Planar Electrical Kalmanson Metrics

• Planar electrical Kalmanson metrics are dissimilarity measures defined on cyclic boundary vertices of resistor networks and satisfy Kalmanson inequalities.
• They unify electrical network theory with algebraic and combinatorial frameworks, linking resistor networks, totally nonnegative Grassmannians, and phylogenetics.
• Reconstruction algorithms leverage Grassmannian positroid cells and weighted split systems to ensure precise network recovery and minimal configurations.

A planar electrical Kalmanson metric is a dissimilarity metric on a finite set of boundary vertices arranged in cyclic order along the boundary of a planar embedding, realized as the effective resistance matrix of a circular planar resistor network. The class of electrical Kalmanson metrics forms a distinguished and geometrically rich subset of all Kalmanson metrics, characterized by rigorous algebraic, combinatorial, and geometric criteria. Their structure underpins contemporary connections between electrical network theory, totally nonnegative Grassmannians, and phylogenetics.

1. Circular Planar Electrical Networks and the Resistance Metric

A circular planar electrical network $\mathcal{G}$ is a connected undirected graph $G=(V,E)$ equipped with strictly positive edge conductances and a distinguished subset of $n$ boundary vertices $V_B=\{1,2,\ldots,n\}$ appearing in (say) clockwise cyclic order along the boundary of a planar embedding. Applying fixed voltages $U:V_B\to\mathbb{R}$ at the boundary, Kirchhoff’s and Ohm’s laws yield a unique harmonic extension $U:V\to\mathbb{R}$ and a resulting vector of boundary currents $I$. The linear relationship $I=M_R U$ defines the $n\times n$ response (Dirichlet-to-Neumann) matrix $M_R$.

The effective resistance matrix $R=(R_{ij})$ on $V_B$ is defined by

$R_{ij} := |(M_R^{-1})_{ii} - (M_R^{-1})_{ij}|$

for $i, j \in V_B$, $i \neq j$, which quantifies the potential drop when unit current enters at $i$ and exits at $j$. Standard arguments guarantee $R_{ij}=R_{ji}$, $R_{ii}=0$, and triangle inequalities, hence $R$ is a metric on $\{1,\ldots,n\}$ (Gorbounov et al., 2024, Gorbounov et al., 13 Jan 2026).

2. Kalmanson (Circular Four-Point) Inequalities

A symmetric $n \times n$ matrix $D=(d_{ij})$, $d_{ii}=0$, indexed by labels in cyclic order, is a Kalmanson metric if for every quadruple $i < j < k < \ell$ (mod $n$) the inequalities

$$d_{ij} + d_{k\ell} \leq \max\{\,d_{ik} + d_{j\ell},\, d_{i\ell} + d_{jk}\,\}$$

hold. These circular four-point conditions generalize the condition for tree metrics and encode combinatorial convexity in the cyclic labeling (Devadoss et al., 2024, Forcey, 2021).

For any circular planar electrical network, the resistance matrix $R$ always satisfies the Kalmanson inequalities. This follows from total positivity principles of circular minors of the response matrix—Curtis–Ingerman–Morrow, Kenyon–Wilson, and Lam’s theory of Groves (Gorbounov et al., 2024, Gorbounov et al., 13 Jan 2026).

3. Algebraic and Geometric Characterization

Given a Kalmanson metric $D=(d_{ij})$, the corresponding “dual” response matrix $M(D)$ is constructed by

$$M(D)_{ij} = -\frac{1}{2} \left( d_{i,j} + d_{i+1,j+1} - d_{i,j+1} - d_{i+1,j} \right)$$

for $i\neq j$, cyclic mod $n$, and $M(D)_{ii} = -\sum_{j \neq i} M(D)_{ij}$.

Form the $n \times 2n$ boundary measurement matrix $\Omega_D$ in the explicit “$\Omega_R$” pattern. Then, $D$ is a planar electrical Kalmanson metric if and only if:

• $\Omega_D$ has row-rank $n-1$;
• all $(n-1)\times(n-1)$ circular minors of $\Omega_D$ are nonnegative, i.e., $\Omega_D$ defines a point in the totally nonnegative Grassmannian $Gr_{\geq 0}(n-1,2n)$;
• the central Plücker coordinate $\Delta_{2,4,\ldots,2n-2}(\Omega_D)$ is nonzero.

This characterization is both necessary and sufficient: every such $D$ arises as the effective resistance metric of a unique (modulo star–triangle reductions and dualities) circular planar electrical network. This embedding realizes electrical Kalmanson metrics as a well-defined positroid cell within the Grassmannian (Gorbounov et al., 2024, Gorbounov et al., 13 Jan 2026).

4. Combinatorial Structure and Split Systems

A split of $[n]$ is a bipartition $A \sqcup B$, and a weighted circular split system assigns nonnegative weights to splits compatible with the cyclic order. There is a bijection between Kalmanson metrics and weighted circular split systems: for any $D$, there is a unique set $\Sigma$ of such splits and weights $w(S)$ so that

$$d_{ij} = \sum_{S \in \Sigma: i \text{ and } j \text{ different parts of } S} w(S)$$

This is the classic Dress–Huson–Pachter–Speyer Kalmanson–split bijection.

For a circular planar electrical network, the Kron reduction of the Laplacian to the boundary yields a weighted clique–sum graph whose conductances enumerate split-weights, connecting electrical and phylogenetic representations. The induced split system from the resistance metric matches the graphical split system after Kron reduction and half-step label rotation (Devadoss et al., 2024, Forcey, 2021).

5. Reconstruction Algorithms

The recovery of a network from its electrical Kalmanson metric proceeds algorithmically:

1. Given $D$ (satisfying Kalmanson conditions), compute $M(D)$ via the Gromov–Farris formula.
2. Build $\Omega_D$.
3. Ensure $\Omega_D$ meets the Grassmannian and Plücker constraints.
4. Deduce the strand-permutation: for each column of $\Omega_D$, find the minimal span permutation, yielding the strand-pairing matching that underlies the medial graph.
5. From the strand-matching, reconstruct the medial graph and thus the minimal circular planar electrical network.
6. Optionally, apply star–triangle reductions or dualizations to obtain the minimal resistor graph (Gorbounov et al., 2024, Gorbounov et al., 13 Jan 2026).

6. Concrete Example

For the four-node cycle network $1$–$2$–$3$–$4$–$1$ with unit conductances: the resistance metric is

$$R_{12} = R_{23} = R_{34} = R_{41} = \frac{3}{4}, \quad R_{13} = R_{24} = 1$$

The Kalmanson inequalities for the cyclic quadruple $(1,2,3,4)$ hold: $$R_{13} + R_{24} = 2 \geq \max\{ R_{12} + R_{34}, R_{23} + R_{14} \} = 1.5$$ The associated circular split system is the collection of splits of the $4$-gon, with split weights derived either directly from the pseudoinverse formulas or by Kron reduction (Gorbounov et al., 13 Jan 2026, Devadoss et al., 2024, Forcey, 2021).

7. Applications and Further Directions

Planar electrical Kalmanson metrics play a pivotal role in phylogenetic network theory, generalizing tree metrics to circular split networks. Their Grassmannian embedding reveals deeper cluster algebraic structures in network reconstruction and data analysis. Key consequences include:

• Unified treatment of tree and circular network metrics in phylogenetics.
• Explicit reconstruction algorithms relying on Grassmannian positroid combinatorics.
• Guarantees of planarity via Plücker coordinates and detection of minimal network configurations.
• Extension to “cactus networks” via conductance coalescence, compactifying the moduli of networks and correspondences to non-crossing partition systems.
• Enumerative and species-theoretic correspondences support global and compactified versions, linking the theory to CW complexes and combinatorial enumeration (Gorbounov et al., 2024, Devadoss et al., 2024, Forcey, 2021).

A plausible implication is that the $L_2$-embeddability of $\sqrt{D}$ for such metrics connects multitree metric inference with classical Euclidean network algorithms, providing robust procedures for network estimation under noise. The interplay between electrical, combinatorial, and geometric structures in planar electrical Kalmanson metrics constitutes an active area integrating algebraic, topological, and statistical methodologies.