The ratio and product of the multiplicative Zagreb‎ ‎indices

Abstract

‎The 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 ‎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.