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Sombor index: review of extremal results and bounds

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Abstract

The Sombor index was introduced in 2020, and was soon followed by a remarkably large number of studies, both chemical and mathematical. In this review we collect the existing bounds and extremal results related to the Sombor index and its variants.

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References

  1. A. Aashtab, S. Akbari, S. Madadinia, M. Noei, F. Salehi, On the graphs with minimum Sombor index. MATCH Commun. Math. Comput. Chem. 88, 1 (2022)

    Google Scholar 

  2. R. Aguilar-Sánchez, J.A. Méndez-Bermúdez, J.M. Rodríguez, J.M. Sigarreta, Normalized Sombor indices as complexity measures of random networks. Entropy 23, 976 (2021)

    PubMed  PubMed Central  Google Scholar 

  3. M.O. Albertson, The irregularity of a graph. ARS Comb. 46, 219–225 (1997)

    Google Scholar 

  4. S. Alikhani, N. Ghanbari, Sombor index of polymers. MATCH Commun. Math. Comput. Chem. 86, 715–728 (2021)

    Google Scholar 

  5. S. Amin, A.U.R. Virk, M.A. Rehman, N.A. Shah, Analysis of dendrimer generation by Sombor indices. Hindawi J. Chem. 2021, 9930645 (2021)

    Google Scholar 

  6. J.A. Bondy, U.S.R. Murty, Graph Theory (Springer, New York, 2008)

    Google Scholar 

  7. H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees. MATCH Commun. Math. Comput. Chem. 87, 23–49 (2022)

    Google Scholar 

  8. P. Chinglensana, S. M. Mawiong, On Sombor coindex of graphs, arXiv: 2109.03108v2

  9. R. Cruz, J. Rada, Extremal values of the Sombor index in unicyclic and bicyclic graphs. J. Math. Chem. 59, 1098–1116 (2021)

    CAS  Google Scholar 

  10. R. Cruz, J. Rada, J.M. Sigarreta, Sombor index of trees with at most three branch vertices. Appl. Math. Comput. 409, 126414 (2021)

    Google Scholar 

  11. R. Cruz, I. Gutman, J. Rada, Sombor index of chemical graphs. Appl. Math. Comput. 399, 126018 (2021)

    Google Scholar 

  12. K.C. Das, I. Gutman, On Sombor index of trees. Appl. Math. Comput. 412, 126575 (2022)

    Google Scholar 

  13. K.C. Das, Y. Shang, Some extremal graphs with respect to Sombor index. Mathematics 9, 1202 (2021)

    Google Scholar 

  14. K.C. Das, I. Gutman, I. Milovanović, E. Milovanović, B. Furtula, Degree-based energies of graphs. Linear Algebra Appl. 554, 185–204 (2018)

    Google Scholar 

  15. K.C. Das, A.S. Çevik, I.N. Cangul, Y. Shang, On Sombor index. Symmetry 13, 140 (2021)

    Google Scholar 

  16. K.C. Das, A. Ghalavand, A.R. Ashraf, On a conjecture about the Sombor index of graphs. Symmetry 13, 1830 (2021)

    Google Scholar 

  17. H. Deng, Z. Tang, R. Wu, Molecular trees with extremal values of Sombor indices. Int. J. Quantum Chem. 121, e26622 (2021)

    CAS  Google Scholar 

  18. S. Dorjsembe, B. Horoldagva, Reduced Sombor index of bicyclic graphs. Asian-Europ. J. Math. (2021). https://doi.org/10.1142/S1793557122501285

    Article  Google Scholar 

  19. T. Došlić, T. Réti, A. Ali, On the structure of graphs with integer Sombor indices. Discr. Math. Lett. 7, 1–4 (2021)

    Google Scholar 

  20. M. Eliasi, A. Iranmanesh, On ordinary generalized geometric–arithmetic index. Appl. Math. Lett. 24, 582–587 (2011)

    Google Scholar 

  21. M. Eliasi, A. Iranmanesh, I. Gutman, Multiplicative versions of first Zagreb index. MATCH Commun. Math. Comput. Chem. 68, 217–230 (2012)

    Google Scholar 

  22. S. Fajtlowicz, On conjectures of Graffiti-II. Congr. Numer. 60, 187–197 (1987)

    Google Scholar 

  23. X. Fang, L. You, 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, e26740 (2021)

    CAS  Google Scholar 

  24. S. Filipovski, Relations between Sombor index and some degree-based topological indices. Iran. J. Math. Chem. 12, 19–26 (2021)

    Google Scholar 

  25. B. Furtula, I. Gutman, A forgotten topological index. J. Math. Chem. 53, 1184–1190 (2015)

    CAS  Google Scholar 

  26. N. Ghanbari, On the Sombor characteristic polynomial and Sombor energy of a graph, arXiv: 2108.08552v1

  27. N. Ghanbari, S. Alikhani, Sombor index of certain graphs. Iran. J. Math. Chem. 12, 27–37 (2021)

    Google Scholar 

  28. K.J. Gowtham, I. Gutman, On characteristic polynomial and energy of Sombor matrix. Open J. Discrete Appl. Math. 4, 29–35 (2021)

    Google Scholar 

  29. K.J. Gowtham, N.N. Swamy, On Sombor energy of graphs. Nanosyst. Phys. Chem. Math. 12, 411–417 (2021)

    CAS  Google Scholar 

  30. I. Gutman, Sombor index-one year later. Bull. Acad. Serb. Sci. Arts 153, 43–55 (2020)

    Google Scholar 

  31. I. Gutman, Geometric approach to degree-based topological indices: Sombor indices. MATCH Commun. Math. Comput. Chem. 86, 11–16 (2021)

    Google Scholar 

  32. I. Gutman, Some basic properties of Sombor indices. Open J. Discrete Appl. Math. 4, 1–3 (2021)

    Google Scholar 

  33. I. Gutman, Spectrum and energy of the Sombor matrix. Military Tech. Courier 69, 551–561 (2021)

    Google Scholar 

  34. I. Gutman, Comparing degree-based energies of trees. Contrib. Math. 4, 1–5 (2021)

    Google Scholar 

  35. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals, Total \(\pi\)-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)

    CAS  Google Scholar 

  36. I. Gutman, B. Ruščić, N. Trinajstić, C.F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Chem. Phys. 62, 3399–3405 (1975)

    CAS  Google Scholar 

  37. I. Gutman, B. Furtula, C. Elphick, Three new/old vertex-degree-based topological indices. MATCH Commun. Math. Comput. Chem. 72, 617–632 (2014)

    Google Scholar 

  38. I. Gutman, V.R. Kulli, I. Redžepović, Sombor index of Kragujevac trees. Sci. Publ. Univ. Novi Pazar Ser. A 13, 61–70 (2021)

    Google Scholar 

  39. I. Gutman, J. Monsalve, J. Rada, A relation between a vertex-degree-based topological index and its energy. Linear Algebra Appl. 636, 134–142 (2022)

    Google Scholar 

  40. J. C. Hernández, J. M. Rodríguez, O. Rosario, J. M. Sigarreta, Optimal inequalities and extremal problems on the general Sombor index, arXiv: 2108.05224v1

  41. B. Horoldagva, C. Xu, On Sombor index of graphs. MATCH Commun. Math. Comput. Chem. 86, 703–713 (2021)

    Google Scholar 

  42. Y. Huang, H. Liu, On the modified Sombor indices of some aromatic compounds. J. South China Normal Univ. Nat. Sci. Ed. 53, 91–99 (2021)

    Google Scholar 

  43. Y. Huang, H. Liu, Bounds of modified Sombor index, spectral radius and energy. AIMS Math. 6, 11263–11274 (2021)

    Google Scholar 

  44. X. Li, Z. Wang, Trees with extremal spectral radius of weighted adjacency matrices among trees weighted by degree-based indices. Linear Algebra Appl. 620, 61–75 (2021)

    Google Scholar 

  45. S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter. Appl. Math. Comput. 416, 126731 (2022)

    Google Scholar 

  46. Z. Lin, L. Miao, On the spectral radius, energy and Estrada index of the Sombor matrix of graphs. arXiv:2102.03960v2

  47. H. Liu, Extremal problems on Sombor indices of unicyclic graphs with a given diameter. arXiv:2107.10673v2

  48. H. Liu, Extremal cacti with respect to Sombor index. Iran. J. Math. Chem. 12, 197–208 (2021)

    Google Scholar 

  49. H. Liu, H. Chen, Q. Xiao, X. Fang, Z. Tang, More on Sombor indices of chemical graphs and their applications to the boiling point of benzenoid hydrocarbons. Int. J. Quantum Chem. 121, e26689 (2021)

    CAS  Google Scholar 

  50. H. Liu, L. You, Z. Tang, J.B. Liu, On the reduced Sombor index and its applications. MATCH Commun. Math. Comput. Chem. 86, 729–753 (2021)

    Google Scholar 

  51. H. Liu, L. You, Y. Huang, Extremal Sombor indices of tetracyclic (chemical) graphs. MATCH Commun. Math. Comput. Chem. 88, 1 (2022)

    Google Scholar 

  52. H. Liu, L. You, Y. Huang, Ordering chemical graphs by Sombor indices and its applications. MATCH Commun. Math. Comput. Chem. 87, 5–22 (2022)

    Google Scholar 

  53. H. Liu, L. You, Y. Huang, X. Fang, Spectral properties of p-Sombor matrices and beyond. MATCH Commun. Math. Comput. Chem. 87, 59–87 (2022)

    Google Scholar 

  54. J.A. Méndez-Bermúdez, R. Aguilar-Sánchez, E.D. Molina, J.M. Rodríguez, Mean Sombor index. Discrete Math. Lett. 9, 18–25 (2022)

    Google Scholar 

  55. I. Milovanović, E. Milovanović, A. Ali, M. Matejić, Some results on the Sombor indices of graphs. Contrib. Math. 3, 59–67 (2021)

    Google Scholar 

  56. I. Milovanović, E. Milovanović, M. Matejić, On some mathematical properties of Sombor indices. Bull. Int. Math. Virtual Inst. 11, 341–353 (2021)

    Google Scholar 

  57. W. Ning, Y. Song, K. Wang, More on Sombor index of graphs. Mathematics 10, 301 (2022)

    Google Scholar 

  58. M.R. Oboudi, Non-semiregular bipartite graphs with integer Sombor index. Discrete Math. Lett. 8, 38–40 (2022)

    Google Scholar 

  59. C. Phanjoubam, S.M. Mawiong, On Sombor index and some topological indices. Iran. J. Math. Chem. 12, 209–215 (2021)

    Google Scholar 

  60. C. Phanjoubam, S. M. Mawiong, On general Sombor index. arXiv:2110.03225v1

  61. J. Rada, J.M. Rodriguez, J.M. Sigarreta, General properties on Sombor indices. Discrete Appl. Math. 299, 87–97 (2021)

    Google Scholar 

  62. M. Randić, On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975)

    Google Scholar 

  63. I. Redžepović, Chemical applicability of Sombor indices. J. Serb. Chem. Soc. 86, 445–457 (2021)

    Google Scholar 

  64. I. Redžepović, I. Gutman, Relating energy and Sombor energy–An empirical study. MATCH Commun. Math. Comput. Chem. 88, 133–140 (2022)

    Google Scholar 

  65. T. Réti, T. Došlić, A. Ali, On the Sombor index of graphs. Contrib. Math. 3, 11–18 (2021)

    Google Scholar 

  66. Y. Shang, Sombor index and degree-related properties of simplicial networks. Appl. Math. Comput. 419, 126881 (2022)

    Google Scholar 

  67. Y. Shao, Y. Gao, W. Gao, X. Zhao, Degree-based energies of trees. Linear Algebra Appl. 621, 18–28 (2021)

    Google Scholar 

  68. X. Sun, J. Du, On Sombor index of trees with fixed domination number. Appl. Math. Comput. 421, 126946 (2022)

    Google Scholar 

  69. A. Ülker, A. Gürsoy, N.K. Gürsoy, The energy and Sombor index of graphs. MATCH Commun. Math. Comput. Chem. 87, 51–58 (2022)

    Google Scholar 

  70. A. Ülker, A. Gürsoy, N.K. Gürsoy, I. Gutman, Relating graph energy and Sombor index. Discr. Math. Lett. 8, 6–9 (2022)

    Google Scholar 

  71. S.O. Ünal, An application of Sombor index over a special class of semigroup graph. Hindawi J. Math. 2021, 3273117 (2021)

    Google Scholar 

  72. Z.K. Vukićević, On the Sombor index of chemical trees. Mathematica Montisnigri 50, 5–14 (2021)

    Google Scholar 

  73. D. Vukičević, B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J. Math. Chem. 46, 1369–1376 (2009)

    Google Scholar 

  74. D. Vukičević, M. Gašperov, Bond aditive modeling 1. Adriatic indices. Croat. Chem. Acta 83, 243–260 (2010)

    Google Scholar 

  75. Z. Wang, Y. Mao, Y. Li, B. Furtula, On relations between Sombor and other degree-based indices. J. Appl. Math. Comput. 68, 1–17 (2022)

    Google Scholar 

  76. J. Ye, J. Qian, Extremal polygonal cacti for general Sombor index. arXiv:2108.12775v2

  77. W. Zhang, L. You, H. Liu, Y. Huang, The expected values and variances for Sombor indices in a general random chain. Appl. Math. Comput. 411, 126521 (2021)

    Google Scholar 

  78. T. Zhou, Z. Lin, L. Miao, The Sombor index of trees and unicyclic graphs with given maximum degree. Discrete Math. Lett. 7, 24–29 (2021)

    Google Scholar 

  79. T. Zhou, Z. Lin, L. Miao, The Sombor index of trees and unicyclic graphs with given matching number. arXiv:2103.04645v1

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Acknowledgements

We would like to thank the anonymous referees for invaluable comments and suggestions. This paper is supported by the National Natural Science Foundation of China (Grant Nos. 11971180, 11501139), the Guangdong Provincial Natural Science Foundation (Grant No. 2019A1515012052), the Hunan Provincial Natural Science Foundation of China (Grant No. 2020JJ4423), and the Department of Education of Hunan Province (Grant No. 19A318).

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Hechao Liu & Lihua You

  2. Faculty of Science, University of Kragujevac, P.O.Box 60, 34000, Kragujevac, Serbia

    Ivan Gutman

  3. Department of Mathematics Teaching, Guangzhou Civil Aviation College, Guangzhou, 510403, People’s Republic of China

    Yufei Huang

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  1. Hechao Liu
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  2. Ivan Gutman
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Correspondence to Lihua You.

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Liu, H., Gutman, I., You, L. et al. Sombor index: review of extremal results and bounds. J Math Chem 60, 771–798 (2022). https://doi.org/10.1007/s10910-022-01333-y

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