Computing stable homology representations of graph configuration spaces

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.