  We study a new invariant of circular planar electrical networks, well known
to phylogeneticists: the circular split system. We use our invariant to answer
some open questions about levels of complexity of networks and their related
Kalmanson metrics. The key to our analysis is the realization that certain
matrices arising from weighted split systems are studied in another guise: the
Kron reductions of Laplacian matrices of planar electrical networks.
Specifically we show that a response matrix of a circular planar electrical
network corresponds to a unique resistance metric obeying the Kalmanson
condition, and thus a unique weighted circular split system. Our results allow
interchange of methods: phylogenetic reconstruction using theorems about
electrical networks, and circuit reconstruction using phylogenetic techniques.
