Basis criteria for generalized spline modules via determinant

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Altınok Bhupal S., Sarıoğlan S.

DISCRETE MATHEMATICS, vol.344, no.2, 2021 (SCI-Expanded) identifier identifier identifier

  • Publication Type: Article / Article
  • Volume: 344 Issue: 2
  • Publication Date: 2021
  • Doi Number: 10.1016/j.disc.2020.112223
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, MathSciNet, zbMATH
  • Keywords: Generalized splines, Graphs, Modules, Basis
  • Hacettepe University Affiliated: Yes


Given a graph whose edges are labeled by ideals of a commutative ring R with identity, a generalized spline is a vertex labeling by the elements of R such that the difference of the labels on adjacent vertices lies in the ideal associated to the edge. The set of generalized splines has a ring and an R-module structure. We study the module structure of generalized splines where the base ring is a greatest common divisor domain. We give basis criteria for generalized splines on cycles, diamond graphs and trees by using determinantal techniques. In the last section of the paper, we define a graded module structure for generalized splines and give some applications of the basis criteria for cycles, diamond graphs and trees. (C) 2020 Elsevier B.V. All rights reserved.