Graphs admitting only constant splines

TL;DR

This paper characterizes graphs that only admit constant splines over various rings, using cutset properties to decompose spline spaces, and applies these findings to specific triangulations.

Contribution

It provides a complete characterization of graphs with only constant splines for a broad class of rings, extending prior work and applying to triangulations.

Findings

Graphs with certain cutsets admit only constant splines.

Spline spaces decompose as direct sums based on graph cutsets.

Results apply to triangulations over different rings.

Abstract

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.