On relationship between reformulated Sombor and other vertex--degree indices

Document Type : Research Paper

Authors

1 Faculty of Electronic Engineering, University of Niš, Niš, Serbia

2 Department of Mathematics, Faculty of Science, Selçuk University, Konya, Turkey

3 Faculty of Electronic Engineering, University of Nis, Serbia

10.22108/toc.2024.136304.2036

Abstract

Let $G=(V,E)$, $V=\{v_1, v_2,\ldots,v_n\}$, $E=\{e_1, e_2,\ldots,e_m\}$, be a simple connected graph with $n\ge 2$ vertices and $m$ edges, with vertex degree sequence $\Delta=d_1\ge d_2\ge \cdots \ge d_n=\delta$, $ d_i=d(v_i)$, and edge degree sequence $\Delta_e=d(e_1)\ge d(e_2)\ge \cdots \ge d(e_n)=\delta_e$. The reformulated Sombor index is defined as $RS(G) =\sum_{e_i\sim e_j}\sqrt{d(e_i)^2+d(e_j)^2}$. We consider a relationship between reformulated Sombor index and some of the vertex--degree-based indices.

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References

[1] M. O. Albertson, The irregularity of graphs, Ars Combin., 46 (1997) 219–225.
[2] A. Ali, Z. Iqbal and Z. Iqbal, Two physicochemical properties of benzenoid chains: solvent accessible molecular volume and molar refraction, Canadian Journal of Physics, 97 5 (2018) 524–528.
[3] M. Bhanumathi, K. E. J. Rani and S. Balachandran, The edge version of inverse sum indeg index of connected graph, Int. J. Math., 7 (2016) 8–12.
[4] B. Bollobas and P. Erdos, Graphs of extremal weights, Ars. Comb., 50 (1998) 225–233.
5] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math., 285 no. 1–3 (2004) 57–66.
[6] N. De, Some bounds of reformulated Zagreb indices, Appl. Math. Sci, 6 no. 101 (2012) 5005–5012.
[7] T. Došlić, T. Reti and D. Vukičević, On the vertex degree indices of connected graphs, Chem. Phys. Lett., 512 (2011) 283–286.
[8] C. S. Edwards, The largest vertex degree sum for a triangle in a graph, Bull. London Math. Soc., 9 no. 2 (1977) 203–208.
[9] B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem., 53 no. 4 (2015) 1184–1190.
[10] I. Gutman, Degree–based topological indices, Croat. Chem. Acta, 86 (2013) 351–361.
[11] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 (1972) 535–538.
[12] I. Gutman, B. Ruščić, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys., 62 (1975) 3399–3405.
[13] I. Gutman, Geometric approach to degree–based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem., 86 no. 1 (2021) 11–16.
[14] N. Harish, B. Sarveshkumar and B. Chaluvaraju, The reformulated Sombor index of a graph, Trans. Combin., 13 no. 1 (2024) 1–16.
[15] B. Hollas, The covariance of topological indices thta depend on the degree of a vertex, MATCH Commun. Math. Comput. Chem., 54 (2005) 177–187.
[16] V. R. Kulli, On k edge index and coindex of graphs, Int. J. Fazzy Math. Arch., 10 no. 2 (2016) 111–116.
[17] Z. Lin, On Aα –eigenvalues of graphs and topological indices, Contrib. Math., 5 (2022) 17–24.
[18] A. Miličević, S. Nikolić and N. Trinajstić, On reformulated Zagreb indices, Mol. Divers., 8 (2004) 393–399.
[19] D. S. Mitrinović and P. M. Vasić, Analytic inequalities, Springer Verlag, Berlin–Heidelberg–New York, 1970.
[20] S. Nikolić, G. Kovačević, A. Miličević and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta, 76 (2003) 113–124.
[21] K. Pattabiraman and T. Suganya, Edge version of some degree based topological descriptors of graphs, J. Math. Nanosci., 8 no. 1 (2018) 1–12.
[22] J. Radon, Über die absolut additiven Mengenfunktionen, Sitzungsber. Acad. Wissen. Wien, 122 (1913) 1295–1438.
[23] V. S. Shegehalli and R. Kanabur, Arithmetic–geometric indices of path graph, J. Comput. Math. Sci., 6 no. 1 (2015) 19–24.
[24] R. Todeschini and V. Consonni, Handbook of molecular descriptors, Wiley – VCH, Weinheim, 2000.
[25] D. Vukičević and M. Gašperov, Bond additive modeling 1. Adriatic indices, Croat. Chem. Acta, 83 (2010) 243–260.
[26] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end vertex degree of edges, J. Math. Chem., 46 (2009) 1369–1376.
[27] D. Vukičević, Bond additive modelling 2. Mathematical properties of max–min rodeg index, Croat. Chem. Acta, 83 (2010) 261–273.
[28] D. Vukičević and J. Durdević, Bond additive modeling 10. Upper and lower bounds of bond incident degree indices of catacondensed fluoranthenes, Chem. Phys. Lett., 515 (2011) 186–189.
Transactions on Combinatorics

Articles in Press, Corrected Proof
Available Online from 30 September 2024

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History

  • Receive Date: 02 January 2023
  • Revise Date: 10 November 2023
  • Accept Date: 11 September 2024
  • Published Online: 30 September 2024

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