\[ d^2(i,j)=\begin{cases} 0,&\text{if $i=j$}\\ 1,&\text{if $i$ and $j$ are non adjacent}\\ 1-\frac{1}{\sqrt{d(i)d(j)}},&\text{otherwise} \end{cases}\]
Remark that the matrix $ \left(\frac{A_{i,j}}{\sqrt{d(i)d(j)}}\right)_{i,j} $ is symmetric, stochastic and hence have real eigenvalues, the largest one being equal to $ 1$. Nevertheless, a multigraph is not reconstructible from this distance as replacing every edge by $ k $ parallel edges does not affect the distances between the vertices.