Pacific Journal of Mathematics
We study generalized graph splines, introduced by Gilbert, Tymoczko, and Viel (2016). 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.
Generalized graph splines, Splines
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Anders, Katie; Crans, Alissa S.; Foster-Greenwood, Briana; Mellor, Blake; and Tymoczko, Julianna, "Graphs Admitting Only Constant Splines" (2020). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.