Let $G_{1i} = G_1 $ and $G_{2i} = G _2 (1 \le i \le k)$ be $k$ copies of graphs $G_1$ and $G_2$.
Let $G_3$ be an arbitrary graph.
The first operation $G_1\sim _k (G_3 \times G _2 )$ of $G_1$ , $G_2$ and $G_3$ is obtained by making the Cartesian product of two graphs $G_ 3$ and $G_ 2$ , thus produces $k$ copies $G_{ 2i}; (1 \le i \le k)$ of $G_ 2$ , then makes $k$ joins $G _{1i} \vee G _{2i} $ where $i = 1,2,...,k.$
I dont understand the above definition and how the graph $G_1\sim _k (G_3 \times G _2 )$ is done for arbitrary graphs.
Can someone kindly explain the above graph for $K_2\sim _k (C_4 \times P_3 )$ where $K_2$ is complete graph on $2$ vertices ,$C_4$ is the cycle on $4 $ vertices and $P_3$ is path on $3$ vertices.