Amply regular graphs with μ close to half the valency and group divisible designs

@misc{jin2026amply,
  title = {{Amply regular graphs with $μ$ close to half the valency and group divisible designs}},
  author = {Wei Jin and Jack H. Koolen and Chenhui Lv},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.25754},
  eprint = {2605.25754},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.25754},
}

TY  - UNPB
TI  - Amply regular graphs with μ close to half the valency and group divisible designs
AU  - Wei Jin
AU  - Jack H. Koolen
AU  - Chenhui Lv
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2605.25754
UR  - https://arxiv.org/abs/2605.25754
AB  - In this paper, we classify connected amply regular graphs with diameter d ≥ 4 and parameters $(v, k, λ, μ)$ satisfying μ= (k−1)/2, where k≥ 5 is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph K_2 □ Λ, where Λ is the unique bipartite (0,2)-graph on $14$ vertices, or the point--block incidence graph of a group divisible design with the dual property, namely a GDDDP(2, k+1; k; 0, (k−1)/2). For the last family, we give equivalent characterizations in terms of bipartite Q-regular graphs and relation graphs of symmetric association schemes with five classes. Furthermore, we present constructions of such amply regular graphs, yielding infinite families of examples derived from Paley graphs, Peisert graphs, and Paley digraphs.
ER  -