Construction of directed strongly regular graphs using finite incidence structures

TL;DR

This paper introduces new methods to construct infinite families of directed strongly regular graphs using finite incidence structures, expanding known parameter sets and providing explicit examples with feasible parameters.

Contribution

The paper presents novel constructions of directed strongly regular graphs from finite incidence structures, including explicit parameter sets and a review of feasible parameters.

Findings

Constructed new infinite families of directed strongly regular graphs.

Provided explicit examples with feasible parameters.

Demonstrated how methods can generate other families.

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.