Extremal trees for the general Sombor index

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, North-Eastern Hill University, Shillong, India

2 Department of Basic Sciences and Social Sciences, North-Eastern Hill University, Shillong, India

Abstract

Recently, the Sombor index of a graph has been extended to general Sombor index. The general Sombor index of a simple graph $G$ is defined as $SO_\alpha(G)=\displaystyle\sum_{uv\in E(G)}[d_G(u)^2+d_G(v)^2]^{{\alpha}/2}$, where $d_G(u)$ denotes the degree of a vertex $u$ in $G$ and $\alpha$ is a real number. In this paper, we obtain bounds for the general Sombor index of trees. We further determine the trees with the extremal general Sombor indices.

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Main Subjects

References

[1] B. Bollobás and P. Erdös, Graphs of extremal weights, Ars Combin. 50 (1998), 225–233.
[2] R. Cruz, I. Gutman, and J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399 (2021), Article ID: 126018. https://doi.org/10.1016/j.amc.2021.126018
[3] Q. Cui and L. Zhong, The general Randić index of trees with given number of pendent vertices, Appl. Math. Comput. 302 (2017), 111–121. https://doi.org/10.1016/j.amc.2017.01.021
[4] Q. Cui and L. Zhong, On the general sum-connectivity index of trees with given number of pendent vertices, Discrete Appl. Math. 222 (2017), 213–221. https://doi.org/10.1016/j.dam.2017.01.016
[5] K.C. Das, A.S. B. Zhou and N. Trinajstićevik, I.N. Cangul, and Y. Shang, On sombor index, Symmetry 13 (2021), no. 1, Article ID: 140. https://doi.org/10.3390/sym13010140
[6] Z. Du, B. Zhou, and N. Trinajstić, On the general sum-connectivity index of trees, Appl. Math. Lett. 24 (2011), no. 3, 402–405. https://doi.org/10.1016/j.aml.2010.10.038
 [7] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 11–16.
[8] Y. Hu, X. Li, and Y. Yuan, Trees with maximum general Randić index, MATCH Commun. Math. Comput. Chem. 52 (2004), 129–146.
[9] Y. Hu, X. Li, and Y. Yuan, Trees with minimum general Randić index, MATCH Commun. Math. Comput. Chem. 52 (2004), 119–128.
[10] X. Li, J. Liu, and L. Zhong, Trees with a given order and matching number that have maximum general Randi´ c index, Discrete Math. 310 (2010), no. 17-18, 2249–2257. https://doi.org/10.1016/j.disc.2010.04.028
[11] H. Liu, I. Gutman, L. You, and Y. Huang, Sombor index: review of extremal results and bounds, J. Math. Chem. 60 (2022), no. 5, 771–798. https://doi.org/10.1007/s10910-022-01333-y
[12] H. Liu, X. Yan, and Z. Yan, Bounds on the general randić index of trees with a given maximum degree, MATCH Commun. Math. Comput. Chem. 58 (2007), no. 1, 155–166.
[13] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[14] C. Phanjoubam and S.M. Mawiong, On Sombor index and some topological indices, Iranian J. Math. Chem. 12 (2021), no. 4, 209–215. https://doi.org/10.22052/ijmc.2021.243137.1588
[15] C. Phanjoubam, S.M. Mawiong, and A.M. Buhphang, On general Sombor index of graphs, Asian-Eur. J. Math. 16 (2023), no. 3, Article ID: 2350052. https://doi.org/10.1142/S1793557123500523
[16] M. Randić, Characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), no. 23, 6609–6615. https://doi.org/10.1021/ja00856a001
[17] I. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc. 86 (2021), no. 5, 445–457. http://dx.doi.org/10.2298/JSC201215006R
[18] I. Tomescu and M.K. Jamil, Maximum general sum-connectivity index for trees with given independence number, MATCH Commun. Math. Comput. Chem. 72 (2014), no. 3, 715–722.
[19] Z. Wang, Y. Mao, Y. Li, and B. Furtula, On relations between sombor and other degree-based indices, J. Appl. Math. Comput. 68 (2022), no. 1, 1–17. https://doi.org/10.1007/s12190-021-01516-x.
[20] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), no. 1, 17–20. https://doi.org/10.1021/ja01193a005
[21] L.Z. Zhang, M. Lu, and F. Tian, Maximum Randić index on trees with k-pendant vertices, J. Math. Chem. 41 (2007), no. 2, 161–171. https://doi.org/10.1007/s10910-006-9066-0
[22] L. Zhong, General Randić index on trees with a given order diameter, MATCH Commun. Math. Comput. Chem. 62 (2009), no. 1, 177–187.
[23] L. Zhong and Q. Qian, The minimum general sum-connectivity index of trees with given matching number, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 2, 15271544. https://doi.org/10.1007/s40840-019-00755-3
[24] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009), no. 4, 1252–1270. https://doi.org/10.1007/s10910-008-9515-z
[25] B. Zhou and N. Trinajstić, On general sum-connectivity index, J. Math. Chem. 47 (2010), no. 1, 210–218. https://doi.org/10.1007/s10910-009-9542-4
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 28 November 2024

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