  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.
