On minimal trees with respect to hyper-Zagreb indices

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_۱(\Gamma)=\sum_{\vartheta \omega\in E(\Gamma)}(d_{\Gamma}(\vartheta ) +d_{\Gamma}(\omega))^{۲} \ \ {\rm and} \ \ HM_۲(\Gamma)=\sum_{\vartheta \omega\in E(\Gamma)}(d_{\Gamma}(\vartheta) d_{\Gamma}( \omega))^{۲}, 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.

کلیدواژه ها:

Hyper-Zagreb indices ، maximum degree ، tree ، extremal problems ، Spider