University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 8 4 2017 12 01 The Ratio and Product of the Multiplicative Zagreb‎ Indices 377 390 45116 10.22052/ijmc.2017.53731.1198 EN R. Kazemi Imam Khomeini international university Journal Article 2016 05 09 ‎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<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