Graph invariants from the topology of rigid isotopy classes

TL;DR

This paper introduces a new family of graph invariants based on the topology of the space of geometric realizations, revealing stabilization phenomena and asymptotic behavior of rigid isotopy classes in Euclidean spaces.

Contribution

It defines novel topological invariants of graphs derived from the topology of their realization spaces and analyzes their stability and asymptotic properties.

Findings

The realization space $W_{G,d}$ is $k$-connected for $d \\geq n + k + 1$.

The homology of $W_{G,d}$ stabilizes as $d \\to \\infty$, forming consistent clusters.

The sum of Betti numbers, called the Floer number, is invariant for large $d$.

Abstract

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.