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Abstract
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Suppose $G$ is a molecular graph with edge set $E(G)$, the Hyper-Zagreb index of $G$ is defined as $\displaystyle HM(G)=\sum_{uv\in E(G)}[deg_{G}(u) deg_{G}(v)]^2$, where $deg_G(u)$ is the degree of a vertex $u$ in $G$. In this paper, we characterize the chemical trees of order $n\ge 12$ with the first twenty smallest Hyper Zagreb index.
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