Designs over regular graphs with least eigenvalue $$-2$$

Designs over edge-regular, co-edge-regular and amply regular graphs are investigated. Using linear algebra, we obtain lower bounds in certain inequalities involving the parameters of the designs. Some results on designs meeting the bounds are obtained. These designs are over connected regular graphs with least eigenvalue $$-2$$ , have the minimal number of blocks and do not appear in an earlier work. Partial classification such designs over strongly regular graphs with least eigenvalue $$-2$$ is given.