Metric properties of electrical networks and the graph reconstruction problems

Using the generalized Temperley trick, we demonstrate the explicit embedding of circular electrical networks into totally non-negative Grassmannians. Building on this result, we show that the effective resistances between boundary nodes of circular electrical networks satisfy the Kalmanson property, and we provide the full characterization of planar electrical Kalmanson metrics. Additionally, we present a graph reconstruction algorithm with applications in phylogenetic network analysis as well as the numerical solution of the Calderon problem.