the ratio and product of the multiplicative zagreb‎ ‎indices

The first multiplicative zagreb index 1(g) is equal to the product of squares of the degree of the vertices and the second multiplicative zagreb index 2(g) is equal to the product of the products of the degree of pairs of adjacent vertices of the underlying molecular graphs g . also, the multiplicative sum zagreb index 3(g) is equal to the product of the sums of the degree of pairs of adjacent vertices of g . in this paper, weintroduce a new version of the multiplicative sum zagreb index and study the moments of the ratio and product of all indices in a randomly chosen molecular graph with tree structure of order n . also, a supermartingale is introduced by doob’s supermartingale inequality.