Some results in the multiplicative version of Sombor index

Movahedi, Fateme

doi:10.47974/JDMSC-2619

Open Access Research Article

Some results in the multiplicative version of Sombor index

^*Fateme MovahediCorresponding authorf.movahedi@gu.ac.irDepartment of MathematicsFaculty of SciencesGolestan UniversityGorgan, 4913815759, Iran0000-0001-7863-7915View full profile →

* Corresponding author · click or hover a name for details

pp. 1–15Online FirstMay 2026DOI: 10.47974/JDMSC-2619 Crossmark XML

Received:: 02 Jun 2025
Published Online:: 22 May 2026
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2619
Pages:: 1–15

Abstract

For a graph 𝒜 including vertex set V𝒜 and edge set E𝒜, the Sombor index is defined as SO(𝒜) = ∑wz∈E𝒜‍ √η𝒜 (w)2 + η𝒜 (z)2 where η𝒜 (w) is the degree of w in 𝒜. This study focus on the multiplicative version of SO, namely the multiplicative Sombor index (𝔐SO) which defined as 𝔐SO (𝒜) = ∏wz∈E𝒜‍ √η𝒜(w)2 + η𝒜(z)2. In this work, sharp upper and lower bounds on 𝔐SO of trees in terms of order and given diameter are obtained. We identify the extremal trees for these bounds. Furthermore, the maximal trees are characterized with respect to order and given number of leaves. We determine sharp bounds of simple graphs for 𝔐SO in terms of graph parameters.

Keywords

Multiplicative sombor index Tree Bound Connected graph

Subject Classifications

05C3505C0705C90

References

[1] F. Harary, Graph Theory. Reading, MA: Addison-Wesley (1972).
[2] J. Devillers and A. T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam: Gordon and Breach (1999).
[3] I. Gutman, “Geometric approach to degree-based topological indices: Sombor indices,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 86, pp. 11-16 (2021).
[4] Z. Samiei and F. Movahedi, “Investigating graph invariants for predicting properties of chemical structures of antiviral drugs,” Polycyclic Aromatic Compounds, vol. 44, no. 10, pp. 6696-6713 (2024).
[5] F. Movahedi and M. H. Akhbari, “Degree-based topological indices of the molecular structure of hyaluronic acid-methotrexate conjugates in cancer treatment,” International Journal of Quantum Chemistry, vol. 123, no. 7, Art. no. e27106 (2023).
[6] J. Rada, J. M. Rodríguez, and J. M. Sigarreta, “Sombor index and elliptic Sombor index of benzenoid systems,” Applied Mathematics and Computation, vol. 475, Art. no. 128756 (2024).
[7] I. Radžeporic’, “Chemical applicability of Sombor indices,” Journal of the Serbian Chemical Society, vol. 86, no. 5, pp. 445-457 (2021).
[8] F. Movahedi, M. Zameni, M. H. Akhbari, M. H. Shahavi, and S. Rajabnezhad, “Topological indices and data analysis techniques modeling to predict the physicochemical properties of tetracycline antibiotics,’’ Journal of Pharmaceutical Sciences, vol. 114, no. 8, Art. no. 103871 (2025), doi: 10.1016/j.xphs.2025.103871.
[9] S. Lal, S. Negi, and V. K. Bhat, “Degree-based topological indices of boron nanotubes,” AIP Advances, vol. 13, no. 10, Art. no. 105321 (2023).
[10] F. Movahedi and M. H. Akhbari, “Investigating Sombor-type topological indices in Titania nanotube structures,” Nanomeghyas, vol. 11, no. 3, pp. 46-58 (2024).
[11] R. Cruz, I. Gutman, and J. Rada, “Sombor index of chemical graphs,” Applied Mathematics and Computation, vol. 399, Art. no. 126018 (2021).
[12] J. Rada, J. M. Rodríguez, and J. M. Sigarreta, “General properties on Sombor indices,” Discrete Applied Mathematics, vol. 299, pp. 87-97 (2021).
[13] H. Liu, “Proof of an open problem on the Sombor index,” Journal of Applied Mathematics and Computing, vol. 69, pp. 2465-2471 (2023).
[14] I. Gutman, N. Gürsoy, A. Gürsoy, and A. Ülker, “New bounds on Sombor index,” Communications in Combinatorics and Optimization, vol. 8, no. 2, pp. 305-311 (2023).
[15] H. Liu, I. Gutman, L. You, and Y. Huang, “Sombor index: review of extremal results and bounds,” Journal of Mathematical Chemistry, vol. 60, pp. 771-798 (2022).
[16] F. Movahedi and M. H. Akhbari, “Entire Sombor index of graphs,” Iranian Journal of Mathematical Chemistry, vol. 14, no. 1, pp. 33-45 (2023).
[17] W. Ning, Y. Song, and K. Wang, “More on Sombor index of graphs,” Mathematics, vol. 10, no. 3, Art. no. 301 (2022).
[18] F. Movahedi, “Extremal trees for Sombor index with given degree sequence,” Iranian Journal of Mathematical Chemistry, vol. 13, no. 4, pp. 281-290 (2022).
[19] T. Zhou, Z. Lin, and L. Miao, “The extremal Sombor index of trees and unicyclic graphs with given matching number,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 26, no. 8, pp. 2205-2216 (2023), doi: 10.1080/09720529.2021.2015090.
[20] S. Dorjsembe, L. Buyantogtokh, and B. Horoldagva, “Reduced Sombor index of graphs,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 1, pp. 77-86 (2025), doi: 10.47974/JDMSC-1845.
[21] K. C. Das, M. Imran, and T. Vetrík, “General Sombor index of graphs and trees,” Journal of Discrete Mathematical Sciences and Cryptography, vol. 28, no. 1, pp. 101-111 (2025), doi: 10.47974/JDMSC-1918.
[22] R. Todeschini and V. Consonni, “New local vertex invariants and molecular descriptors based on functions of the vertex degrees,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 64, no. 2, pp. 359-372 (2010).
[23] H. Liu, “Multiplicative Sombor index of graphs,” Discrete Mathematics Letters, vol. 9, pp. 80-85 (2022).
[24] S. Sharma, V. K. Bhat, and S. Lal, “Multiplicative topological indices of the crystal cubic carbon structure,” Crystal Research and Technology, vol. 58, no. 3, Art. no. 2200222 (2023).
[25] A. Mahboob, S. Mahboob, M. M. Jaradat, N. Nigar, and I. Siddique, “On some properties of multiplicative topological indices in Silicon-Carbon,” Journal of Mathematics, vol. 2021, Art. no. 4611199 (2021).
[26] M. M. Hassan, “Connection number-based multiplicative Zagreb indices of chemical structures,” Current Organic Chemistry, vol. 27, no. 21, pp. 1873-1881 (2023).
[27] Z. Samiei and F. Movahedi, “Some degree-based topological indices of the molecular structure of drugs in the treatment RNA-viruses,” Journal of Information and Optimization Sciences, vol. 47, no. 2, pp. 465-486 (2026).
[28] S. Amin, M. A. Rehman, A. Naseem, J. Younis, and S. S. Asghar, “Analysis of bismuth (III) iodide and dendrimers in drug applications,” Journal of Chemistry, vol. 2022, Art. no. 3163294 (2022).
[29] B. Zhou and Z. Du, “On eccentric connectivity index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 63, pp. 181-198 (2010).
[30] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities. Cambridge: Cambridge University Press (1952).

Views: 185Downloads: 132Citations: 0