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

Abstract

The first ($Pi_1$) and the second $(Pi_2$) multiplicative Zagreb indices of a connected graph $G$‎, ‎with vertex set‎ ‎$V(G)$ and edge set $E(G)$‎, ‎are defined as $Pi_1(G) = prod_{u in‎ ‎V(G)} {d_u}^2$ and $Pi_2(G) = prod_{uv in‎‎E(G)} {d_u}d_{v}$‎, ‎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.