We study the Universal Difference Property (UDP) introduced by Altınok, Anders, Arreola, Asencio, Ireland, Sarıoğlan, and Smith, focusing on the relationship between the structural properties of a graph and UDP. We present condtions for when UDP must hold on unicyclic graphs. We then prove that if UDP does not hold on an edge-labeled graph, then it cannot hold on any subdivision of that graph. Additionally, we show that if an edge-labeled graph satisfies the pairwise edge-disjoint path property, then the graph satisfies UDP. Lastly, we explore the relationship between UDP and subgraphs and prove that trees and cycles are the only two families of connected graphs for which UDP must hold for any edge-labeling over any ring.