Zero forcing number for Cartesian product of some graphs

The zero forcing number of a graph G, denoted Z(G), is a graph parameter  which is based on a color change rule that describes how to color the vertices. Zero forcing is useful in several branches of science such as electrical engineering, computational complexity and quantum control.  In this paper, we investigate the zero forcing number for Cartesian products of some graphs. The main contribution of this paper is to introduce a new presentation of the Cartesian product of two complete bipartite graphs and to obtain the zero forcing number of these graphs.  We also introduce a purely graph theoretical method to prove $Z(K_n \Box K_m)=mn-m-n+2$.