# On trees and the multiplicative sum Zagreb index

Document Type : Original paper

Authors

1 Dept. of Mathematics, Khansar Faculty of Mathematics and Computer Science, Khansar, Iran,

2 Dept. of Mathematics, Khansar Faculty of Mathematics and Computer Science, Khansar, Iran

Abstract

For a  graph $G$ with  edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as $\Pi^*(G)=\Pi_{uv\in E(G)}[d_G(u)+d_G(v)]$,  where $d_G(v)$ is the degree of vertex $v$ in $G$. In this paper, we first introduce some graph transformations that  decrease this index. In application,  we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb indices  among all trees of  order $n\geq 13$.

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