Amply regular graphs with $\mu$ close to half the valency and group divisible designs

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.