The distance spectrum of corona and cluster of two graphs

Abstract

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