  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.
