basic graphs. Using the techniques of this paper, one can construct two graphs $G,G'$ with the same number of vertices and edges for which the homologies of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ and $\operatorname{Conf}_n^{\operatorname{sink}}(G',V')$ are not isomorphic. For instance, take $G$ to be the graph which looks like the capital letter ’Q’ and take $G'$ to be the circle with two vertices.\
Finally, one might immediately observe that Theorems \[treepi1\] and \[dmtmainthm\] are applications of our cellular models to the cases where $G$ is a tree. Similar to the classical theory of configuration spaces of graphs, it appears that studying $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ for general graphs $G$ is quite difficult. We therefore will approach these cases from an entirely different perspective; that of asymptotic algebra.\

Sink configuration spaces and representation stability
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While it appears to be quite difficult to say anything about the homology groups of $\operatorname{Conf}_n^{\operatorname{sink}}(G,V)$ for general graphs, it is often times easier to deduce facts about these groups while allowing $n$ to vary. Many of the great recent advancements in the study of configuration spaces of manifolds have been achieved through the use of **representation stability** techniques (see, [@M][@HR][@CEF][@MW] for a small sampling). The language of representation stability was first introduced by 