On graphs with at least three distance eigenvalues less than −1

Let G be a connected graph with order n   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-eigenvalue of G  . In this paper, we show that λn−1(D(G))≤−1λn−1(D(G))≤−1 if n≥4n≥4 and λn−2(D(G))≤−1λn−2(D(G))≤−1 if n≥7n≥7. We also characterize all connected graphs with λn−1(D(G))=−1λn−1(D(G))=−1, moreover it is shown that these graphs are determined by their distance spectra.