Let us have two Groups $G$ and $H$. Then, is the Cayley graph associated with $G\times H$, the direct product of the groups with respect to some generating set with cardinality $mn$; a product of the Cayley graphs associated to groups $G$ and $H$ with generating sets of cardinalities $m$ and $n$ respectively? If so, is the graph product a cartesian product?
I think yes, because the product group acts transitively on the product of Cayley graphs, I think? Thanks beforehand.
Let $G$ and $H$ be groups, with generating sets $S$ and $T$. Then $(S\times \{e_H\})\cup (\{e_G\}\times T)$ is a generating set for $G\times H$. Moreover, using Wikipedia's notation, $$ \Gamma(G\times H, (S\times \{e_H\})\cup (\{e_G\}\times T))\cong \Gamma(G,S)\times \Gamma(H,T) $$ where the rightmost $\times$ is a Cartesian product. This is because the Cayley graph for $G\times H$ has an edge of the form $(g,h)\to (gs,h)$ and $(g,h)\to (g,ht)$, which exactly fits the definition of the Cartesian product of $\Gamma(G,S)$ and $\Gamma(H,T)$.
It may help to look at some examples, like $(G,S)=(H,T)=(\mathbb Z_2,\{1\})$, where the generating set for $G\times H=\mathbb Z_2\times \mathbb Z_2$ is $\{(1,0),(0,1)\}$.