Hamming图及相关图的自同态
Endomorphisms of The Hamming Graph and Related Graphs
摘要：
本文确定了Hamming图及其它相关图的所有奇异自同态。汉明图具有顶点$\mathbb{Z}^{m}_n$，其中两个顶点相邻，如果它们的汉明距离是$1$。我们证明了它的奇异自同态是一致的（每个核具有相同的大小），并且它们是由拉丁超立方体诱导的（这基本上决定了奇异自同态的数量）。然而，我们做了同样的，它的补充和一些相关的图，其中汉明距离是允许的$1,...,k$之一，对于一些$1\leq k\leq m-1$。最后，我们考虑相同的情况，其中顶点是$\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\times\cdots \times\mathbb{Z}_{n_m}$中的元组（并非所有$n_i$都相等）。
英文摘要：
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).
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