The Ratio and Product of the Multiplicative Zagreb Indices

The first multiplicative Zagreb index1(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.