What does the Alon-Boppana bound do?

The Alon-Boppana bound establishes the theoretical limit for eigenvalue separation in d-regular graphs, stating that the second eigenvalue cannot be smaller than 2√(d-1) for large graphs.

What is a Ramanujan graph?

A Ramanujan graph is a d-regular graph that is an optimal spectral expander, meaning its second-largest eigenvalue is at most 2√(d-1), providing the best possible expansion properties.

How does the skill handle edge growth?

It provides rules like 2-switching to preserve graph regularity and greedy spectral monotonicity heuristics to maintain the Ramanujan property while adding edges.

Why is mixing time important in this context?

In Ramanujan graphs, mixing time is O(log n), meaning a random walk reaches its stationary distribution very quickly, which is critical for efficient search and data propagation.

Ramanujan Expander

Name: Ramanujan Expander
Author: plurigrid

byplurigrid

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数据科学与机器学习

Analyzes and constructs Ramanujan graphs to achieve optimal spectral expansion and network efficiency within graph-based systems.

关于

The Ramanujan Expander skill provides specialized tools for working with optimal spectral expanders, ensuring your graph structures meet the theoretical Alon-Boppana bound for eigenvalue separation. It enables developers to verify the Ramanujan property, perform edge growth while preserving regularity, and calculate mixing times for rapid information dissemination. This skill is essential for researchers and engineers building high-efficiency communication networks, distributed systems, or performing advanced spectral clustering where connectivity and expansion are critical.

主要功能

Spectral and non-backtracking centrality computation
O(log n) mixing time estimation based on spectral gaps
2 GitHub stars
Alon-Boppana bound verification for spectral optimality
Regularity-preserving edge growth via 2-switch algorithms
Lubotzky-Phillips-Sarnak (LPS) graph construction

使用场景

Simulating rapid information mixing and propagation in distributed systems
Designing high-performance peer-to-peer network topologies
Verifying the spectral efficiency and robustness of existing graph datasets

关于

主要功能

Spectral and non-backtracking centrality computation
O(log n) mixing time estimation based on spectral gaps
2 GitHub stars
Alon-Boppana bound verification for spectral optimality
Regularity-preserving edge growth via 2-switch algorithms
Lubotzky-Phillips-Sarnak (LPS) graph construction

使用场景

Simulating rapid information mixing and propagation in distributed systems
Designing high-performance peer-to-peer network topologies
Verifying the spectral efficiency and robustness of existing graph datasets