Let $G$ be a graph whose edges are labeled by ideals of a commutative ring $R$ with identity. Such a graph is called an edge-labeled graph over $R$. A generalized spline is a vertex labeling so that the difference between the labels of any two adjacent vertices lies in the ideal corresponding to the edge. These generalized splines form a module over $R$. In this paper, we consider complete graphs whose edges are labeled with proper ideals of $\mathbb{ Z} / m\mathbb{ Z}$. We compute minimum generating sets of constant flow-up classes for spline modules on edge-labeled complete graphs over $\mathbb{ Z} / m\mathbb{ Z}$ and their rank under some restrictions.
Categories: math.AC math.CO