On Harary-Euler Sombor index of a graph

Articles in Press

Document Type : Original paper

Author

Department of Mathematics, Tafresh University

Abstract

Let $G$ be an arbitrary undirected simple connected graph. In this paper, we introduce the modification of the Harary index of $G$ in which the contribution of each edge $uv$ is weighted by $d_u^2 + d_v^2 + d_ud_v$ - a term inspired by the geometry of ellipses - rather than a constant unit weight. Then we compute the values of the Harary-Euler Sombor index of some familiar classes of graphs. Also, we establish mathematical relations between the Harary-Euler Sombor index and other classic indices. Moreover, we state an upper bound for the Harary-Euler Sombor index of bipartite graphs. In addition, we state an upper bound for the Harary-Euler Sombor index of $G$ in terms of the order of $G$ and the largest (smallest) eigenvalue of the Harary-Euler Sombor matrix of $G$ and we introduce a family of graphs for which the given bound is sharp. Finally, we determine the extremum values of the Harary-Euler Sombor index of trees.

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References

[1] N.A. Abd Aziz, A note on graphs with integer Sombor index, Commun. Comb. Optim. 11 (2026), no. 1, 205–209.
https://doi.org/10.22049/cco.2024.29474.2013
[2] S. Akbari, M. Habibi, and S. Rabizadeh, Relations between energy and Sombor index, MATCH Commun. Math. Comput. Chem. 92 (2024), no. 2, 425–435. https://doi.org/10.46793/match.92-2.425A
[3] S. Akbari, M. Habibi, and S. Rouhani, A note on an inequality between energy and Sombor index of a graph, MATCH Commun. Math. Comp. Chem. 90 (2023), no. 3, 765–771. https://doi.org/10.46793/match.90-3.765A
[4] Y. Alizadeh, A. Iranmanesh, and T. Došlić, Additively weighted Harary index of some composite graphs, Discrete Math. 313 (2013), no. 1, 26–34. https://doi.org/10.1016/j.disc.2012.09.011
[5] M. An and L. Xiong, Multiplicatively weighted Harary index of some composite graphs, Filomat 29 (2015), no. 4, 795–805. https://doi.org/10.2298/FIL1504795A
[6] B. Borovićanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comp. Chem. 78 (2017), no. 1, 17–100.
[7] C. Espinal, I. Gutman, and J. Rada, Elliptic Sombor index of chemical graphs, Commun. Comb. Optim. 10 (2025), no. 4, 989–999. https://doi.org/10.22049/cco.2024.29404.1977.
[8] H. Faheem, S. Ahmad, and R. Farooq, Maximal and minimal Zagreb indices of trees with fixed number of vertices of maximum degree, MATCH Commun. Math. Comp. Chem. 95 (2026), no. 1, 233–264. https://doi.org/10.46793/match.95-1.03825
[9] I. Gutman, Geometric approach to degree-based topological indices: Sombor indices, MATCH Commun. Math. Comput. Chem. 86 (2021), no. 1, 11–16.
[10] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), no. 1, 83–92.
[11] M. Habibi, A. Alidadi, and H. Arianpoor, On Harary-Sombor index of graphs, Commun. Comb. Optim. (2025), In press.
https://doi.org/10.22049/cco.2025.29942.2234
[12] M. Habibi and R. Singh, Proof of a conjecture on Sombor index, J. Discrete Math. Appl. 10 (2025), no. 2, 157–160.
https://doi.org/10.22061/jdma.2025.11904.1126
[13] S. Ilanko, Comments on the historical bases of the Rayleigh and Ritz methods, Journal of Sound and Vibration 319 (2009), no. 1-2, 731–733. https://doi.org/10.1016/j.jsv.2008.06.001
[14] A.W. Leissa, The historical bases of the Rayleigh and Ritz methods, Journal of Sound and Vibration 287 (2005), no. 4-5, 961–978. https://doi.org/10.1016/j.jsv.2004.12.021
[15] H. Liu, I. Gutman, L. You, and Y. Huang, Sombor index: review of extremal results and bounds, J. Math. Chem. 60 (2022), no. 5, 771–798. https://doi.org/10.1007/s10910-022-01333-y
[16] H. Lu and Z. Zhu, Product of Wiener and Harary indices of uniform hypergraphs, MATCH Commun. Math. Comput. Chem. 94 (2025), no. 1, 215–228. https://doi.org/10.46793/match.94-1.215L
[17] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[18] S. Rabizadeh, M. Habibi, and I. Gutman, Some notes on Sombor index of graphs, MATCH Commun. Math. Comput. Chem. 93 (2025), no. 3, 853–859. https://doi.org/10.46793/match.93-3.853R
[19] Z. Tang, Y. Li, and H. Deng, The Euler Sombor index of a graph, Int. J. Quantum Chem. 124 (2024), no. 9, e27387.
https://doi.org/10.1002/qua.27387
[20] M. You and H. Deng, The higher-order Sombor index, Commun. Comb. Optim. 9 (2024), no. 3, 579–594. https://doi.org/10.22049/cco.2023.28658.1654
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 11 June 2026

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