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Given a commutative ring R with identity and a finite graph G = (V, E), an edge labeling of G is a function that labels each edge of G by an ideal of R. The pair (G, α) is called an edge labeled graph. A generalized spline on an edge labeled graph (G, α) is a vertex labeling F ∈ R|V | such that, for each edge uv, the difference F(u) − F(v) ∈ α(uv). The set of all generalized splines on (G, α) over R is denoted by R(G,α) . The set R(G,α) has a ring and an R-module structure.

In this talk, we first give basic definitions, notations and properties of generalized splines and present some examples. We also explain the topological and algebraic geometric backgrounds of the theory. Then, we focus on the module structure of R(G,α) and discuss basis conditions.