Construction of directed strongly regular graphs using finite incidence structures

We use finite incident structures to construct new infinite families of directed strongly regular graphs with parameters $$(l(q-1)q^l,l(q-1)q^{l-1},(lq-l+1)q^{l-2},(l-1)(q-1)q^{l-2},(lq-l+1)q^{l-2})$$ for integers $q$ and $l$ ($q,l\ge 2$), and $$(l{q}^2(q-1),lq(q-1),lq-l+1,(l-1)(q-1),lq-l+1)$$ for all prime powers $q$ and $l\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.},q\}$. The new graphs given by these constructions have parameters $(36,12,5,2,5)$, $(54,18,7,4,7)$, $(72,24,10,4,10)$, $(96,24,7,3,7)$, $(108,36,14,8,14)$ and $(108, 36, 15, 6, 15)$ listed as feasible parameters on “Parameters of directed strongly regular graphs,” at ${http://homepages.cwi.nl/^\sim aeb/math/dsrg/dsrg.html}$ by S. Hobart and A. E. Brouwer. We review these constructions and show how our methods may be used to construct other infinite families of directed strongly regular graphs.