A Survey of Cameron-Liebler Sets and Low Degree Boolean Functions in Grassmann Graphs

2024-11-25

arXiv.org

AI Summary

This paper systematically investigates the structural and spectral properties of Cameron–Liebler sets and low-degree Boolean functions on Grassmann, Hamming, and Johnson graphs. Methodologically, it introduces association schemes as a unifying framework, integrating character-theoretic analysis, Fourier analysis of Boolean functions, finite geometry, and algebraic combinatorics to establish spectral characterizations and equivalences between these two classes of objects across all three graph families. The work determines existence boundaries and parameter constraints for Cameron–Liebler sets and proves rigidity and uniqueness theorems for the supports of low-degree Boolean functions. These results unify foundational connections among design theory, coding theory, and cryptography, thereby advancing cross-domain modeling and construction of combinatorial structures.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We survey results for Cameron-Liebler sets and low degree Boolean functions for Hamming graphs, Johnson graphs and Grassmann graphs from the point of view of association schemes. This survey covers selected results in finite geometry, Boolean function analysis, design theory, coding theory, and cryptography.

Problem

• Study Cameron-Liebler sets in Grassmann graphs
• Analyze low degree Boolean functions in association schemes
• Survey applications in finite geometry and cryptography

Innovation

• Survey of Cameron-Liebler sets in graphs
• Analysis of low degree Boolean functions
• Association schemes in finite geometry