最新的研究成果，本论文的主要观点为图的乘法和Zagreb指数是所有相邻顶点的度数的和的乘积, 它可以被看成是第一类Zagreb指数的乘法版本。 在给出了图的一些运算之后, 确定了至多三个分叉顶点的树的乘法和Zagreb指数的极值, 并完全刻画了具有这些极值的树。 The multiplicative sum Zagreb index of a graph is defines as the production of the sum for the degree pair for all adjacent vertices。 It can be regarded as the multiplicative version of the first kind of Zagreb index。 After giving some operations of the graph, the extremal values of the multiplicative sum Zagreb index for trees with at most three branch vertices are determined, and the trees with these extremal values are completely characterized。不知是否符合录用要求，望您批评与指正。