Reduced second Zagreb index of product graphs

The reduced second Zagreb index of a graph $G$ is defined as $RM_2(G)=\sum\limits_{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)$, where d$_{G}(v)$ denotes the degree of the vertex $v$ of graph $G$. Recently Furtula et al. (Furtula B., Gutman I., Ediz S. Discrete Appl. Math., 2014) characterized the maximum trees with respect to reduced second Zagreb index. The aim of this paper is to compute reduced second Zagreb index of the Cartesian product of $k\ (\ge 2)$ number of graphs and hence as a consequence the reduced second Zagreb index of some special graphs applicable in various real world problems are computed. Topological properties of different nanomaterials like nanotube, nanotorus etc. are studied here graphically in terms of the aforesaid aforementioned index.