On quasi-strongly regular graphs with parameters (n,k,a;k − 2,c2) for a < k − 5

Abstract

A quasi-strongly regular graph of grade 2 with parameters (n,k,a;c1,c2) is a k-regular graph on n vertices such that every pair of adjacent vertices have a common neighbours, every pair of distinct nonadjacent vertices have c1 or c2 common neighbours, and for each ci(i=1,2), there exists a pair of non-adjacent vertices sharing ci common neighbours. If a quasi-strongly regular graph is neither a strongly regular graph nor a Deza graph, then it is called a strictly quasi-strongly regular graph. In earlier papers, we characterised the strictly quasi-strongly regular graphs with parameters (n,k,a;k−2,c2) satisfying a≥k−5. In this paper, we characterize strictly quasi-strongly regular graphs with parameters (n,k,a;k−2,c2) satisfying a<k−5 and l1>2, where l1 is the number of vertices with k−2 common neighbours with a given vertex.