# The Constructions of directed strongly regular graph by algebraic method

## TL;DR

This paper introduces new classes of directed strongly regular graphs using algebraic methods like Kronecker product, semidirect product, and Cayley graphs, and analyzes their properties and automorphisms.

## Contribution

It presents novel constructions of DSRGs via algebraic techniques and provides conditions for Cayley graphs to be DSRGs, expanding the understanding of their structure.

## Key findings

- New classes of DSRGs constructed using algebraic methods
- Conditions for Cayley graphs to be DSRGs established
- Analysis of automorphism groups and neighbor structures in DSRGs

## Abstract

The concept of directed strongly regular graphs (DSRG) was introduced by Duval in "A Directed Graph Version of Strongly Regular Graphs" [Journal of Combinatorial Theory, Series A 47(1988)71-100]. Duval also provided several construction methods for directed strongly regular graphs. In this paper, We construct several new classes of directed strongly regular graphs which are obtained by using Kronecker matrix product, Semidirect product and Cayley coset graph. At the same time, using group representation, for two special cases, we give some other sufficient and necessary conditions of Cayley graphs to be DSRG. At last, we finish this paper with a discussion of some propositions of in(out)-neighbours and automorphism group in directed strongly regular graphs.