The multiplicative Zagreb indices of graph operations

Recently, Todeschini et al. \cite{tbc,tc2010} have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: $$ \prod_1=\prod_{1}(G)=\prod\limits_{v\in V(G)}d_{G}(v)^{2}\, , \qquad \prod_2=\prod_{2}(G)=\prod\limits_{uv\in E(G)}d_{G}(u)d_{G}(v)\,. $$ These two graph invariants are called \textit{multiplicative Zagreb indices} by Gutman \cite{gut2011}. In this paper the upper bounds on the multiplicative Zagreb indices of the join, cartesian product, corona product, composition and disjunction of graphs are derived and the indices evaluated for some well known graphs.