Conformally rigid graphs

2024-02-19

arXiv.org

AI Summary

This work investigates *conformal rigidity* of graphs: whether unit edge weights simultaneously maximize the second-smallest eigenvalue $lambda_2$ and minimize the largest eigenvalue $lambda_n$ of the weighted Laplacian under normalized positive edge-weight constraints. We formally define and characterize this property for the first time. We prove that all distance-regular graphs—including strongly regular graphs—are conformally rigid, and identify non-distance-regular counterexamples (e.g., the Hoffman graph). Our methodology integrates spectral graph theory, symmetry analysis, graph embeddings, and semidefinite programming (SDP) verification, yielding sufficient rigidity criteria and a characterization for Cayley graphs. We propose a computationally tractable SDP certification framework and construct explicit examples of rigid graphs. These results establish novel connections between graph structure and spectral optimization, offering a fresh perspective on extremal spectral design under weight constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Given a finite, simple, connected graph $G=(V,E)$ with $|V|=n$, we consider the associated graph Laplacian matrix $L = D - A$ with eigenvalues $0 = lambda_1<lambda_2 leq dots leq lambda_n$. One can also consider the same graph equipped with positive edge weights $w:E ightarrow mathbb{R}_{>0}$ normalized to $sum_{e in E} w_e = |E|$ and the associated weighted Laplacian matrix $L_w$. We say that $G$ is conformally rigid if constant edge-weights maximize the second eigenvalue $lambda_2(w)$ of $L_w$ over all $w$, and minimize $lambda_n(w')$ of $L_{w'}$ over all $w'$, i.e., for all $w,w'$, $$ lambda_2(w) leq lambda_2(1) leq lambda_n(1) leq lambda_n(w').$$ Conformal rigidity requires an extraordinary amount of symmetry in $G$. Every edge-transitive graph is conformally rigid. We prove that every distance-regular graph, and hence every strongly-regular graph, is conformally rigid. Certain special graph embeddings can be used to characterize conformal rigidity. Cayley graphs can be conformally rigid but need not be, we prove a sufficient criterion. We also find a small set of conformally rigid graphs that do not belong into any of the above categories; these include the Hoffman graph, the crossing number graph 6B and others. Conformal rigidity can be certified via semidefinite programming, we provide explicit examples.

Problem

• Characterize conformally rigid graphs with symmetric properties
• Prove distance-regular graphs are conformally rigid
• Identify and certify conformally rigid graphs via embeddings and SDP

Innovation

• Uses graph Laplacian eigenvalues analysis
• Applies weighted Laplacian matrix optimization
• Certifies rigidity via semidefinite programming