Distance matrices on the H-join of graphs: A general result and applications

Given a graph H with vertices 1,…,s and a set of pairwise vertex disjoint graphs G1,…,Gs, the vertex i of H is assigned to Gi. Let G be the graph obtained from the graphs G1,…,Gs and the edges connecting each vertex of Gi with all the vertices of Gj for all edge ij of H. The graph G is called the H-join of G1,…,Gs. Let M(G) be a matrix on a graph G. A general result on the eigenvalues of M(G), when the all ones vector is an eigenvector of M(Gi) for i=1,2,…,s, is given. This result is applied to obtain the distance eigenvalues, the distance Laplacian eigenvalues and as well as the distance signless Laplacian eigenvalues of G when G1,…,Gs are regular graphs. Finally, we introduce the notions of the distance incidence energy and distance Laplacian-energy like of a graph and we derive sharp lower bounds on these two distance energies among all the connected graphs of prescribed order in terms of the vertex connectivity. The graphs for which those bounds are attained are characterized.