On trees and the multiplicative sum Zagreb index

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 $n\geq 13$‎.

Keywords

• Multiplicative sum Zagreb index‎
• ‎Graph transformation‎
• ‎Branching point‎
• ‎Trees