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Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distancematrix of $G$. Let $\partial_1(G)\ge\partial_2(G)\ge\cdots\ge\partial_n(G)$denote the eigenvalues of $D(G)$. In this paper, we characterize all connectedgraphs with $\partial_{3}(G)\leq -1$ and $\partial_{n-1}(G)\geq -2$. By theway, we determine all connected graphs with at most three distance eigenvaluesdifferent from $-1$ and $-2$.

Graphs with at most three distance eigenvalues different from $-1$ and $-2$