Graphs associated with matrices over finite fields and their endomorphisms

Let Fm×n be the set of m×n matrices over a field F. Consider a graph G=(Fm×n,∼) with Fm×n as the vertex set such that two vertices A,B∈Fm×n are adjacent if rank(A−B)=1. We study graph properties of G when F is a finite field. In particular, G is a regular connected graph with diameter equal to min{m,n}; it is always Hamiltonian. Furthermore, we determine the independence number, chromatic number and clique number of G. These results are used to characterize the graph endomorphisms of G, which extends Hua's fundamental theorem of geometry on Fm×n.