The inertia and energy of the distance matrix of a connected graph

Let G   be a connected graph and D(G)D(G) be the distance matrix of G  . Suppose that λ1(D)≥λ2(D)≥⋯≥λn(D)λ1(D)≥λ2(D)≥⋯≥λn(D) are the D-eigenvalues of G. In this paper, we show that the distance matrix of a clique tree is non-singular. Moreover, we also prove that the distance matrix of a clique tree has exactly one positive D-eigenvalue. In addition, we determine the extremal graphs with maximum and minimum distance energy among all clique trees.