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
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 21 اردیبهشت 1404 اصل مقاله (459.5 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, Gareziaeddin center, Khoy Branch, Islamic Azad University, Khoy, Iran
³Department of Mathematics, Kazerun Branch, Islamic Azad University, P. O. Box: 73135-168, 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

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On minimal trees with respect to hyper-Zagreb indices