The distance spectrum of corona and cluster of two graphs

Let GG be a connected graph with a distance matrix DD. The DD-eigenvalues {μ1,μ2,…,…,μp}{μ1,μ2,…,…,μp} of GG are the eigenvalues of DD and form the distance spectrum or DD-spectrum of GG. Given two graphs GG with vertex set {v1,v2,……,vp}{v1,v2,……,vp} and HH, the corona G∘HG∘H is defined as the graph obtained by taking pp copies of HH and for each ii, joining the iith vertex of GG to all the vertices in the iith copy of HH. Let HH be a rooted graph rooted at uu. Then the cluster G{H}G{H} is defined as the graph obtained by taking pp copies of HH and for each ii, joining the iith vertex of GG to the root in the iith copy of HH. In this paper we describe the distance spectrum of G∘HG∘H, for a connected distance regular graph GG and any rr-regular graph HH in terms of the distance spectrum of GG and adjacency spectrum of HH. We also describe the distance spectrum of G{Kn}G{Kn}, where GG is a connected distance regular graph.