Minimum Generating Sets for Complete Graphs

Abstract

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.

References in corpus (3)

• Integer Generalized Splines on Cycles
• Bases and Structure Constants of Generalized Splines with Integer Coefficients on Cycles
• Splines over integer quotient rings