  In 2022, the second author found a prolific construction of strongly regular
graphs, which is based on joining a coclique and a divisible design graph with
certain parameters. The construction produces strongly regular graphs with the
same parameters as the complement of the symplectic graph $\mathsf{Sp}(2d,q)$.
  In this paper, we determine the parameters of strongly regular graphs which
admit a decomposition into a divisible design graph and a coclique attaining
the Hoffman bound. In particular, it is shown that when the least eigenvalue of
such a strongly regular graph is a prime power, its parameters coincide with
those of the complement of $\mathsf{Sp}(2d,q)$. Furthermore, a generalization
of the construction is discussed.
