  We define a new family of graph invariants, studying the topology of the
moduli space of their geometric realizations in Euclidean spaces, using a
limiting procedure reminiscent of Floer homology. Given a labeled graph $G$ on
$n$ vertices and $d \geq 1$, $W_{G, d} \subseteq \mathbb{R}^{d \times n}$
denotes the space of nondegenerate realizations of $G$ in $\mathbb{R}^d$.The
set $W_{G, d}$ might not be connected, even when it is nonempty, and we refer
to its connected components as rigid isotopy classes of $G$ in $\mathbb{R}^d$.
We study the topology of these rigid isotopy classes. First, regarding the
connectivity of $W_{G, d}$, we generalize a result of Maehara that $W_{G, d}$
is nonempty for $d \geq n$ to show that $W_{G, d}$ is $k$-connected for $d \geq
n + k + 1$, and so $W_{G, \infty}$ is always contractible. While $\pi_k(W_{G,
d}) = 0$ for $G$, $k$ fixed and $d$ large enough, we also prove that, in spite
of this, when $d\to \infty$ the structure of the nonvanishing homology of
$W_{G, d}$ exhibits a stabilization phenomenon: it consists of $(n-1)$ equally
spaced clusters whose shape does not depend on $d$, for $d$ large enough. This
leads to the definition of a family of graph invariants, capturing this
structure. For instance, the sum of the Betti numbers of $W_{G,d}$ does not
depend on $d$, for $d$ large enough; we call this number the Floer number of
the graph $G$. Finally, we give asymptotic estimates on the number of rigid
isotopy classes of $\mathbb{R}^d$--geometric graphs on $n$ vertices for $d$
fixed and $n$ tending to infinity. When $d=1$ we show that asymptotically as
$n\to \infty$ each isomorphism class corresponds to a constant number of rigid
isotopy classes, on average. For $d>1$ we prove a similar statement at the
logarithmic scale.
