University of Isfahan Transactions on Combinatorics 2251-8657 15 2 2025 05 11 On minimal trees with respect to hyper-Zagreb indices 77 90 29499 10.22108/toc.2025.140617.2146 EN Nasrin Dehgardi Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran Hamideh Aram Department of Mathematics, Khoy.C., Islamic Azad University, Khoy, Iran Mahdieh Azari Department of Mathematics, Kaz.C., Islamic Azad University, Kazerun, Iran 0000-0002-0919-0598 Journal Article 2024 02 04 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. https://toc.ui.ac.ir/article_29499_57537069f1ae6f51859d5ff09fd07f06.pdf