Construction of directed strongly regular graphs using finite incidence structures

Abstract

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 \[(lq^2(q-1),\ lq(q-1),\ lq-l+1,\ (l-1)(q-1),\ lq-l+1)\] for all prime powers $q$ and $l\in \{1,2,...,q\}$. 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.

13 pages