Minimum generating sets for complete graphs

Abstract

Let G be a graph with edges 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 Z/mZ. We compute minimum generating sets of constant flow-up classes for spline modules on edge-labeled complete graphs over Z/mZ and determine their rank under some restrictions.