Endomorphisms of The Hamming Graph and Related Graphs

In this paper we determine all singular endomorphisms of the Hamming graph and other related graphs. The Hamming graph has vertices $\mathbb{Z}^{m}_n$ where two vertices are adjacent, if their Hamming distance is $1$. We show that its singular endomorphisms are uniform (each kernel has the same size) and that they are induced by Latin hypercubes (which essentially determines the number of singular endomorphisms). However, we do the same for its complement and some related graphs where the Hamming distance is allowed to be one of $1,...,k$, for some $1\leq k\leq m-1$. Ultimately, we consider the same situation where the vertices are tuples in $\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\times\cdots \times\mathbb{Z}_{n_m}$ (not all $n_i$ are equal).