Sharp upper bounds on $KG$-Sombor index of graphs

Articles in Press

Document Type : Original paper

Authors

Department of Mathematics, Mongolian National University of Education, Baga toiruu-14, Ulaanbaatar, Mongolia

Abstract

The $KG$-Sombor index of a graph $G$ is defined as $$KG(G)=\sum_{\substack{u \in V(G) \\ e \in E(G) \\ u \sim e}}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(e)},$$ where the summation goes over pairs of vertices $u$ and edges $e$ such that $e$ is incident to $u$. In this paper, we establish sharp upper bounds for the $KG$-Sombor index in three classes of graphs: graphs of order $n$ with $k$ pendent vertices, graphs of order $n$ with $k$ cut edges, and unicyclic graphs of order $n$ with girth $g$. Moreover, in each case, the extremal graphs attaining these bounds are completely characterized.

Keywords

Main Subjects

References

[1] O. Altangoo, D. Bolormaa, B. Gantuya, and T.A. Selenge, On the KG-Sombor index, Vojnotehnički glasnik/Military Technical Courier 72 (2024), no. 4, 1493–1508. https://doi.org/10.5937/vojtehg72-49839
[2] R. Cruz, I. Gutman, and J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399 (2021), 126018. https://doi.org/10.1016/j.amc.2021.126018
[3] K.C. Das, A.S. Çevik, I.N. Cangul, and Y. Shang, On sombor index, Symmetry 13 (2021), no. 1, 140. https://doi.org/10.3390/sym13010140
[4] S. Dorjsembe and B. Horoldagva, Reduced Sombor index of bicyclic graphs, Asian-Eur. J. Math. 15 (2022), no. 7, 2250128. https://doi.org/10.1142/S1793557122501285
[5] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 11–16.
[6] I. Gutman, Some basic properties of Sombor indices, Open J. Discrete Appl. Math. 4 (2021), no. 1, 1–3. https://www.doi.org/10.30538/psrp-odam2021.0047
[7] I. Gutman, V.R. Kulli, and I. Redžepović, Sombor index of Kragujevac trees, Ser. A: Appl. Math. Inform. Mech. 13 (2021), no. 2, 61–70.
[8] I. Gutman, I. Redžepović, and V.R. Kulli, KG-Sombor index of Kragujevac trees, Open J. Discrete Appl. Math. 5 (2022), no. 2, 19–25. https://www.doi.org/10.30538/psrp–odam2022.0075
[9] B. Horoldagva and C. Xu, On Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 3, 703–713.
[10] S. Kosari, N. Dehgardi, and A. Khan, Lower bound on the KG-Sombor index, Commun. Comb. Optim. 8 (2023), no. 4, 751–757. https://doi.org/10.22049/cco.2023.28666.1662
[11] V.R. Kulli, KG-Sombor indices of certain chemical drugs, Int. J. Eng. Sci. Res. Tec. 11 (2022), no. 6, 27–35. https://www.doi.org/10.5281/zenodo.6840092
[12] V.R. Kulli and I. Gutman, Computation of Sombor indices of certain networks, SSRG Int. J. Appl. Chem 8 (2021), no. 1, 1–5. https://www.doi.org/10.14445/23939133/IJAC-V8I1P101
[13] V.R. Kulli, N. Harish, B. Chaluvaraju, and I. Gutman, Mathematical properties of KG-Sombor index, Bull. Int. Math. Virtual Inst. 12 (2022), no. 2, 379–386. https://www.doi.org/10.7251/BIMVI2202379K
[14] 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
[15] I. Milovanović, E. Milovanović, and M. Matejić, On some mathematical properties of Sombor indices, Bull. Int. Math. Virtual Inst. 11 (2021), no. 2, 341–353. http://dx.doi.org/10.7251/BIMVI2102341M
[16] I. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc. 86 (2021), no. 5, 445–457. http://dx.doi.org/10.2298/JSC201215006R
[17] T. Réti, T. Došlić, and A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021), 11–18. https://doi.org/10.47443/cm.2021.0006
[18] T.A. Selenge and B. Horoldagva, Extremal Kragujevac trees with respect to Sombor indices, Commun. Comb. Optim. 9 (2024), no. 1, 177–183. https://doi.org/10.22049/cco.2023.28058.1430
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 06 January 2026

Files

Share

How to cite

Statistics