一类2级一般强正则图  

A Family of Generalized Strongly Regular Graphs of Grade 2

作　　者：宋思敏 杨丽芳 张更生[1,3] SONG Simin;YANG Lifang;ZHANG Gengsheng(College of Mathematical Sciences,Hebei Normal University,Shijiazhuang,Hebei,050024,P.R.China;Department of Basic,Shijiazhuang Engineering Vocational College,Shijiazhuang,Hebei,050061,P.R.China;Hebei Key Laboratory of Computational Mathematics and Applications,Shijiazhuang,Hebei,050024,P.R.China)

机构地区：[1]河北师范大学数学科学学院,石家庄河北050024 [2]石家庄工程职业学院基础部,石家庄河北050061 [3]河北省计算数学与应用重点实验室,石家庄河北050024

出　　处：《数学进展》2021年第4期511-518,共8页Advances in Mathematics（China）

基　　金：国家自然科学基金(No.11571091);河北省自然科学基金(No.F2019205147);河北师范大学研究生创新资助项目(No.CXZZSS2020050)。

摘　　要：作为强正则图的一种新推广,p级一般强正则图是使得任意两个邻接的顶点和任意两个不邻接的顶点的公共邻接点数都有p种不同取值的非空k-正则图.对于参数为(n,k;a1,a2;c1,c2)的2级一般强正则图中任一顶点,如果与这个顶点邻接且有ai(i=1,2)个公共邻接点或者与这个顶点不邻接且有c1(i=1,2)个公共邻接点的顶点数与该点的选取无关,则称这个2级一般强正则图为自由的.本文研究了参数为(n,k;k-1,a2;k-1,c2)的2级一般强正则图,得到一类自由的2级一般强正则图存在的充要条件.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.For any vertex v of a generalized strongly regular graph of grade 2 with parameters(n,k;a1,a2;c1,c2),if the number of the vertices that are adjacent to v and share ai(i=1,2)common neighbours with v,or are non-adjacent to v and share ci(i=1,2)common neighbours with v is independent of the choice of the vertex v,then the generalized strongly regular graph of grade 2 is free.In this paper,we investigate the generalized strongly regular graph of grade 2 with parameters(n,k;k-1,a2;k-1,c2)and provide the sufficient and necessary conditions for the existence of a family of free generalized strongly regular graphs of grade 2.

关 键 词：强正则图 一般强正则图 合成图 同构 

分 类 号：O157.5[理学—数学]