Vol. 304, No. 2, 2020

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Graphs admitting only constant splines

Katie Anders, Alissa S. Crans, Briana Foster-Greenwood, Blake Mellor and Julianna Tymoczko

Vol. 304 (2020), No. 2, 385–400
Abstract

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.

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Keywords
splines, generalized graph splines
Mathematical Subject Classification 2010
Primary: 05C25, 05C78, 05E15
Milestones
Received: 8 August 2018
Revised: 25 July 2019
Accepted: 2 October 2019
Published: 12 February 2020
Authors
Katie Anders
Department of Mathematics
University of Texas at Tyler
Tyler, TX
United States
Alissa S. Crans
Department of Mathematics
Loyola Marymount University
Los Angeles, CA
United States
Briana Foster-Greenwood
Department of Mathematics and Statistics
California State Polytechnic University
Pomona, CA
United States
Blake Mellor
Department of Mathematics
Loyola Marymount University
Los Angeles, CA
United States
Julianna Tymoczko
Department of Mathematics and Statistics
Smith College
Northampton, MA
United States