Sombor index of thorny graphs
Abstract
Introduction/purpose: The thorny graph of a graph G is obtained by attaching pendent vertices to the vertices of G. A mathematical study of the Sombor index of thorny graphs is undertaken.
Methods: Combinatorial graph theory is applied.
Results: A general expression for the Sombor index of thorny graphs is obtained, as well as lower and upper bounds. Several special cases of this general expression are pointed out.
Conclusion: The paper contributes to the theory of the Sombor index.
References
Alqahtani, M., Kaviyarasu, M. & Rajeshwari, M. 2024. Site Selection for Thermal Power Plant Based on Sombor Index in Neutrosophic Graph. European Journal of Pure and Applied Mathematics, 17(4), pp. 2586–2620. Available at: https://doi.org/10.29020/nybg.ejpam.v17i4.5461.
Anwar, S., Azeem, M., Jamil, M.K., Almohsen, B. & Shang, Y. 2024. Single-valued neutrosophic fuzzy Sombor numbers and their applications in trade flows between different countries via sea route. The Journal of Supercomputing, 80(14), pp. 19976–20019. Available at: https://doi.org/10.1007/s11227-024-06169-8.
Bonchev, D. & Klein, D.J. 2002. On the Wiener Number of Thorn Trees, Stars, Rings, and Rods. Croatica Chemica Acta, 75(2), pp. 613–620 [online]. Available at: https://hrcak.srce.hr/127540 [Accessed: 25 January 2025].
Bondy, J.A. & Murty, U.S.R. 1976. Graph theory with applications. The Macmillan Press Ltd. ISBN: 0-444-19451-70.
De, N. 2012. On Eccentric Connectivity Index and Polynomial of Thorn Graph. Applied Mathematics, 3(8), art.ID:21477. Available at: https://doi.org/10.4236/am.2012.38139.
Došlić, T., Réti, T. & Vukičević, D. 2011. On the vertex degree indices of connected graphs. Chemical Physics Letters, 512(4-6), pp. 283–286. Available at: https://doi.org/10.1016/j.cplett.2011.07.040.
Gutman, I. 2021a. 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: 25 January 2025].
Gutman, I. 2021b. Some basic properties of Sombor indices. Open Journal of Discrete Applied Mathematics (ODAM), 4(1), pp. 1–3. Available at: https://www.doi.org/10.30538/psrp-odam2021.0047.
Gutman, I. 2024. Improved Estimates of Sombor Index. Iranian Journal of Mathematical Chemistry, 15(1), pp. 1–5. Available at: https://www.doi.org/10.22052/ijmc.2023.253825.1782.
Gutman, I. & Das, K.C. 2004. The first Zagreb index 30 years after. MATCH Communications in Mathematical and in Computer Chemistry, 50(1), pp. 83–92 [online]. Available at: https://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf [Accessed: 25 January 2025].
Gutman, I., Popović, L. et al. 1998. Graph representation of organic molecules Cayley’s plerograms vs. his kenograms. Journal of the Chemical Society, Faraday Transactions, 94(7), pp. 857–860. Available at: https://www.doi.org/10.1039/A708076J.
Gutman, I. & Trinajstić, N. 1972. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chemical Physics Letters, 17(4), pp. 535–538. Available at: https://doi.org/10.1016/0009-2614(72)85099-1.
Gutman, I. & Vidovic, D. 1998. Relations between Wiener-type topological indices of plerograms and kenograms. Journal of the Serbian Chemical Society, 63(9), pp. 695–702 [online]. Available at: https://www.shd.org.rs/JSCS/Start.html [Accessed: 25 January 2025].
Hamid, K., Waseem Iqbal, M., Abbas, Q., Arif, M., Brezulianu, A. & Geman, O. 2022. Discovering irregularities from computer networks by topological mapping. Applied Sciences, 12(8), art.number:12051. Available at: https://doi.org/10.3390/app122312051.
Harary, F. 1969. Graph Theory (on Demand Printing Of 02787), 1st Edition. Boca Raton: CRC Press. Available at: https://doi.org/10.1201/9780429493768.
Hayat, S., Arshad, M. & Khan, A. 2024. Graphs with given connectivity and their minimum Sombor index having applications to QSPR studies of monocarboxylic acids. Heliyon, 10(1), e23392. Available at: https://doi.org/10.1016/j.heliyon.2023.e23392.
Imran, M., Azeem, M., Jamil, M.K. & Deveci, M. 2024. Some operations on intuitionistic fuzzy graphs via novel versions of the sombor index for internet routing. Granular Computing, 9(2), art.number:53. Available at: https://doi.org/10.1007/s41066-024-00467-5.
Jamil, M.K., Anwer, S., Azeem, M. & Gutman, I. 2025. Intuitionistic fuzzy Sombor indices: A novel approach for improving the performance of vaccination centers. Communications in Combinatorics and Optimization, 10(3), pp. 563–593. Available at: https://doi.org/10.22049/cco.2023.28767.1709.
Krishnan, V.L. & Narayan, P. 2023. Analysis of Sombor and Harmonic Indices of Thorn Cog-Graphs. Mathematical Modelling of Engineering Problems, 10(5), pp. 1777–1784. Available at: https://doi.org/10.18280/mmep.100529.
Lakshmi, K.V. & Parvathi, N. 2023. An Analysis of Thorn Graph on Topological Indices. IAENG International Journal of Applied Mathematics, 53(3), pp. 313–322 [online]. Available at: https://www.iaeng.org/IJAM/issues_v53/issue_3/IJAM_53_3_38.pdf [Accessed: 25 January 2025].
Li, Y., Deng, H. & Tang, Z. 2024. Sombor index of maximal outerplanar graphs. Discrete Applied Mathematics, 356, pp. 96–103. Available at: https://doi.org/10.1016/j.dam.2024.05.019.
Liu, H., Gutman, I., You, L. & Huang, Y. 2022. Sombor index: review of extremal results and bounds. Journal of Mathematical Chemistry, 60(5), pp. 771–798. Available at: https://doi.org/10.1007/s10910-022-01333-y.
Marinescu-Ghemeci, R. 2010. Radio number for some thorn graphs. Discussiones Mathematicae Graph Theory, 30(2), pp. 201–222. Available at: https://doi.org/10.7151/dmgt.1487.
Milovanović, I., Milovanović, E. & Matejić, M. 2021. On some mathematical properties of Sombor indices. Bulletin of the International Mathematical Virtual Institute, 11(2), pp. 341–353 [online]. Available at: http://www.imvibl.org/buletin/b ulletin_imvi_11_2_2021/bulletin_imvi_10_2_2020_341_353.pdf [Accessed: 25 January 2025].
Nikolić, S., Kovačević, G., Miličević, A. & Trinajstić, N. 2003. The Zagreb indices 30 years after. Croatica Chemica Acta, 76(2), pp. 113–124 [online]. Available at: https://hrcak.srce.hr/103086 [Accessed: 25 January 2025].
Rather, B.A., Imran, M. & Pirzada, S. 2024. Sombor index and eigenvalues of comaximal graphs of commutative rings. Journal of Algebra and its Applications, 23(06), art.number:2450115. Available at: https://doi.org/10.1142/S02194988245011599.
Rauf, A. & Ahmad, S. 2024. On sombor indices of tetraphenylethylene, terpyridine rosettes and QSPR analysis on fluorescence properties of several aromatic hetero-cyclic species. International Journal of Quantum Chemistry, 124(1), e27261. Available at: https://doi.org/10.1002/qua.27261.
Walikar, H.B., Ramane, H.S., Sindagi, L., Shirakol, S.S. & Gutman, I. 2006. Hosoya polynomial of thorn trees, rods, rings, and stars. Kragujevac Journal of Science, 28, pp. 47–56 [online]. Available at: https://www.pmf.kg.ac.rs/KJS/en/volumes/kjs28/kjs28walikarramane47.pdf [Accessed: 25 January 2025].
Wang, Z., Gao, F., Zhao, D. & Liu, H. 2024. Sharp upper bound on the Sombor index of bipartite graphs with a given diameter. Journal of Applied Mathematics and Computing, 70(1), pp. 27–46. Available at: https://doi.org/10.1007/s12190-023-01955-8.
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