Homomorphisms on the Coxeter-like graphs

 

Let  ${tildeGamma}_n$ 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 ${tildeGamma}_n$ that is induced by the set of all symmetric invertible matrices  $SGL_n(F2)$. It was shown that both of the graphs ${tildeGamma}_n$  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  ${tildeGamma}_m$.