarity index of bicyclic
networks, Sci. Rep.6 (2016) #19066.
29. S. Ji, S. Wang, On the sharp lower bounds of Zagreb indices of graphs with
given number of cut vertices, J. Math. Anal. Appl. 458 (2018) 21–29.