Minimum Generating Sets for Complete Graphs

TL;DR

This paper investigates the structure of spline modules on complete graphs with edges labeled by ideals in modular rings, providing minimal generating sets and rank calculations under specific conditions.

Contribution

It introduces methods to compute minimum generating sets for spline modules on complete graphs over modular rings, advancing understanding of their algebraic structure.

Findings

Computed minimum generating sets for spline modules

Determined the rank of these modules under certain restrictions

Analyzed the structure of spline modules over  rings

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.