  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.
