Algebraic characterizations of generating and affinely generating Γ-magic maps and Γ-distance magic labelings on regular graphs
Abstract
For an abelian group Γ of order n, a graph G of order n is Γ-distance magic if it admits a bijection V(G)→Γ whose open-neighborhood sums are constant, and group distance magic if this holds for every such Γ. More generally, for any finite abelian Γ, a Γ-magic map is a map f: V(G)→Γ with constant open-neighborhood sums; we call f generating if its labels generate Γ, and affinely generating if its pairwise differences do. Let G be regular, let Γ≅ℤ/d1⊕⋯⊕ℤ/dr with d1|⋯| dr, put BΓ=⨁i<rℤ/di, and let A̅m be the adjacency operator induced on (ℤ/m)V(G) modulo constants. We prove that G admits a generating Γ-magic map iff BΓ↪kerA̅dr, and an affinely generating one iff Γ↪kerA̅dr; when |V(G)|=|Γ|, it is Γ-distance magic iff the latter embedding has vertex-separating image. If the reduced adjacency operator is nonsingular over ℚ, these become subgroup conditions in the reduced adjacency Smith group. Cichacz and Froncek conjectured that every distance magic graph is group distance magic. Using the affine criterion we construct a $6$-regular distance magic graph of order $27$ admitting no affinely generating (ℤ/3)3-magic map, so the conjecture fails even for affine generation. We propose the generating group distance magic conjecture, and prove it for regular distance magic graphs whenever Γ is $2$-generated, hence for cube-free order. Further applications concern Cayley graphs on elementary abelian p-groups, Hamming relation graphs, strongly regular graphs, and symmetric designs.
Related
- Structural Characterizations and Algebraic Realizations of a Family of Regular Integral Graphs
- Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance
- Direct Generation of a Somos-4 Sequence from an Algebraic Generating Function
- Differential Recursions, Projective Discriminants, and Algebraic Generating Functions
- Approximating CDTW Distance of Piecewise Algebraic Curves
BibTeX
@misc{batal2026algebraic,
title = {{Algebraic characterizations of generating and affinely generating $Γ$-magic maps and $Γ$-distance magic labelings on regular graphs}},
author = {Ahmet Batal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05934},
eprint = {2609.05934},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05934},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Algebraic characterizations of generating and affinely generating Γ-magic maps and Γ-distance magic labelings on regular graphs
AU - Ahmet Batal
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05934
UR - https://arxiv.org/abs/2609.05934
AB - For an abelian group Γ of order n, a graph G of order n is Γ-distance magic if it admits a bijection V(G)→Γ whose open-neighborhood sums are constant, and group distance magic if this holds for every such Γ. More generally, for any finite abelian Γ, a Γ-magic map is a map f: V(G)→Γ with constant open-neighborhood sums; we call f generating if its labels generate Γ, and affinely generating if its pairwise differences do. Let G be regular, let Γ≅ℤ/d_1⊕⋯⊕ℤ/d_r with d_1|⋯| d_r, put B_Γ=⨁_i<rℤ/d_i, and let A̅_m be the adjacency operator induced on (ℤ/m)^V(G) modulo constants. We prove that G admits a generating Γ-magic map iff B_Γ↪kerA̅_d_r, and an affinely generating one iff Γ↪kerA̅_d_r; when |V(G)|=|Γ|, it is Γ-distance magic iff the latter embedding has vertex-separating image. If the reduced adjacency operator is nonsingular over ℚ, these become subgroup conditions in the reduced adjacency Smith group. Cichacz and Froncek conjectured that every distance magic graph is group distance magic. Using the affine criterion we construct a $6$-regular distance magic graph of order $27$ admitting no affinely generating (ℤ/3)^3-magic map, so the conjecture fails even for affine generation. We propose the generating group distance magic conjecture, and prove it for regular distance magic graphs whenever Γ is $2$-generated, hence for cube-free order. Further applications concern Cayley graphs on elementary abelian p-groups, Hamming relation graphs, strongly regular graphs, and symmetric designs.
ER -