The ratio and product of the multiplicative Zagreb‎ ‎indices

ýThe first multiplicative Zagreb index $\Pi_1(G)$ is equal to theý ýproduct of squares of the degree of the vertices and the secondý ýmultiplicative Zagreb index $\Pi_2(G)$ is equal to the product ofý ýthe products of the degree of pairs of adjacent vertices of theý ýunderlying molecular graphs $G$ý. ýAlsoý, ýthe multiplicative sum Zagreb index $\Pi_3(G)$ is equal to the product ofý ýthe sums of the degree of pairs of adjacent vertices of $G$ý. ýIný ýthis paperý, ýwe introduce a new version of the multiplicative sumý ýZagreb index and study the moments of the ratio and product of ýall aboveý indices in a randomly chosen molecular graph with tree structure of order $n$. þAlso, a ýsupermartingale is introduced by ýýDoob's supermartingaleý ýinequality.