Directed Strongly Regular Cayley Graphs Of Order Pq

Directed strongly regular graph was first proposed by Duval in 1988.In the past decades,many scholars at home and abroad have studied it,and directed strongly regular Cayley graph is more special.Domestic scholar Feng Rongquan has researched directed strongly regular Cayley graphs on dihedral groups D2n=<x,y|xn?1?y2,y-1xy=x-1>.This paper mainly researches the necessary and sufficient conditions for constructing strongly directed regular Cayley graphs on pq order non-abelian group G=<a,b|ap?|?1?bq,b-1ab=ar>.This paper first gives some preliminary knowledge,including the knowledge of group theory and graph theory,then introduces the construction method,operation properties and necessary and sufficient conditions of the construction of pq order non-abelian groups,and obtains some related properties.This paper mainly considers the necessity of taking this structure into account by taking advantage of its properties and necessary and sufficient conditions,isomorphism knowledge of groups.In the key part of this paper,we first introduce the sufficiency of the construction,and then use the computing characteristics of group rings to transform the idea of coset construction to the operation of polynomials,and then prove the necessity of the construction by irreducible polynomials.In this paper,we give the process of prove,and give an example at the end.