Conformal Rigidity and Spectral Embeddings of Graphs

TL;DR

This paper explores the concept of conformal rigidity in graphs, linking it to spectral embeddings and symmetries, and provides new characterizations and examples of conformally rigid graphs.

Contribution

It introduces new characterizations of conformal rigidity using spectral embeddings and symmetries, and identifies infinite families of conformally rigid circulant graphs.

Findings

All 1-walk regular graphs are conformally rigid.

Vertex-transitive graphs have a characterization for conformal rigidity.

An infinite family of conformally rigid circulant graphs is constructed.

Abstract

We investigate the structure of conformally rigid graphs. Graphs are conformally rigid if introducing edge weights cannot increase (decrease) the second (last) eigenvalue of the Graph Laplacian. Edge-transitive graphs and distance-regular graphs are known to be conformally rigid. We establish new results using the connection between conformal rigidity and edge-isometric spectral embeddings of the graph. All $1$-walk regular graphs are conformally rigid, a consequence of a stronger property of their embeddings. Using symmetries of the graph, we establish two related characterizations of when a vertex-transitive graph is conformally rigid. This provides a necessary and sufficient condition for a Cayley graph on an abelian group to be conformally rigid. As an application we exhibit an infinite family of conformally rigid circulants. Our symmetry technique can be interpreted in the language of semidefinite programming which provides another criterion for conformal rigidity in terms of edge orbits. The paper also describes a number of explicit conformally rigid graphs whose conformal rigidity is not yet explained by the existing theory.