Distance Matrices and the Graph Curvature Equation

Take a finite, simple, connected graph $G = (V,E)$. Consider it's distance matrix $D$, whose $(i,j)$th entry is $d(v_i,v_j)$. In previous work by Steinerberger (J. Graph Theory, 2023), it was empirically seen that the all-ones vector is frequently lies in the image of $D$, even when $D$ is not invertible. We present some findings towards understanding this phenomenon, including counterexamples for graphs of all sizes $n \geq 7$ and a proof that $D$ is invertible with high probability.