Geodesic
Trees with the Minimal Second Zagreb Index
Kragujevac Journal of Mathematics, Tome 42 (2018) no. 3, p. 325 .

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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~$n\geq11$.
Classification : 05C07 05C05
Keywords: Second Zagreb index, subdivision, tree 
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M. Eliasi; A. Ghalav. Trees with the Minimal Second Zagreb Index. Kragujevac Journal of Mathematics, Tome 42 (2018) no. 3, p. 325 . http://geodesic.mathdoc.fr/item/KJM_2018_42_3_a0/