Spectra of quasi-strongly regular graphs

Abstract

A quasi-strongly regular graph G 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 neighbors, every pair of distinct nonadjacent vertices have c1 or c2 common neighbors, and for each ci(i=1,2), there exists a pair of non-adjacent vertices sharing ci common neighbors. The children G1 and G2 of the graph G are defined on the vertex set of G such that every two distinct vertices are adjacent in G1 or G2 if and only if they share c1 or c2 neighbors, respectively. The graph G is a quasi-strongly regular graph of type A or B if it is of grade 2 and has a child that is a connected strongly regular graph or a disjoint union of cliques of order m(m>1), respectively. In this paper, we investigate the spectral properties of the graph G if it is a quasi-strongly regular graph of type A or B. Several examples of distance-regular graphs which are quasi-strongly regular graphs of type A are given. Moreover, we give several constructions of quasi-strongly regular graphs and calculate their spectrum.