Characterizing graphs with the second largest distance eigenvalue less than -1/2

Let $G$ be a connected graph with vertex set $V$. The distance, $d_G(u, v)$, between vertices $u$ and $v$ of $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $\mathbf{D}(G) =[d_G(u, v)]_{u,v\in V}$. The second largest distance eigenvalue $\lambda_2(G)$ of $G$ is the second largest one in the spectrum of $\mathbf{D}(G)$. In this work, we completely characterize the connected graphs $G$ for which $\lambda_2(G)<-1/2$ through approaches both spectral and structural.