Corpus ID: 236772393

Circular planar electrical networks, Split systems, and Phylogenetic networks

  title={Circular planar electrical networks, Split systems, and Phylogenetic networks},
  author={S. Forcey},
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… Expand


The Space of Circular Planar Electrical Networks
This work discusses several parametrizations of the space of circular planar electrical networks, and shows how to test if a network with n nodes is well-connected by checking that $\binom{n}{2}$ minors of the $n\times n$ response matrix are positive. Expand
Circular planar electrical networks: Posets and positivity
This work constructs a poset EP n of electrical networks with n boundary vertices, and proves that it is graded by number of edges of critical representatives, and answers various enumerative questions related to EP n. Expand
Reconstructibility of level-2 unrooted phylogenetic networks from shortest distances
Recently it was shown that a certain class of phylogenetic networks, called level-2 networks, cannot be reconstructed from their associated distance matrices. In this paper, we show that they can beExpand
Electroid varieties and a compactification of the space of electrical networks
  • T. Lam
  • Mathematics
  • Advances in Mathematics
  • 2018
We construct a compactification of the space of circular planar electrical networks studied by Curtis-Ingerman-Morrow and De Verdiere-Gitler-Vertigan, using cactus networks. We embed thisExpand
Galois connections for phylogenetic networks and their polytopes
We describe Galois connections which arise between two kinds of combinatorial structures, both of which generalize trees with labelled leaves, and then apply those connections to a family ofExpand
Kron Reduction of Graphs With Applications to Electrical Networks
  • F. Dörfler, F. Bullo
  • Computer Science, Mathematics
  • IEEE Transactions on Circuits and Systems I: Regular Papers
  • 2013
This paper provides a comprehensive and detailed graph-theoretic analysis of Kron reduction encompassing topological, algebraic, spectral, resistive, and sensitivity analyses and leads to novel insights both on the mathematical and the physical side. Expand
Circular planar graphs and resistor networks
Abstract We consider circular planar graphs and circular planar resistor networks. Associated with each circular planar graph Γ there is a set π ( Γ ) = {( P ; Q )} of pairs of sequences of boundaryExpand
A Space of Phylogenetic Networks
A circular split network is considered, a generalization of a tree in which multiple parallel edges signify divergence, and a geometric space of such networks is introduced, forming a natural extension of the work by Billera, Holmes, and Vogtmann on tree space. Expand
Level-1 phylogenetic networks and their balanced minimum evolution polytopes
This work investigates a two-parameter family of polytopes that arise from phylogenetic networks, and which specialize to the Balanced Minimum Evolution poly topes as well as the Symmetric Travelling Salesmanpolytopes, and describes lower bound faces and a family of faces for every dimension. Expand
Uprooted Phylogenetic Networks
It is shown that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph associated with the split system $$Sigma (N)$$Σ(N) induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Expand