Abstract: Let $G=(V,E)$ be a simplified connected graph with $n$ vertices. For a simplified connected graph with $n$ vertices, it has $n$ distance eigenvalues. The second small distance eigenvalue is studied mainly in this paper. And method of disabling the subgraph is used to characterize trees with $\lambda_{n-1}(D(G))\in [-2.4295,0]$, unicyclic and bicyclic graphs with $\lambda_{n-1}(D(G))\in [-2,0]$.

Key words: the second smallest distance eigenvalue, trees, unicyclic graphs, bicyclic graphs