  Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance
matrix of $G.$ Suppose that
$\lambda_{1}(D(G))\geq\lambda_{2}(D(G))\geq\cdots\geq\lambda_{n}(D(G))$ are the
distance spectrum of $G$. A graph $G$ is said to be determined by its
$D$-spectrum if with respect to the distance matrix $D(G)$, any graph with the
same spectrum as $G$ is isomorphic to $G$. In this paper, we consider spectral
characterization on the second largest distance eigenvalue $\lambda_{2}(D(G))$
of graphs, and prove that the graphs with
$\lambda_{2}(D(G))\leq\frac{17-\sqrt{329}}{2}\approx-0.5692$ are determined by
their $D$-spectra.
