Generalized Integer Splines on Arbitrary Graphs

TL;DR

This paper introduces a framework for generalized integer splines on arbitrary graphs, using collapsing operations to analyze their structure and construct module bases based on edge weights.

Contribution

It presents a novel collapsing technique that reduces graphs to single vertices, enabling explicit basis construction for spline modules.

Findings

Collapse operations preserve spline module structure

Explicit basis construction from edge weights

Applicable to arbitrary graphs

Abstract

Generalized integer splines on a graph $G$ with integer edge weights are integer vertex labelings such that if two vertices share an edge in $G$, the vertex labels are congruent modulo the edge weight. We introduce collapsing operations that reduce any simple graph to a single vertex, carrying with it the edge weight information. This corresponds to a sequence of surjective maps between the associated spline modules, leading to an explicit construction of a module basis in terms of the edge weights.