Some Constructions of Quasi-strongly Regular Digraphs

Abstract

Quasi-strongly regular digraphs are a combinatorial generalization of strongly regular digraphs and quasi-strongly regular graphs. A quasi-strongly regular digraph $\Gamma$ with parameters $(n,k,t,a;c_{1},c_{2},\ldots ,c_{p})$ is a k-regular digraph on n vertices such that each vertex is incident with t undirected edges, for any two distinct vertices x, y the number of paths of length 2 from x to y is a if $x\rightarrow y$ and $c_{i}$ otherwise, and for each $c_{i}$ there exist two distinct vertices $x\nrightarrow y$ such that the number of paths of length 2 from x to y is $c_{i}$, where $i\in \{1,2,\ldots ,p\}$. We call p the grade of $\Gamma$. In this paper, we first study quasi-strongly regular digraphs of grade 2 and obtain some constraints between the parameters. Moreover, we present some constructions of quasi-strongly regular digraphs from different combinatorial objects. Finally, we introduce a variation of quasi-strongly regular digraphs and provide some constructions.