The cycle has this property. For instance, the distance matrix for a 6-cycle is:

$A=\begin{bmatrix} 0 & 1 & 2 & 3 & 2 & 1 \\\\ 1 & 0 & 1 & 2 & 3 & 2 \\\\ 2 & 1 & 0 & 1 & 2 & 3 \\\\ 3 & 2 & 1 & 0 & 1 & 2 \\\\ 2 & 3 & 2 & 1 & 0 & 1 \\\\ 1 & 2 & 3 & 2 & 1 & 0 \\\\ \end{bmatrix}$

The question is: is the cycle the only graph with this property?