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Let $G$ be a connected graph with vertex set $V(G)$. The distance,$d_G(u,v)$, between vertices $u$ and $v$ in $G$ is defined as the length of ashortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is thematrix $D(G)=(d_G(u,v))_{u,v\in V(G)}$. The second largest distance eigenvalueof $G$ is the second largest one in the spectrum of $D(G)$. We show that anyconnected graph with the second largest distance eigenvalue less than$\frac{-3+\sqrt{5}}{2}$ is chordal, and characterize those bicyclic graphs andsplit graphs with the second largest distance eigenvalue less than$-\frac{1}{2}$.

A graph for which the second largest distance eigenvalue is less than $\frac{-3+\sqrt{5}}{2}$ is chordal