Relationships between Randic index and other topological indices

Document Type : Original paper

Authors

1 School of Mathematics and Statistics, Zhaoqing University, Zhaoqing 526061, China

2 Department of Mathematics, Azarbaijan Shahid Madani University Tabriz, Iran

3 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and let $d_u$ denote the degree of vertex $u$ in $G$. The Randi'c index of $G$ is defined as ${R}(G) =\sum_{uv\in E(G)} 1/\sqrt{d_ud_v}.$ In this paper, we investigate the relationships between Randi'c index and several topological indices.

Keywords

Main Subjects

References

[1] M.O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219–225.
[2] O. Araujo and J.A. De La Peña, The connectivity index of a weighted graph, Linear Algebra Appl. 283 (1998), no. 1-3, 171–177.
[3] B. Bollobás and P. Erdős, Graphs of extremal weights, Ars Combin. 50 (1998), 225–233.
[4] B. Borovicanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), no. 1, 17–100.
[5] ‪Ş. Burcu Bozkurt Altındağ, A.D. Güngör, I. Gutman, and A.S. Cevik, Randić matrix and Randić energy, MATCH Commun. Math. Comput. Chem. 64 (2010), 239–250.
[6] C. Delorme, O. Favaron, and D. Rautenbach, On the Randić index, Discrete Math. 257 (2002), no. 1, 29–38.
[7] S.S. Dragomir, A survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities, J. Inequal. Pure Appl. Math. 4 (2003), no. 3, 1–142.
[8] E. Estrada, L. Torres, L. Rodríguez, and I. Gutman, An atom-bond connectivity index: Modelling the enthalpy of formation of alkanes, Indian J. Chem. A 37 (1998), 849–855.
[9] S. Fajtlowicz, On conjectures of Graffiti-II, Congr. Numer. 60 (1987), 187–197.
[10] B. Furtula, A. Graovac, and D. Vukičević, Augmented Zagreb index, J. Math. Chem. 48 (2010), no. 2, 370–380.
[11] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015), no. 4, 1184–1190.
[12] I. Gutman, Degree-based topological indices, Croat. Chem. Acta 86 (2013), no. 4, 351–361.
[13] I. Gutman, E. Milovanović, and I. Milovanović, Beyond the Zagreb indices, AKCE Int. J. Graphs Comb. (2018), in press.
[14] I. Gutman, B. Ruščić, N. Trinajstić, and C.F. Wilcox Jr, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975), no. 9, 3399–3405.
[15] I. Gutman, M. Togan, A. Yurttas, A.S. Cevik, and I.N. Cangul, Inverse problem for sigma index, MATCH Commun. Math. Comput. Chem. 79 (2018), no. 2, 491–508.
[16] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π−electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538.
[17] L.B. Kier and L.H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, New York, 1976.
[18] V.R. Kulli, F-indices of chemical networks, Int. J. Math. Archive 10 (2019), no. 3, 21–30.
[19] X. Li and Y. Shi, A survey on the Randić index, MATCH Commun. Math. Comput. Chem. 59 (2008), no. 1, 127–156.
[20] I.Z. Milovanović, E.I. Milovanović, and M.M. Matejić,  On upper bounds for the geometric–arithmetic topological index, MATCH Commun. Math. Comput. Chem. 80 (2018), no. 1, 109–127.
[21] D.S. Mitrinović, J.E. Pečarić, and A.M. Fink, Classical and New Inequalities in Analysis, Springer, Netherlands, 1993.
[22] M.R. Oboudi, A new lower bound for the energy of graphs, Linear Algebra Appl. 580 (2019), 384–395.
[23] L. Pogliani, From molecular connectivity indices to semiempirical connectivity terms: Recent trends in graph theoretical descriptors, Chem. Rev. 100 (2000), no. 10, 3827–3858.
[24] A. Portilla, J.M. Rodríguez, and J.M. Sigaretta, Recent lower bounds for geometric-arithmetic index, Discrete Math. Lett. 1 (2019), 59–82.
[25] J. Radon, Theorie und Anwendungen der absolut additiven Mengenfunktionen, Sitzungsber. Acad. Wissen. Wien. 112 (1913), 1295–1438.
[26] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615.
[27] M. Randić, The connectivity index 25 years after, J. Mol. Graphics Model. 20 (2001), no. 1, 19–35.
[28] , On history of the Randić index and emerging hostility toward chemical graph theory, MATCH Commun. Math. Comput. Chem. 59 (2008), no. 1, 5–124.
[29] M. Randić, M. Nović, and D. Plavšić, Solved and Unsolved Problems of Structural Chemistry, CRC Press, Boca Raton, 2016.
[30] P.S. Ranjini, V. Lokesha, A.R. Bindusree, and M.P. Raju, New bounds on Zagreb indices and the Zagreb co-indices, Bol. Soc. Parana. Mat. SPM. 31 (2013), no. 1, 51–55.
[31] P.B. Sarasija and R. Binthiya, Bounds on the Seidel energy of strongly quotient graphs, J. Chem. Pharm. Sci. 10 (2017), no. 1, 1–4.
[32] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, John Wiley & Sons, 2008.
[33] R. Todeschini and V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley-VCH, Weinheim, 2009.
[34] D. Vukičević, Bond additive modeling 2. Mathematical properties of max-min rodeg index, Croat. Chem. Acta. 83 (2010), no. 3, 261–273.
[35] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46 (2009), no. 4, 1369–1376.
[36] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), no. 1, 210–218.
Communications in Combinatorics and Optimization
Volume 6, Issue 1
June 2021
Pages 137-154

Files

Share

How to cite

Statistics