A simple approach to order the multiplicative Zagreb indices of connected graphs

The first (\Pi_۱) and the second (\Pi_۲) multiplicative Zagreb indices of a connected graph G‎, ‎with vertex set‎ ‎V(G) and edge set E(G)‎, ‎are defined as \Pi_۱(G) = \prod_{u \in‎ ‎V(G)} {d_u}^۲ and \Pi_۲(G) = \prod_{uv \in‎‎E(G)} {d_u}d_{v}‎, ‎respectively‎, ‎where {d_u} denotes the degree of the vertex u‎. ‎In this paper we present a simple approach to order these indices for connected graphs on the same number of vertices‎. ‎Moreover‎, ‎as an application of this‎ ‎simple approach‎, ‎we extend the known ordering of the first and the second multiplicative Zagreb indices‎ ‎for some classes of connected graphs‎.

Keywords:

multiplicative Zagreb index , Majorization , Unicyclic graphs , bicyclic graphs