A Family Of Generalized Strongly Regular Graphs Of Grade 2

A generalized strongly regular graph of grade p,as a new generalization of strongly regular graphs,is a regular graph such that the number of common neighbours of both any two adjacent vertices and any two non-adjacent vertices takes on p distinct values,and was first proposed by Huo Lijun and Zhang Gengsheng in 2017.In this paper,we study a family of generalized strongly regular graphs of order 2 with parameters(n,k;k-1,a2;k-1,c2)and describe its structure.We first divide generalized strongly regular graph of grade 2 into four classes,according to whether the number of vertices whose common adjacency points are 1 and 2 with the vertex v in the graph G is related to the selection of v.Then we study some properties of generalized strongly regular graphs of grade 2 with parameters(n,k;k-1,a2;k-1,c2)based on the classification.Finally,we obtain the existence of every class of graphs with parameters(n,k;k-1,a2;k-1,c2)by using these properties,and give the necessary and sufficient conditions for the existence of generalized strongly regular graphs of grade 2 with parameters(n,k;k-1,a2;k-1,c2).The main result of the present paper is in the follow:Let G be a generalized strongly regular graphs of grade 2.Then a1=c1k-1 if and only if G is isomorphic to the composition of G1 and G2,where G1 is a GSR(n1,k1,?,?)and G2 is n2/2K2 Moreover,the parameters satisfy:(1)??k1-1 when n2=2;(2)??k1.