Endomorphisms of quadratic forms graph in characteristic two

Let q be a power of 2, and let Fq be the finite field with q elements. The quadratic forms graph, denoted by Q(n, q) where n >= 2, has all quadratic forms on Fqn as vertices and two vertices f and g are adjacent if rk(f-g)=1 or 2. A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. A graph G is a core if every endomorphism of G is an automorphism. We prove that Q(n, q) is a pseudo-core and Q(2m, q) is a core. Moreover, we gave the smallest eigenvalue of Q(n, q).