University of KashanIranian Journal of Mathematical Chemistry2228-64898420171201The ratio and product of the multiplicative Zagreb indices3773904511610.22052/ijmc.2017.53731.1198ENR.KazemiImam Khomeini international universityJournal Article20160509The first multiplicative Zagreb index $Pi_1(G)$ is equal to the product of squares of the degree of the vertices and the second multiplicative Zagreb index $Pi_2(G)$ is equal to the product of<br /> the products of the degree of pairs of adjacent vertices of the underlying molecular graphs $G$. Also, the multiplicative sum Zagreb index $Pi_3(G)$ is equal to the product of the sums of the degree of pairs of adjacent vertices of $G$. In this paper, we introduce a new version of the multiplicative sum Zagreb index and study the moments of the ratio and product of all above indices in a randomly chosen molecular graph with tree structure of order $n$. Also, a supermartingale is introduced by Doob's supermartingale inequality.https://ijmc.kashanu.ac.ir/article_45116_c080bfbf95b3706d865e19550282e4e3.pdf