  Recent work of An, Drummond-Cole, and Knudsen, as well as the author, has
shown that the homology groups of configuration spaces of graphs can be
equipped with the structure of a finitely generated graded module over a
polynomial ring. In this work we study this module structure in certain
families of graphs using the language of FI-algebras recently explored by Nagel
and R\"omer. As an application we prove that the syzygies of the modules in
these families exhibit a range of stable behaviors.
