On the KG-Sombor index
Abstract
Introduction/purpose: Degree-based graph invariants are a type of molecular descriptor that represent the connectivity of atoms (vertices) in a molecule through bonds (edges). They are used to model structural properties of molecules and provide valuable information for fields such as physical chemistry, pharmacology, environmental science, and material science. Recently, novel degree-based molecular structure descriptors, known as Sombor index-like graph invariants, have been explored from a geometrical perspective. These graph invariants have found applications in network science, where they are used to model dynamic effects in biological, social, and technological complex systems. There is also emerging interest in their potential military applications. Among these descriptors is the KG-Sombor index which is defined using both vertex and edge degrees.
Methods: The study uses combinatorial graph theory to identify and analyze extremal graphs that either maximize or minimize the KG-Sombor index.
Results: The extremal graphs are characterized concerning the KGSombor index, with a particular focus on trees, molecular trees, and unicyclic graphs.
Conclusion: This research advances the theoretical understanding of Sombor index-like graph invariants.
References
Cruz, R., Gutman, I. & Rada, J. 2021. Sombor index of chemical graphs. Applied Mathematics and Computation, 399, art.number:126018. Available at: https://doi.org/10.1016/j.amc.2021.126018.
Cruz, R. & Rada, J. 2021. Extremal values of the Sombor index in unicyclic and bicyclic graphs. Journal of Mathematical Chemistry, 59, pp.1098-1116. Available at: https://doi.org/10.1007/s10910-021-01232-8.
Damnjanović, I., Milošević, M. & Stevanović, D. 2023. A Note on Extremal Sombor Indices of Trees with a Given Degree Sequence. MATCH Communications in Mathematical and in Computer Chemistry, 90(1), pp.197-202. Available at: https://doi.org/10.46793/match.90-1.197D.
Das, K.C., Cevik, A.S., Cangul, I.N. & Shang, Y. 2021. On Sombor Index. Symmetry, 13(1), art.number:140. Available at: https://doi.org/10.3390/sym13010140.
Dorjsembe, Sh. & Horoldagva, B. 2022. Reduced Sombor index of bicyclic graphs. Asian-European Journal of Mathematics, 15(07), art.number:2250128. Available at: https://doi.org/10.1142/S1793557122501285.
Gutman, I. 2021. Geometric Approach to Degree–Based Topological Indices: Sombor Indices. MATCH Communications in Mathematical and in Computer Chemistry, 86(1), pp.11-16 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match86/n1/match86n1_11-16.pdf [Accessed: 14 March 2024].
Gutman, I. 2022. Sombor indices - back to geometry. Open Journal of Discrete Applied Mathematics, 5(2), pp.1-5. Available at: https://doi.org/10.30538/psrp-odam2022.0072.
Gutman, I. 2024. Relating Sombor and Euler indices. Vojnotehnički glasnik/Military Technical Courier, 72(1), pp.1-12. Available at: https://doi.org/10.5937/vojtehg72-48818.
Gutman, I. & Das, K.Ch. 2004. The first Zagreb index 30 years after. MATCH Communications in Mathematical and in Computer Chemistry, 50, pp.83-92 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf [Accessed: 14 March 2024].
Horoldagva, B., Selenge, T.-A., Buyantogtokh, L. & Dorjsembe, Sh. 2021. Upper bounds for the reduced second Zagreb index of graphs. Transactions on Combinatorics, 10(3), pp.137-148. Available at: https://doi.org/10.22108/toc.2020.125478.1774.
Horoldagva, B. & Xu, C. 2021. On Sombor index of graphs. MATCH Communications in Mathematical and in Computer Chemistry, 86(3), pp.703-713 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match86/n3/match86n3_703-713.pdf [Accessed: 14 March 2024].
Kosari, S., Dehgardi, N. & Khan, A. 2023. Lower bound on the KG-Sombor index. Communications in Combinatorics and Optimization, 8(4), pp.751-757. Available at: https://doi.org/10.22049/cco.2023.28666.1662.
Kulli, V.R., Harish, N., Chaluvaraju, B. & Gutman, I. 2022. Mathematical properties of KG-Sombor index. Bulletin of International Mathematical Virtual Institute, 12(2), pp.379-386 [online]. Available at: http://www.imvibl.org/buletin/bulletin_imvi_12_2_22/bulletin_imvi_12_2_22_379_386.pdf [Accessed: 14 March 2024].
Liu, H., Gutman, I., You, H. & Huang, Y. 2022. Sombor index: review of extremal results and bounds. Journal of Mathematical Chemistry, 60, pp.771-798. Available at: https://doi.org/10.1007/s10910-022-01333-y.
Rada, J., Rodríguez, J.M. & Sigarreta, J.M. 2021. General properties on Sombor indices. Discrete Applied Mathematics, 299, pp.87-97. Available at: https://doi.org/10.1016/j.dam.2021.04.014.
Selenge, T.-A. & Horoldagva, B. 2024. Extremal Kragujevac trees with respect to Sombor indices. Communications in Combinatorics and Optimization, 9(1), pp.177-183. Available at: https://doi.org/10.22049/cco.2023.28058.1430.
Selenge, T.-A. & Horoldagva, B. 2015. Maximum Zagreb indices in the class of k-apex trees. Korean Journal of Mathematics, 23(3), pp.401-408. Available at: https://doi.org/10.11568/kjm.2015.23.3.401.
Tang, Z., Li, Q. & Deng, H. 2023. Trees with Extremal Values of the Sombor–Index–Like Graph Invariants. MATCH Communications in Mathematical and in Computer Chemistry, 90(1), pp.203-222. Available at: https://doi.org/10.46793/match.90-1.203T.
Zhang, H. & Zhang, S. 2006. Uncyclic graphs with the first three smallest and largest first general Zagreb index. MATCH Communications in Mathematical and in Computer Chemistry, 55(2), pp.427-438 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match55/n2/match55n2_427-438.pdf [Accessed: 14 March 2024].
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