Homomorphisms from the Coxeter graph

Let $S_n\left(F_2\right)$ be the set of all $n\times n$ symmetric matrices with coefficients in the binary field $F_2=\{0,1\}$, and let $SG{L}_n\left(F_2\right)$ be its subset formed by invertible matrices. Let ${\hat{\Gamma }}_n$ be the graph with the vertex set $S_n\left(F_2\right)$ where a pair of vertices $\{A,B\}$ form an edge if and only if $\mathrm{rank}(A-B)=1$. Similarly, let ${\Gamma }_n$ be the subgraph in ${\hat{\Gamma }}_n$, which is induced by the set $SG{L}_n\left(F_2\right)$. Graph ${\Gamma }_n$ generalizes the well-known Coxeter graph, which is isomorphic to ${\Gamma }_3$. Motivated by research topics in coding theory, matrix theory, and graph theory, this paper represents the first step towards the characterization of all graph homomorphisms $\Phi :{\Gamma }_n\to {\hat{\Gamma }}_m$ where $n,m$ are positive integers. Here, the case $n=3$ is solved.