We consider two graphs, the toroidal knight's move graph, given by $K(p,q)$, and the toroidal triangular grid, given by $T(p,q)$. These graphs are simply the Knight's graph and triangular grid extended to a torus so edges can "wrap" around.
The vertices of $K(p,q)$ are elements of $\mathbb Z_p \times \mathbb Z_q$, where $(v,w) \sim (s,t)$ if and only if either $(v-s \in \{\pm 1\}, w-t\in\{\pm2\})$ or $(v-s \in \{\pm 2\}, w-t\in\{\pm1\})$.
The vertices of $T(p,q)$ are elements of $\mathbb Z_p \times \mathbb Z_q$, where $(v,w) \sim (s,t)$ if and only if either $(v=s, w-t\in\{\pm1\})$ or $(v-s \in \{\pm 1\}, w=t)$, or $(v-s = w-t\in\{\pm1\})$.
How might we go about computing the spectrum of $K(8,8)$ and $T(8,8)$, or more generally $K(p,q)$ and $T(p,q)$? My understanding would be to maybe convert the graphs to more known graphs by cartesian, direct, strong products, and determine the spectrum that way, but I don't quite see how to do so.