Effective resistance metric and geometry of electrical networks

It is known that the effective resistance defines a metric on the nodes of the electrical networks. Moreover for a circular planar electrical network this metric obeys the Kalmanson property as it was shown recently. We will call such a metric an electrical Kalmanson metric. The main results in the talk will be a complete description of the electrical Kalmanson metrics in the set of all Kalmanson metrics in terms of geometry of the positive Isotropic Grassmannian whose connection to the theory of electrical networks was discovered earlier.

One important area of applications where the Kalmanson metrics are actively used is the theory of phylogenetic networks which are a generalisation of phylogenetic trees. Our results allow to use the powerful methods of reconstruction of the minimal graphs of electrical networks in phylogenetics.

This is a joint work with Anton Kazakov.