Relations between Sombor Index and some Degree-Based Topological Indices

Document Type : Research Paper

Author

Famnit, University of Primorska, Glagoljaška 8 Koper, Slovenia

Abstract

In [13] Gutman introduced a novel graph invariant called Sombor index SO, defined via $\sqrt{\deg(u)^{2}+\deg(v)^{2}}.$
In this paper we provide relations between Sombor index and some degree-based topological indices: Zagreb indices, Forgotten index and Randi' {c} index.
Similar relations are established in the class of triangle-free graphs.

Keywords


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    1. A. Ali, I. Gutman, E. Milovanović, I. Milovanović, Sum of powers of the degrees of graphs: Extremal results and bounds, MATCH Commun. Math. Comput. Chem. 80 (2018) 5−84.
    2. B. Borovićanin, K. C. Das, B. Furtula, I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017) 17−100.
    3. K. C. Das, Sharp bounds for the sum of the squares of degrees of a graph, Kragujevac J. Math.25 (2003) 31−49.
    4. K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285 (2004) 57−66.
    5. K. C. Das, On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63 (2010) 433−440.
    6. K. C. Das, I. Gutman, B. Zhou, New upper bounds on Zagreb indices, J. Math. Chem. 46 (2009) 514−521.
    7. K. C. Das, K. Xu, J. Nam, On Zagreb indices of graphs, Front. Math. China 10 (2015) 567−582.
    8. S. Filipovski, New bounds for the first Zagreb index, MATCH Commun. Math. Comput. Chem. 85 (2) (2021) 303−312.
    9. B. Furtula and I. Gutman, A forgotten topological index, J. Math. Chem. 53 (4) (2015) 1184−1190.
    10. G. H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 64 (2011) 79−84.
    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, 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 (2021) 11−16.
    14. V. R. Kulli, Graph indices, in: M. Pal, S. Samanta, A. Pal (Eds.), Handbook of Research of Advanced Applications of Graph Theory in Modern Society, Global, Hershey, 2020, pp. 66−91.
    15. X. Li, H. Zhao, Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004) 57−62.
    16. X. Li, J. Zheng, A unified approach to the extremal trees for different indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 195−208.
    17. J. Rada, S. Bermudo, Is Every Graph the Extremal Value of a Vertex–Degree– Based Topological Index? MATCH Commun. Math. Comput. Chem. 81 (2) (2019) 315−323.
    18. M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609−6615.
    19. R. Todeschini, V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley−VCH, Weinheim, 2009.
    20. 20. Y. Yao, M. Liu, K. C. Das, Y. Ye,  Some extremal results for vertex-degree-based invariants, MATCH Commun. Math. Comput. Chem. 81 (2) (2019) 325−344.
    21. 21. B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52 (2004) 113−118.