Bases for Integer Generalized Splines on Graphs, an Investigation
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Let R be a ring and G a graph whose edges are labeled by ideals of R. A vertex labeling of G by elements of the ring is called a generalized spline if for each edge, the difference of the labels of the two connected vertices lies in the edge's ideal. The author examines the algebraic structure of the set of generalized splines -it is actually an R-module- and present results of Handschy, Melnick and Reinders concerning the existence of splines where the graph is a cycle and the ring is Z. The author also summarizes more general results of Gilbert, Polster and Tymoczcko regarding existence of nontrivial splines on arbitrary graphs. They show that when working over an integral domain, the set of all splines on a graph contains a free submodule whose rank is the number of vertices. The author gives examples of their construction working over the ring of integers.