Graphs admitting only constant splines
Katie A. Anders, Alissa S. Crans, Briana Foster-Greenwood, Blake Mellor, Julianna Tymoczko · Pacific Journal of Mathematics · 2020
We study {\em generalized graph splines,} introduced by Gilbert, Viel, and the last author. For a large class of rings, we characterize the graphs that only admit constant splines. To do this, we prove that if a graph has a particular type of cutset (e.g., a bridge), then the space of splines naturally decomposes as a certain direct sum of submodules. As an application, we use these results to describe splines on a triangulation studied by Zhou and Lai, but over a different ring than they used.