Effective resistance in planar graphs and continued fractions

TL;DR

The paper solves the inverse problem for effective resistance on planar graphs by showing that for coprime $t>c\ge 1$ there exists a simple planar graph $G$ with an edge $e$ such that $\rho(G,e)=\frac{c}{t}$ and the vertex count satisfies $|V| \le C\max\{\frac{t}{c},\frac{t}{t-c},\log t\}$, with a matching lower bound up to constants. The authors develop a marked-graph framework with operations like subdivision, duplication, marked sum, and duality to assemble target $\zeta$-values (where $\rho=1/(1+\zeta)$) and then convert these into $\rho$ via the duality and simplification steps. A central technique is decomposing the target fraction using Bourgain–Kontorovich results on continued fractions, which reduces the construction to combining small $\zeta$-blocks and controlling the total continued-fraction weight via the sum $S(q)$. This extends Sedláček-type inverse problems from spanning-tree counts to ratios of spanning-tree quantities in planar graphs, yielding near-optimal size bounds and connecting graph-theoretic constructions with analytic number theory methods.

Abstract

For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{τ(G/e)}{τ(G)}$, where $τ(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedláček's inverse problem for the number of spanning trees.

Theorems & Definitions (19)

• Theorem 1.1: Main theorem
• Proposition 1.2
• Theorem 1.3: Alon--Bucić--Gishboliner ABG
• Theorem 1.4: Bourgain Bou12
• Theorem 1.5
• Remark 1.6
• Proposition 2.1: CP-SY
• Lemma 2.2: CP-SY
• Lemma 2.3
• proof