On quasi-strongly regular graphs with parameters (n,k,a;k−1,c2)

Quasi-strongly regular graphs are a combinatorial generalization of strongly regular graphs. 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 any two adjacent vertices have a common neighbours, any two non-adjacent 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 of grade 2 is neither a strongly regular graph nor a Deza graph, then it is called a strictly quasi-strongly regular graph. In this paper, we characterize strictly quasi-strongly regular graphs with parameters satisfying c1=k−1.