A note on three-quarters circulant digraphs
Abstract
We introduce and study a new family of circulant digraphs associated with the cyclic group ZN, obtained by restricting admissible combinations of two generators a and b to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.
Related
- Automorphism groups of endomorphism monoids of $0$-unit $3$-valent circulant digraphs
- On 2-distance-transitive circulant digraphs
- A study on Type-2 isomorphic circulant graphs. Part 3: 384 pairs of Type-2 isomorphic circulant graphs C32(R)
- A study on Type-2 isomorphic circulant graphs. Part 4: 960 triples of Type-2 isomorphic circulant graphs C54(R)
- A Study on Type-2 Isomorphic Circulant Graphs: Part 7: C432(R), C6750(S) - each has 2 types of Type-2 isomorphic circulant graphs
BibTeX
@misc{dalfo2026note,
title = {{A note on three-quarters circulant digraphs}},
author = {C. Dalfó and M. A. Fiol and M. A. Reyes},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33718},
eprint = {2609.33718},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33718},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A note on three-quarters circulant digraphs
AU - C. Dalfó
AU - M. A. Fiol
AU - M. A. Reyes
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33718
UR - https://arxiv.org/abs/2609.33718
AB - We introduce and study a new family of circulant digraphs associated with the cyclic group Z_N, obtained by restricting admissible combinations of two generators a and b to three coordinate sectors. The resulting distance-like function differs from the standard directed distance in circulant digraphs and gives rise to new geometric and combinatorial phenomena. Using planar lattice representations and periodic tessellations, we analyze the growth of reachable sets and derive Moore-type upper bounds for the corresponding order/diameter problem. We construct explicit infinite families of three-quarters circulant structures with the prescribed diameter and provide lattice-based methods for determining admissible generator pairs. Separate constructions are obtained for even and odd diameters. In addition, computational experiments for small and moderate orders suggest improved families for even diameters and motivate a conjectural asymptotic formula for the maximum attainable order. The paper highlights the interplay between constrained lattice representations, periodic tilings, and extremal problems for circulant networks.
ER -