Abstract:In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, \lambda, \mu)$ satisfying $\mu = \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\K_2 \square \Lambda$, where $\Lambda$ 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 $GDDDP\left(2, k+1;\, k;\, 0, \frac{k-1}{2}\right)$. 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.
| Comments: | arXiv admin note: substantial text overlap with arXiv:2508.02010 |
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.25754 [math.CO] |
| (or arXiv:2605.25754v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25754 arXiv-issued DOI via DataCite (pending registration) |
From: Chenhui Lv [view email]
[v1]
Mon, 25 May 2026 12:06:38 UTC (298 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。