Given a graph whose edges are labeled by ideals of a commutative ring R with
identity, a generalized spline is a vertex labeling by the elements of R such
that the difference of the labels on adjacent vertices lies in the ideal
associated to the edge. The set of generalized splines has a ring and an
R-module structure. We study the module structure of generalized splines where
the base ring is a greatest common divisor domain. We give basis criteria for
generalized splines on cycles, diamond graphs and trees by using determinantal
techniques. In the last section of the paper, we define a graded module
structure for generalized splines and give some applications of the basis
criteria for cycles, diamond graphs and trees.Comment: 20 pages, 10 figure