On minimal trees with respect to hyper-Zagreb indices

Dehgardi, Nasrin; Aram, Hamideh; Azari, Mahdieh

doi:10.22108/toc.2025.140617.2146

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	On minimal trees with respect to hyper-Zagreb indices
Transactions on Combinatorics
دوره 15، شماره 2، شهریور 2026، صفحه 77-90 اصل مقاله (460.2 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2025.140617.2146
نویسندگان
Nasrin Dehgardi¹؛ Hamideh Aram²؛ Mahdieh Azari^* ³
¹Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran
²Department of Mathematics, Khoy.C., Islamic Azad University, Khoy, Iran
³Department of Mathematics, Kaz.C., Islamic Azad University, Kazerun, Iran
چکیده
Zagreb indices are among the foremost topological indices in mathematical chemistry. These indices are crucial for investigating the total $\pi$-electron energy of alternant hydrocarbons and are utilized to study various aspects of molecular properties, including complexity, chirality, ZE-isomerism, and hetero-systems. In this paper, we focus on two well-known modifications of these indices: the first and second hyper-Zagreb indices. For a finite simple graph $\Gamma$, these indices are expressed as $$HM_1(\Gamma)=\sum_{\vartheta \omega\in E(\Gamma)}(d_{\Gamma}(\vartheta ) +d_{\Gamma}(\omega))^{2} \ \ {\rm and} \ \ HM_2(\Gamma)=\sum_{\vartheta \omega\in E(\Gamma)}(d_{\Gamma}(\vartheta) d_{\Gamma}( \omega))^{2},$$ where $E(\Gamma)$ denotes the edge set of $\Gamma$ and $d_{\Gamma}(\vartheta)$ indicates the degree of the vertex $\vartheta$ in $\Gamma$. In this paper, we introduce graph transformations on trees and connected graphs that minimize the first and second hyper-Zagreb indices. Accordingly, we determine the minimum values of these indices within the class of all trees with a given number of vertices and a specified maximum vertex degree. Additionally, we characterize the corresponding minimal trees. Our results will be extended to all connected graphs with a given order and maximum vertex degree.
کلیدواژه‌ها
Hyper-Zagreb indices؛ maximum degree؛ tree؛ extremal problems؛ spider

مراجع
[1] A. Ali, I. Gutman, E. Milovanović and I. Milovanović, Sum of powers of the degrees of graphs: Extremal results and bounds, MATCH Commun. Math. Comput. Chem., 80 (2018) 5–84. [2] A. Ali, Y. Shang, D. Dimitrov and T. Réti, Ad-hoc Lanzhou index, Mathematics, 11 no. 20 (2023) 4256. [3] A. R. Ashrafi, T. Došlić and A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math., 158 no. 15 (2010) 1571–1578. [4] A. Alwardi, A. Alqesmah, R. Rangarajan and I. N. Cangul, Entire Zagreb indices of graphs, Discrete Math. Algorithm. Appl., 10 no. 3 (2018) 1850037 p.16. [5] M. Azari, Generalized Zagreb index of product graphs, Trans. Comb., 8 no. 4 (2019) 35–48. [6] M. Azari and A. Iranmanesh, Generalized Zagreb index of graphs, Studia Univ. Babes Bolyai Chem., 56 no. 3 (2011) 59–70. [7] S. Balachandran and T. Vetrı́k, Exponential second Zagreb index of chemical trees, Trans. Comb., 10 no. 2 (2021) 97–106. [8] B. Basavanagoud and S. Patil, The hyper-Zagreb index of four operations on graphs, Math. Sci. Lett., 6 no. 2 (2017) 193–198. [9] B. Borovićanin, K. C. Das, B. Furtula and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem., 78 no. 1 (2017) 17–100. [10] M. V. Diudea, QSPR/QSAR Studies by Molecular Descriptors, Nova Science, Huntingdon, New York, USA, 2000. [11] T. Došlić, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp., 1 (2008) 66–80. [12] M. Eliasi and A. Ghalavand, Ordering of trees by multiplicative second Zagreb index, Trans. Comb., 5 no. 1 (2016) 49–55. [13] S. Elumalai, T. Mansour and M. A. Rostami, New bounds on the hyper-Zagreb index for the simple connected graphs, Electron. J. Graph Theory Appl. (EJGTA), 6 no. 1 (2018) 166–177. [14] F. Falahati-Nezhad and M. Azari, Bounds on the hyper-Zagreb index, J. Appl. Math. Inform., 34 no. 3-4 (2016) 319–330. [15] B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. Chem., 48 (2010) 370–380. [16] W. Gao, M. K. Jamil and M. R. Farahani, The hyper-Zagreb index and some graph operations, J. Appl. Math. Comput., 54 no. 1-2 (2017) 263–275. [17] W. Gao, M. K. Jamil, A. Javed, M. R. Farahani, Sh. Wang and J. B. Liu, Sharp bounds of the hyper-Zagreb index on acyclic, unicylic, and bicyclic graphs, Discrete Dyn. Nat. Soc., 2017 6079450 5 pp. [18] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Int. Math. Virt. Instit., 1 (2011) 13–19. [19] I. Gutman, E. Milovanović and I. Milovanović, Beyond the Zagreb indices, AKCE Int. J. Graphs Comb., 17 no. 1 (2020) 74–85. [20] A. M. Naji and N. D. Soner, The first leap Zagreb index of some graph operations, Int. J. Appl. Graph Theory, 2 no. 1 (2018) 7–18. [21] I. Gutman, B. Ruščić, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys., 62 (1975) 3399–3405. [22] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17 no. 4 (1972) 535–538. [23] A. Ilić and B. Zhou, On reformulated Zagreb indices, Discrete Appl. Math., 160 no. 3 (2012) 204–209. [24] R. Ismail, M. Azeem, Y. Shang, M. Imran and A. A. Ahmad, A unified approach for extremal general exponential multiplicative Zagreb indices, Axioms, 12 no. 7 (2023) 675 14 pp. [25] G. M. Keerthi, R. D. Bhagyashri and H. B. Huchesh, On corellation of physicochemical properties and the hyper Zagreb index for some molecular structures, South East Asian J. Math. Math. Sci., 17 no. 3 (2021) 331–346. [26] L. Luo, N. Dehgardi and A. Fahad, Lower bounds on the entire Zagreb indices of trees, Discrete Dyn. Nat. Soc., 2020 8616725 8 pp. [27] E. Milovanović, M. Matejić and I. Milovanović, Some new upper bounds for the hyper-Zagreb index, Discrete Math. Lett., 1 (2019) 30–35. [28] A. Miličević, S. Nikolić and N. Trinajstić, On reformulated Zagreb indices, Mol. Divers., 8 no. 4 (2004) 393–399. [29] A. Modabish, A. Alameri, M. S. Gumaan and M. Alsharafi, The second hyper-Zagreb index of graph operations, J. Math. Comput. Sci., 11 no. 2 (2021) 1455–1469. [30] A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Comb. Optim., 2 no. 2 (2017) 99–117. [31] J. Rada, Exponential vertex-degree-based topological indices and discrimination, MATCH Commun. Math. Comput. Chem., 82 no. 1 (2019) 29–41. [32] G. V. Rajasekharaiah and U. P. Murthy, Hyper-Zagreb indices of graphs and its applications, J. Algebra Comb. Discrete Appl., 8 no. 1 (2021) 9–22. [33] Z. Raza, S. Akhter and Y. Shang, Expected value of first Zagreb connection index in random cyclooctatetraene chain, random polyphenyls chain, and random chain network, Front. Chem., 10 (2023) 1067874. [34] H. Rezapour, R. Nasiri and S. Mousavi, The hyper-Zagreb index of trees and unicyclic graphs, Iran. J. Math. Sci. Inform., 18 no. 1 (2023) 41–54. [35] M. Rizwan, S. Shahab, A. A. Bhatti, M. Javaid and M. Anjum, On the hyper Zagreb index of trees with a specified degree of vertices, Symmetry, 15 no. 7 (2023) 1295. [36] G. H. Shirdel, H. Rezapour and A. M. Sayadi, The hyper-Zagreb index of graph operations, Iranian J. Math. Chem., 4 no. 2 (2013) 213–220. [37] N. D. Soner and A. M. Naji, The k-distance neighborhood polynomial of a graph, Int. J. Math. Comput. Sci., 3 no. 9 (2016) 2359–2364. [38] R. Todeschini and V. Consonni, New local vertex invariants and molecular descriptors based on functions on the vertex degrees, MATCH Commun. Math. Comput. Chem., 64 no. 2 (2010) 359–372. [39] G. Wei, M. R. Farahani, M. K. Siddiqui and M. K. Jamil, On the first and second Zagreb and first and second hyper-Zagreb indices of carbon nanocones CN Ck [n], J. Comput. Theor. Nanosci., 13 no. 10 (2016) 7475–7482. [40] J-M. Zhu, N. Dehgardi and X. Li, The third leap Zagreb index for trees, J. Chem., 2019 no. 2 (2019) 9296401.
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On minimal trees with respect to hyper-Zagreb indices