Consider the set $\mathcal{G}_v$ of all finite simple graphs on a given set of $v$ vertices. Let $m={v\choose 2}$ for sake of notation. Given an identification of $\{1,\dots,m\}$ with the set of 2-element subsets of $\{1,\dots,v\}$, there is a natural bijection $\Phi$ from $\mathcal{G}_v$ onto the $m$-hypercube, $Q_m=\{0,1\}^m$, where each component of a vector $q\in Q_m$ corresponds to the presence of an edge in the corresponding graph ($0$ and $1$ corresponding to the lack of or presence of respectively). We define $Q^{k}_m=\{q\in Q_m:\displaystyle\sum_{i=1}^m q_i=k\}$. The automorphism group of $Q_m$ is isomorphic to $S_m\ltimes S_2^m$ (the hyperoctohedral group), where $S_n$ is the symmetric group on $n$ letters.
The equivalence relation "being isomorphic as graphs" on $\mathcal{G}_v$ induces an equivalence relation $\simeq$ on $Q_m$ through the above bijection $\Phi$. I am interested in automorphisms $\phi$ of $Q_m$ which correspond to graph isomorphisms, that is, such that $x\simeq \phi(x)$ for every vertex $x\in Q_m$.
It is clear that such automorphisms must take each subset $Q^k_m$ to itself as it must preserve the number of edges (or rather the norm given the Hamming metric). However, it is unclear to me if there is much more structure to be imposed on these automorphisms (maybe by introducing restrictions on the metric?). Are there perhaps some references that may lead to some insight? I apologize for the rather vague ending - please advise if more description is needed. Thank you.
