Turkish Journal of Mathematics
Abstract
Let [[EQUATION]] be a commutative ring with identity and [[EQUATION]] a graph. Fix an [[EQUATION]]-module [[EQUATION]] for each vertex [[EQUATION]] of [[EQUATION]], an extending generalized spline on [[EQUATION]] is a vertex labeling [[EQUATION]], where for each edge [[EQUATION]] there exists an [[EQUATION]]-module [[EQUATION]] together with homomorphisms [[EQUATION]] and [[EQUATION]] such that [[EQUATION]]. Extending generalized splines are further generalizations for generalized splines. They can also be considered as generalized splines over modules.In this paper, we prove that some of the results for splines can be extended to extending generalized splines when [[EQUATION]] for each vertex [[EQUATION]], and we define a method of graph reduction based on graph operations on vertices and edges to produce an explicit [[EQUATION]]-module basis for generalized splines over modules. We show that the latter corresponds to a sequence of surjective homomorphisms between the associated spline modules so that the space of splines decomposes as a direct sum of certain submodules.
DOI
10.55730/1300-0098.3671
Keywords
$\mathbb{Z}$-modules, spline bases, graph reduction, algebraic graph theory
First Page
593
Last Page
612
Publisher
The Scientific and Technological Research Council of Türkiye (TÜBİTAK)
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
DİLAVER, G, & ALTINOK, S (2026). Generalized splines over ℤ-modules on arbitrary graphs. Turkish Journal of Mathematics 50 (4): 593-612. https://doi.org/10.55730/1300-0098.3671
