Generalized graph splines and the Universal Difference Property

TL;DR

This paper investigates the Universal Difference Property (UDP) in generalized graph splines, establishing conditions under which UDP holds for various graph classes and rings, and demonstrating its invariance under graph isomorphisms.

Contribution

It characterizes when UDP holds for edge labeled graphs over rings, linking it to Pr"ufer domains, and proves UDP's preservation under graph isomorphisms.

Findings

Paths, trees, and cycles satisfy UDP.

UDP holds over a ring if and only if it is a Pr"ufer domain.

UDP is preserved by graph isomorphisms.

Abstract

We study the generalized graph splines introduced by Gilbert, Tymoczko, and Viel and focus on an attribute known as the Universal Difference Property (UDP). We prove that paths, trees, and cycles satisfy UDP. We explore UDP on graphs pasted at a single vertex and use Pr\"ufer domains to illustrate that not every edge labeled graph satisfies UDP. We show that UDP must hold for any edge labeled graph over a ring $R$ if and only if $R$ is a Pr\"ufer domain. Lastly, we prove that UDP is preserved by isomorphisms of edge labeled graphs.