Graph homomorphisms on rectangular matrices over division rings II

Abstract: Let ${\mathbb{D}}^{m\times n}$ be the set of $m\times n$ matrices over a division ring $\mathbb{D}$. Two matrices $A,B\in {\mathbb{D}}^{m\times n}$ are adjacent if ${\rm rank}(A-B)=1$. By the adjacency, ${\mathbb{D}}^{m\times n}$ is a connected graph. Suppose $\mathbb{D}, \mathbb{D}'$ are division rings and $m,n,m',n'\geq2$ are integers. We determine additive graph homomorphisms from ${\mathbb{D}}^{m\times n}$ to ${\mathbb{D}'}^{m'\times n'}$. When $|\mathbb{D}|\geq 4$, we characterize the graph homomorphism $\varphi: {\mathbb{D}}^{n\times n}\rightarrow {\mathbb{D}'}^{m'\times n'}$ if $\varphi(0)=0$ and there exists $A_0\in {\mathbb{D}}^{n\times n}$ such that ${\rm rank}(\varphi(A_0))=n$. We also discuss properties and ranges on degenerate graph homomorphisms. If $f:{\mathbb{D}}^{m\times n}\rightarrow {\mathbb{D}'}^{m'\times n'}$ (where ${\rm min}{m,n}=2$) is a degenerate graph homomorphism, we prove that the image of $f$ is contained in a union of two maximal adjacent sets of different types. For the case of finite fields, we obtain two better results on degenerate graph homomorphisms.