Extension of hypercube

Question

I understand the notion of a hypercube as a graph with vertex set $\{0,1\}^{n}$ and an edge between two vertices if their vertices differ in one co-ordinate is there an extensive body of work on the case where we have vertex set $\{1,2,...,r\}^{n}$ with edges between two vertices that differ in exactly one co-ordinate? How about when the vertex set is $\{1,2,...,n\}^{n}$? I am particularly interested in the final case, i've tried different searches but i have no idea what this type of graph maybe called.

Answer 1

Your graph is called the Hamming graph $H(n,r)$. It can also be written $(K_r)^{\Box n}$, the $n$th Cartesian power of the complete graph on $r$ vertices.