Trees with the Minimal Second Zagreb Index

M. Eliasi, A. Ghalav

For simple graph $G$ with edge set $E(G)$, the second Zagreb index of $G$ is defined as $M_{2}(G)=\sum_{uv\in E(G)}[d_G(u)d_G(v)]$, where $d_G(v)$ is the degree of the vertex $v$ in $G$. In this paper, we identify the nine classes of trees, which have the first to the sixth smallest second Zagreb indices, among all the trees of the order~$n\geq11$.