Notes on Various Methods for Constructing Directed Strongly Regular Graphs

TL;DR

This paper develops new methods for constructing directed strongly regular graphs, expanding known parameter sets, and explores their algebraic structures, including Cayley graphs and block matrix representations.

Contribution

It introduces new constructions for directed strongly regular graphs with previously unknown parameters and links these graphs to Cayley graphs and block matrices.

Findings

Constructed directed strongly regular graphs with new parameters.

Established a block matrix characterization for certain directed strongly regular graphs.

Linked directed strongly regular graphs to Cayley graphs with algebraic structures.

Abstract

Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept of directed strongly regular graphs. In this paper we construct several rich families of directed strongly regular graphs with new parameters. Our constructions yielding new parameters are based on extending known explicit constructions to cover more parameter sets. We also explore some of the links between Cayley graphs, block matrices and directed strongly regular graphs with certain parameters. Directed strongly regular graphs which are also Cayley graphs are interesting due to their having more algebraic structure. We construct directed strongly regular Cayley graphs with parameters $((m+1)s,ls,ld,ld-d,ld)$ where $d,l$ and $s$ are integers with $dm=ls$ and $1\leq l<m$. We also give a new block matrix characterization for directed strongly regular graphs with parameters $(m(dm+1),dm,m,m-1,m)$, which were first dicussed by Duval et al. in "Semidirect Product Constructions of Directed Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 104 (2003) 157 - 167].