# Sombor index of certain graphs

Document Type : Research Paper

Authors

1 Department of Informatics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway

2 Department of Mathematics, Yazd University, 89195-741, Yazd, Iran

Abstract

Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we examine the effects on $SO(G)$ when $G$ is modified by operations on vertex and edge of $G$. Also we present bounds for the Sombor index of join and corona product of two graphs.

Keywords

#### References

1.

1. S. Alikhani, S. Jahari, M. Mehryar and R. Hasni, Counting the number of dominating sets of cactus chains, Opt. Adv. Mat. – Rapid Comm. 8, No. 9-10, (2014) 955-960.
2. M. Chellali, Bounds on the 2-domination number in cactus graphs, Opuscula Math. 2 (2006) 5−12.
3. R. Cruz, I. Gutman and J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399 (2021) 126018.
4. K. C. Das, A. S. Cevik, I. N. Cangul and Y. Shang, On Sombor index, Symmetry 13 (2021) #140.
5. H. Deng, Z. Tang and R. Wu, Molecular trees with extremal values of Sombor indices, Int. J. Quantum Chem. DOI: 10.1002/qua.26622.
6. I. Gutman, Geometric approach to degree based topological indices, MATCH Commun. Math. Comput. Chem. 86 (1) (2021) 11−16.
7. F. Harary and B. Uhlenbeck, On the number of Husimi trees, I, Proc. Nat. Acad. Sci. 39 (1953) 315−322.
8. K. Husimi, Note on Mayer’s theory of cluster integrals, J. Chem. Phys. 18 (1950) 682−684.
9. S. Majstorović, T. Došlić and A. Klobučar, -domination on hexagonal cactus chains, Kragujevac J. Math. 36, No 2 (2012) 335−347.
10. Y. Mao, Nordhaus-Gaddum Type Results in Chemical Graph Theory. In Bounds in Chemical Graph Theory–Advances; Gutman, I., Furtula, B., Das, K.C., Milovanović, E., Milovanovixcx, I., Eds.; University of Kragujevac and Faculty of Science Kragujevac: Kragujevac, Serbia, 2017; pp. 3−127.
11. R. J. Riddell, Contributions to the Theory of Condensation, Ph.D. Thesis, University of Michigan, Ann Arbor, 1951.
12. A. Sadeghieh, N. Ghanbari and S. Alikhani, Computation of Gutman index of some cactus chains, Elect. J. Graph Theory Appl. 6 (1) (2018), 138−151.
13. Y. Shang, Bounds of distance Estrada index of graphs, Ars Comb. 128 (2016) 287−294.
14. Z. Wang, Y. Mao, Y.  Li and B. Furtula, On  relations  between  Sombor  and  other degree-based indices, J. Appl. Math. Comput, DOI: https://doi.org/10.1007/s12190-021-01516-x.
15. H. Wiener, Structural determination of the Paraffin boiling points, J. Am. Chem. Soc., 69 (1947) 17−20.