# Unicyclic graphs with maximum Randić indices

Document Type : Original paper

Authors

1 UMT, Malaysia

2 Universiti Pendidikan Sultan Idris

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

Abstract

The Randi'c index $R(G)$ of a graph $G$ is the sum of the weights $(d_u d_v)^{-\frac{1}{2}}$ of all edges $uv$ in $G$, where $d_u$ denotes the degree of vertex $u$. Du and Zhou [On Randi'c indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), 2760--2770] determined the $n$-vertex unicyclic graphs with the third for $n\ge 5$, the fourth for $n\ge 7$ and the fifth for $n\ge 8$ maximum Randi'c indices. Recently, Li et al. [The Randi{' c} indices of trees, unicyclic graphs and bicyclic graphs, Ars Combin. 127 (2016), 409--419] obtained the $n$-vertex unicyclic graphs with the sixth and the seventh for $n\ge 9$ and the eighth for $n\ge 10$ maximum Randi'c indices. In this paper, we characterize the $n$-vertex unicyclic graphs with the ninth, the tenth, the eleventh, the twelfth and the thirteenth maximum Randi'c values.

Keywords

Main Subjects

#### References

[1] B. Bollobás and P. Erdős, Graphs of extremal weights, Ars Combin. 50 (1998), 225–233.
[2] G. Caporossi, I. Gutman, P. Hansen, and L. Pavlović, Graphs with maximum connectivity index, Comput. Biol. Chem. 27 (2003), no. 1, 85–90.
[3] J. Devillers and A.T. Balaban, Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, 1999.
[4] Z. Du, A. Jahanbani, and S.M. Sheikholeslami, Relationships between Randi´c index and other topological indices, Commun. Comb. Optim. 6 (2021), no. 1, 137–154.
[5] Z. Du and B. Zhou, On Randić indices of trees, unicyclic graphs, and bicyclic graphs, Int. J. Quantum Chem. 111 (2011), no. 12, 2760–2770.
[6] Z. Du, B. Zhou, and N. Trinajstić, On Randić indices of chemical trees and chemical unicyclic graphs, MATCH Commun. Math. Comput. Chem. 62 (2009), no. 1, 131–142.
[7] J. Gao and M. Lu, On the Randić index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 53 (2005), no. 2, 377–384.
[8] I. Gutman and O. Miljković, Molecules with smallest connectivity indices, MATCH Commun. Math. Comput. Chem. 41 (2000), 57–70.
[9] I. Gutman, O. Miljković, G. Caporossi, and P. Hansen, Alkanes with small and large Randi´c connectivity indices, Chem. Phys. Lett. 306 (1999), no. 5-6, 366–372.
[10] J. Li, S. Balachandran, S.K. Ayyaswamy, and Y.B. Venkatakrishnan, The Randić indices of trees, unicyclic graphs and bicyclic graphs, Ars Combin. 127 (2016), 409–419.
[11] M. Randić, Characterization of molecular branching, J. Amer. Chem. Soc. 97 (1975), no. 23, 6609–6615.
[12] J. Wang, Y. Zhu, and G. Liu, On the Randić index of bicyclic graphs, Recent results in the theory of Randi´c index (I. Gutman and B. Furtula, eds.), Univ. Kargujevac, 2008, pp. 119–132.
[13] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, New
Jersey, 2001.
14] H. Zhao and X. Li, Trees with small Randi´c connectivity index, MATCH Commun. Math. Comput. Chem. 51 (2004), 167–178.