Orthogonal and unitary signings of cube-like graphs
Abstract
A unitary signing of a d-regular graph is a Hermitian adjacency matrix M whose nonzero entries lie in {±1, ±i} and satisfies M 2 = dI. Motivated by the work of Alon and Zheng on orthogonal and unitary signings of cube-like graphs, we introduce the Θ-property for a generating set S ⊆ Z n 2 : whenever three pairwise disjoint subsets of S have the same sum, at least two of them have even size.
We prove that every zero-free generating set with the Θ-property gives a cube-like graph Q S admitting a unitary signing, and we give an explicit local formula for such a signing. For Sidon sets, the Θ-property is also necessary, yielding a characterization of the Sidon cube-like graphs that admit unitary signings. In this setting there are exactly 2 |S|-n switching-equivalence classes of unitary signings, and we characterize when a unitary signing can be chosen to be orthogonal.
The Θ-property admits a linear-algebraic description in terms of the dependency space D of S:
Equivalently, the map
is linear on D. For the corresponding extremal set problem, in which zero is permitted, this formulation yields the sharp bound |S| ≤ 2n + 1 for generating sets with the Θproperty. We give elementary constructions attaining this bound for every n ≥ 3. Under the additional Sidon condition, the same extremal value is attained for every n ≥ 10 using binary self-dual codes. The exact Sidon maximum is also determined for 3 ≤ n ≤ 9.
Related
- Centralizers of elements in the orthogonal, symplectic and unitary groups
- Local unitary equivalence of orthogonal arrays and related linear codes
- Painlevé representations for joint moments of characteristic polynomials from the orthogonal and unitary symplectic groups
- Fixed-Radius Neighbourhoods in Three-Dimensional Unit-Cube Graphs
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
BibTeX
@misc{chen2026orthogonal,
title = {{Orthogonal and unitary signings of cube-like graphs}},
author = {Meirun Chen and Reza Naserasr},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33819},
eprint = {2609.33819},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05766226v1},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Orthogonal and unitary signings of cube-like graphs
AU - Meirun Chen
AU - Reza Naserasr
PY - 2026
DA - 2026/09/27
PB - HAL
DO - 10.48550/arxiv.2609.33819
UR - https://hal.science/hal-05766226v1
AB - A unitary signing of a d-regular graph is a Hermitian adjacency matrix M whose nonzero entries lie in {±1, ±i} and satisfies M 2 = dI. Motivated by the work of Alon and Zheng on orthogonal and unitary signings of cube-like graphs, we introduce the Θ-property for a generating set S ⊆ Z n 2 : whenever three pairwise disjoint subsets of S have the same sum, at least two of them have even size. We prove that every zero-free generating set with the Θ-property gives a cube-like graph Q S admitting a unitary signing, and we give an explicit local formula for such a signing. For Sidon sets, the Θ-property is also necessary, yielding a characterization of the Sidon cube-like graphs that admit unitary signings. In this setting there are exactly 2 |S|-n switching-equivalence classes of unitary signings, and we characterize when a unitary signing can be chosen to be orthogonal. The Θ-property admits a linear-algebraic description in terms of the dependency space D of S: Equivalently, the map is linear on D. For the corresponding extremal set problem, in which zero is permitted, this formulation yields the sharp bound |S| ≤ 2n + 1 for generating sets with the Θproperty. We give elementary constructions attaining this bound for every n ≥ 3. Under the additional Sidon condition, the same extremal value is attained for every n ≥ 10 using binary self-dual codes. The exact Sidon maximum is also determined for 3 ≤ n ≤ 9.
ER -