Generalized splines on arbitrary graphs
 Citation data:

Pacific Journal of Mathematics, ISSN: 00308730, Vol: 281, Issue: 2, Page: 333364
 Publication Year:
 2016

 Bepress 55
 Bepress 4

 Bepress 13
 Bepress 3

 EBSCO 1
 Repository URL:
 https://scholarworks.smith.edu/mth_facpubs/27; https://scholarworks.smith.edu/mth_facpubs/7; http://arxiv.org/abs/1306.0801
 DOI:
 10.2140/pjm.2016.281.333
 Author(s):
 Publisher(s):
 Tags:
 Mathematics; splines; GKM theory; equivariant cohomology; algebraic graph theory; Discrete Mathematics and Combinatorics; Mathematics  Combinatorics; Mathematics  Algebraic Topology; 05C25 (Primary), 55N91 and 14M25 (Secondary)
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article description
Let G be a graph whose edges are labeled by ideals of a commutative ring. We introduce a generalized spline, which is a vertex labeling of G by elements of the ring so that the difference between the labels of any two adjacent vertices lies in the corresponding edge ideal. Generalized splines arise naturally in combinatorics (algebraic splines of Billera and others) and in algebraic topology (certain equivariant cohomology rings, described by Goresky, Kottwitz, and MacPherson, among others). The central question of this paper asks when an arbitrary edgelabeled graph has nontrivial generalized splines. The answer is "always", and we prove the stronger result that the module of generalized splines contains a free submodule whose rank is the number of vertices in G. We describe the module of generalized splines when G is a tree, and give several ways to describe the ring of generalized splines as an intersection of generalized splines for simpler subgraphs of G. We also present a new tool which we call the GKM matrix, an analogue of the incidence matrix of a graph, and end with open questions.