Extremal Cacti with respect to Sombor index

Document Type : Research Paper


School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P. R. China


Recently, a novel topological index, Sombor index, was introduced by Gutman, defined as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}$ denotes the degree of vertex $u$. In this paper, we first determine the maximum Sombor index among cacti with $n$ vertices and $t$ cycles, then determine the maximum Sombor index among cacti with perfect matchings. We also characterize corresponding maximum cacti.



    1. Alikhani and N. Ghanbari, Sombor index of polymers, MATCH Commun. Math. Comput. Chem. 86 (2021) 715–728.
    2. A. Bondy and U. S. R. Murty, Graph Theory, Springer, New York, 2008.
    3. Chen, W. Li and J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022) 23–49.
    4. Cruz, I. Gutman and J. Rada, Sombor index of chemical graphs, Appl. Math. Comput. 399 (2021) 126018.
    5. Cruz and J. Rada, Extremal values of the Sombor index in unicyclic and bicyclic graphs, J. Math. Chem. 59 (2021) 1098–1116.
    6. C. Das, A. S. Cevik, I. N. Cangul and Y. Shang, On Sombor index, Symmetry 13 (2021) 140.
    7. Deng, Z. Tang and R. Wu, Molecular trees with extremal values of Sombor indices, Int. J. Quantum. Chem. 121 (2021) e26622.
    8. Du, B. Zhou and N. Trinajstic, Minimum sum-connectivity indices of trees and unicyclic graphs of a given matching number, J. Math. Chem. 47 (2010) 842–855.
    9. Fang, L. You and H. Liu, The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs, Int. J. Quantum. Chem. 121 (2021) e26740.
    10. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021) 11–16.
    11. Gutman, Some basic properties of Sombor indices, Open J. Discret. Appl. Math. 4 (2021) 1–3.
    12. Huang and H. Liu, Bounds of modified Sombor index, spectral radius and energy, AIMS Mathematics 6 (2021) 11263–11274.
    13. Horoldagva and C. Xu, On Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 86 (2021) 703–713.
    14. Li and Z. Wang, Trees with extremal spectral radius of weighted adjacency matrices among trees weighted by degree-based indices, Linear Algebra Appl. 620 (2021) 61–75.
    15. Liu, H. Deng and Z. Tang, Minimum Szeged index among unicyclic graphs with

    perfect matchings, J. Comb. Optim. 38 (2019) 443-455.

    1. Liu and Z. Tang, The hyper-Zagreb index of cacti with perfect matchings, AKCE Int. J. Graph Combin. 17 (2020) 422-428.
    2. Liu, L. You and Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem. 87 (2022) 5–22.
    3. Liu, L. You, Y. Huang and X. Fang, Spectral properties of p-Sombor matrices and beyond, MATCH Commun. Math. Comput. Chem. 87 (2022) 59–87.
    4. Liu, L. You, Z. Tang and J. B. Liu, On the reduced Sombor index and its applications, MATCH Commun. Math. Comput. Chem. 86 (2021) 729–753.
    5. Ma and H. Deng, On the sum-connectivity index of cacti, Math. Comput. Model. 54 (2011) 497–507.
    6. Milovanović, E. Milovanović and M. Matejić, On some mathematical properties of Sombor indices, Bull. Int. Math. Virtual Inst. 11 (2021) 341–353.
    7. Redžepović, Chemical applicability of Sombor indices, J. Serb. Chem. Soc. 86 (2021) 445–457.
    8. Réti, T. Došlić and A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021) 11–18.
    9. Wang, Y. Mao, Y. Li and B. Furtula, On relations between Sombor and other degree-based indices, J. Appl. Math. Comput. (2021) DOI:10.1007/s12190-021-01516-x.
    10. Zhou, Z. Lin and L. Miao, The Sombor index of trees and unicyclic graphs with given matching number, arXiv:2103.04645v1.
    11. Zhang, L. You, H. Liu and Y. Huang, The expected values and variances for Sombor indices in a general random chain, Appl. Math. Comput. 411 (2021) 126521.