  Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{
v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and
$v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$,
where $d_{ij}=d( v_i,v_j ) $. Since $D( G )$ is a non-negative real symmetric
matrix, its eigenvalues can be arranged $\lambda_1(G)\ge \lambda_2(G)\ge \cdots
\ge \lambda_n(G)$, where eigenvalues $\lambda_1(G)$ and $\lambda_n(G)$ are
called the distance spectral radius and the least distance eigenvalue of $G$,
respectively. The {\it diameter} of graph $G$ is the farthest distance between
all pairs of vertices. In this paper, we determine the unique graph whose
distance spectral radius attains maximum and minimum among all complements of
graphs of diameter greater than three, respectively. Furthermore, we also
characterize the unique graph whose least distance eigenvalue attains maximum
and minimum among all complements of graphs of diameter greater than three,
respectively.
