The distance function on Coxeter like graphs

Let $S_n(mathbb{F}_2)$ be the set of all $ntimes n$ symmetric matrices with coefficients from the binary field $mathbb{F}_2={0,1}$, and let $SGL_n(mathbb{F}_2)$ be the subset of all invertible matrices.
Let $tilde{Gamma}_n$ be the graph with the vertex set $S_n(mathbb{F}_2)$, where two matrices $A, B in S_n (mathbb{F}_2)$ form an edge if and only if $text{rank}(A-B)=1$.
Let $Gamma_n$ be the subgraph in $tilde{Gamma}_n$, which is induced by the set $SGL_n(mathbb{F}_2)$. If $n=3$, $Gamma_n$ is the Coxeter graph.
It is well-known that is a distance function  on  $tilde{Gamma}_n$ is given by
$$d(A,B) =
  begin{cases}
    text{rank}(A-B),       & quad text{if } A-B  text{ is nonalternate or zero,}\
    text{rank}(A-B)+1,   & quad text{if } A-B text{ is alternate and nonzero.}
  end{cases}
$$
Even the Coxeter graph shows that the distance in $Gamma _n$ must be different.
The main goal is to describe the distance function on $Gamma_n$.
Joint work with Marko Orel.

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