Homomorphisms on the Coxeter-like graphs

 

Let  $\{tilde\Gamma}_n$ be the graph with the vertex set of all symmetric matrices $S_n(F_2)$ with coefficients from binary field $F_2=\{0,1\}$, where two matrices $A,B \in S_n(F_2)$ form an edge if and only if $rank(A – B) = 1$. Let $\Gamma_n$ be the subgraph in $\{tilde\Gamma}_n$ that is induced by the set of all symmetric invertible matrices  $SGL_n(F2)$. It was shown that both of the graphs $\{tilde\Gamma}_n$  and $\Gamma_n$ are cores for $n \qeq 3$, i.e. all their endomorphisms are automorphisms.  The endomorphisms=automorshims of $ \Gamma_3$,  were characterized by a counting argument and by observing that $\Gamma_3$ is isomorphic to the Coxeter graph, which has a well known automorphism group. Our main task  is to characterize all endomorphisms=automorphism of $\Gamma_n$, for $n\ geq 4$, and describe all homomorphisms from $\Gamma_n$ to  $\{tilde\Gamma}_m$.