Let G be a graph with n vertices and positive integer edge weights A. A vertex labeling (g₁,…,g_n) is an integer spline if for every pair of adjacent vertices {i,j} with edge weight a, g_i = g_j mod a. The set of splines on a given weighted graph (G,A) forms a free module of rank n over the integers. We provide an explicit construction for a particularly nice basis for this module, generalizing work done by undergraduates at Smith College and Bard College.

This is joint work with Jeff Suzuki and Jessica Liu.