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 G. The multiplicative reformulated first Zagreb index is defined as Π_{1, e} (G) = Π_x12x2∈E(G) (dG(x1) +dG(x1) - 2)²,where E(G) is the edge set of a graph G and dG(x₁) is the degree of a vertex x₁ in a graph 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 Π_1,e(G) among the collection of all n-vertex graphs. [ABSTRACT FROM AUTHOR]