Construction of directed strongly regular graphs via their orbit matrices and genetic algorithm

TL;DR

This paper introduces a novel method for constructing directed strongly regular graphs using orbit matrices and genetic algorithms, enabling the creation of graphs with specific automorphism groups and parameters.

Contribution

It presents a new approach combining orbit matrices and genetic algorithms for constructing directed strongly regular graphs with prescribed automorphism groups.

Findings

Successfully constructed DSRGs with specified parameters

Demonstrated effectiveness of genetic algorithm in graph construction

Enhanced understanding of automorphism groups in DSRGs

Abstract

In this paper, we introduce orbit matrices of directed strongly regular graphs (DSRGs). Further, we propose a method of constructing directed strongly regular graphs with prescribed automorphism group using genetic algorithm. In the construction, we use orbit matrices, i.e. quotient matrices related to equitable partitions of adjacency matrices of putative directed strongly regular graphs induced by an action of a prescribed automorphism group. Further, we apply this method to construct directed strongly regular graphs with parameters $(36,10,5,2,3)$, $(52,12,3,2,3)$, $(52,15,6,5,6)$, $(55,20,8,6,8)$ and $(55,24,12,11,10)$.