Multiplicative Zagreb indices of cacti

Let $\prod (G)$ be Multiplicative Zagreb index of a graph G. A connected graph is a cactus graph if and only if any two of its cycles have at most one vertex in common, which has been the interest of researchers in the filed of material chemistry and graph theory. In this paper, we use a new tool to the obtain upper and lower bounds of $\prod (G)$ for all cactus graphs and characterize the corresponding extremal graphs.