The distance function on Coxeter like graphs

Let $S_n(\mathbb{F}_2)$ be the set of all $n\times n$ 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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