Graphs associated with matrices over finite fields and their endomorphisms in memory of Professor Michael Neumann and Professor Uri Rothblum

Let Fm×nbe the set of m×n matrices over a field F. Consider a graph G=(Fm×n,∼) with Fm×nas the vertex set such that two vertices A,B∈Fm×nare 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.