On the distance spectrum of graphs

Let G be a connected graph and D(G) be the distance matrix of G. Suppose that λ1(D)⩾λ2(D)⩾⋯⩾λn(D) are the D-eigenvalues of G. In this paper, we characterize all connected graphs with λn(D)=-2. Furthermore, we characterize all connected graphs of diameter 2 with exactly three D-eigenvalues when λ1(D) is not an integer. We also conjecture that the complete k-partite graph is determined by its D-spectrum.