Conformal Rigidity of Graphs: Subdifferentials and Orbit-Isometries

TL;DR

This paper introduces a new subdifferential framework for analyzing conformal rigidity in graphs, linking eigenvalue optimization with spectral embeddings and symmetry considerations.

Contribution

It unifies variational and geometric perspectives on conformal rigidity using subdifferentials and introduces orbit-isometric embeddings to leverage graph symmetries.

Findings

Conformal rigidity can be certified by a single eigenvector for many graphs.

The framework enables algebraically exact certification methods.

Many cases reduce to linear feasibility or quadratic systems solved by Gr"obner bases.

Abstract

A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $\lambda_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge weights minimize the largest eigenvalue $\lambda_n(w)$ over all normalized edge weights; $G$ is conformally rigid if it is lower or upper conformally rigid. This paper establishes a new framework for conformal rigidity through the language of subdifferentials, unifying the variational perspective on eigenvalue optimization with the geometry of edge-isometric spectral embeddings, which are known to characterize conformal rigidity. This subdifferential framework lends itself naturally to techniques of symmetry reduction that motivate the notion of an orbit-isometric embedding - a weaker condition than edge-isometry that accounts for the symmetries of $G$ while remaining sufficient for conformal rigidity. The notion opens the door to tools from representation theory: for a large class of graphs, including all vertex-transitive ones, we show that conformal rigidity is certified by a single eigenvector, resolving an open question and explaining the conformal rigidity of previously unexplained graphs. This extra structure enables a new, algebraically exact certification method for conformal rigidity, bypassing the numerical difficulties of prior approaches. In many cases, the problem reduces to a check of linear feasibility, and in general, to solving a system of quadratic equations via Gr\"{o}bner bases.