On the second neighbourhoods of quasi-strongly regular graphs

Abstract

A quasi-strongly regular graph of grade p with parameters (n,k,λ;μ1,⋯,μp) is a k-regular graph on n vertices such that any two adjacent vertices have λ common neighbours, any two distinct and non-adjacent vertices have μi common neighbours, and for each μi there exist two distinct and non-adjacent vertices sharing μi common neighbours, where i=1,2,⋯,p. This is a combinatorial generalization of a strongly regular graph. In this paper, we focus on quasi-strongly regular graphs of grade 2. We mainly characterize quasi-strongly regular graphs of grade 2 and diameter 2 with disconnected second neighbourhoods of vertices. Moreover, we obtain similar result for the complement of quasi-strongly regular graphs of grade 2.