on the multiplicative‎ ‎reformulated‎ ‎first‎ ‎zagreb index of n-vertex‎ ‎trees ‎ ‎with respect to matching number

‎the multiplicative first zagreb index is the product of the square of the degree of vertices in a graph $mathbb{g}‎. ‎the multiplicative reformulated first zagreb index is defined as prod_{1,e}(mathbb{g})= prod_{x_{1}x_{2}in e(mathbb{g})}(d_{mathbb{g}}(x_{1})+d_{mathbb{g}}(x_{1})-2)^{2}‎, ‎where e(mathbb{g}) is the edge set of a graph mathbb{g}and d_{mathbb{g}}(x_{1}) is the degree of a vertex x_{1} in a graph mathbb{g}‎. ‎in this paper‎, ‎we characterize the minimum and maximum trees and unicyclic graphs with respect to matching and perfect matching using this graph invariant prod_{1,e}(mathbb{g}) among the collection of all n-vertex graphs‎.