Generalized splines over ℤ-modules on arbitrary graphs
Turkish Journal of Mathematics, cilt.50, sa.4, ss.593-612, 2026 (SCI-Expanded, Scopus, TRDizin)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 50 Sayı: 4
- Basım Tarihi: 2026
- Doi Numarası: 10.55730/1300-0098.3671
- Dergi Adı: Turkish Journal of Mathematics
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, TR DİZİN (ULAKBİM), Academic Search Ultimate (EBSCO)
- Sayfa Sayıları: ss.593-612
- Anahtar Kelimeler: algebraic graph theory, graph reduction, spline bases, ℤ-modules
- Hacettepe Üniversitesi Adresli: Evet
Özet
Let R be a commutative ring with identity and G a graph. Fix an R-module Mv for each vertex v of G, an extending generalized spline on G is a vertex labeling f ∈⊔v Mv, where for each edge e = uv there exists an R-module M_uv together with homomorphisms φu : Mu → Muv and φv : Mv → Muv such that φu(fu) = φv(fv). Extending generalized splines are further generalizations of generalized splines. They can also be regarded 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 Mv = mv ℤ for each vertex v, and we define a method of graph reduction based on graph operations on vertices and edges to produce an explicit ℤ-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.