Let $G$ and $G'$ be a finite graph with vertex set of order $n$ and $n'$, respectively.
For the Cartesian product of $G$ and $G'$, the resultant graph is an $\left(n \times n'\right)$ vertex graph. This seems reasonable enough in the sense that each vertex of $G$ map be adjacent to each vertex of $G$ and $G'$.
In the case where there are $m$ different particles moving in this system, the graph is $G^{m} = G \times \cdot \cdot \cdot \times G$ where multiplicative power is through $m$ times.
I would appreciate a physical intuition behind $m$ different particles in the system relates to the graph $G^{m}$. Thanks in advance.
Imagine $m$ copies of $G$ side by side. Inside each $G$, there is one particle. In a time unit, each particle crosses one edge of $G$. A state is determined by the positions of all the $m$ particles. The $m$ copies of $G$ are disconnected from each other.