فهرست نشریات علمی دسترسی آزاد ایران

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.