Configuration spaces of graphs frequently grow factorially in complexity with the number of particles they parametrize. However, for suitable families of nested graphs $G_\bullet$ with compatible symmetric group actions, Ramos and White prove that, for fixed $k$, the rational homology of the $k$\textsuperscript{th} configuration spaces of $G_\bullet$ has multiplicity stability. In the current work, we derive the stable range and use computer algebra to determine the stable representations on homology for $k=2$ and $G_\bullet$ several families of graphs, including the complete graphs, the complete bipartite graphs on $2n$ vertices, the crown graphs on $2n$ vertices, and the complete tripartite graphs on $2n+1$ vertices. We determine the stable multiplicities for certain irreducible components in the case $k=3$ and $G_\bullet$ the complete graphs.

翻译：图的配置空间通常随着其参数化的粒子数量呈阶乘级增长。然而，对于具有相容对称群作用的嵌套图族 $G_\bullet$ 而言，Ramos 和 White 证明，在固定 $k$ 的情况下，$G_\bullet$ 的第 $k$ 个配置空间的有理同调具有重数稳定性。在当前工作中，我们推导出稳定范围，并利用计算机代数确定 $k=2$ 时 $G_\bullet$ 若干图族（包括完全图、$2n$ 个顶点上的完全二分图、$2n$ 个顶点上的皇冠图以及 $2n+1$ 个顶点上的完全三分图）的同调稳定表示。对于 $k=3$ 且 $G_\bullet$ 为完全图的情况，我们确定了某些不可约分量的稳定重数。