| In 1997,Golightly W and others generalized the strongly regular graphs and proposed the concept of quasi-strongly regular graphs.It is a k-regular graph on n vertices such that any two adjacent vertices have a common neighbours and any two non-adjacent vertices have ci common neighbours for some 1≤i ≤p.In this paper,Through the necessary feasibility condition of the quasi-strongly reg-ular graphs parameter sets,we explore the existence of a grade 2 quasi-strongly regular graphs with some special parameters.The following conclusions are drawn:The quasi-strongly regular graphs with the parameter(n,k,k-2;c,c-1)does not exist.When c>3 and c(c-1)/2<k<c2-1/2,the parameter is(n,k,k-3;c,c-1)does not exist,and then we continue to discuss the existence of quasi-strong regular graphs when c=3 and c=2.It is found that only when k=3,a quasi-strong regular graph with the parameter(n,k,k-3;2,1)exists.When k>2a+1 and a≠0,the quasi-strongly regular graph with the parameters(n,k,a;k-1,k-2)does not exist;When k>3a-3 and a>4,the quasi-strongly regular graph with the parameters(n,k,a;k-1,k-3)does not exist.In addition,we also study some properties of quasi-strongly regular graphs when a=0,and construct quasi-strongly regular graphs based on graph product operations. |
