on trees and the multiplicative sum zagreb index

For a graph g with edge set e(g), the multiplicative sum zagreb index of g is defined as π∗(g)=πuv∈e(g)[dg(u)+dg(v)], where dg(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≥13.