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 Pi(1,e) (G) = Pi(x1x2 is an element of E(G)) (d(G)(x(1)) + d(G)(x(1)) - 2)(2), where E(G) is the edge set of a graph G and d(G)(x(1)) is the degree of a vertex x(1) 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 Pi(1,e)(G) among the collection of all n-vertex graphs. (c) 2024 University of Kashan Press. All rights reserved.