# Spectral methods for testing cluster structure of graphs

## TL;DR

This paper introduces a spectral algorithm for testing whether a graph can be partitioned into two clusters with high conductance, improving query complexity bounds and removing a logarithmic factor from previous results.

## Contribution

It presents a spectral method for 2-clusterability testing that enhances previous bounds and simplifies the approach by leveraging eigenvector geometry.

## Key findings

- Achieves a query complexity of $O(n^{1/2+O(1)rac{1}{\u03bc}} \, ext{poly}(1/\u03b5, 1/, \, ext{log} n))$
- Removes a log n factor from the previous bound on ^* for k=2
- Provides an algorithm with high probability guarantees for acceptance and rejection in property testing

## Abstract

In the framework of graph property testing, we study the problem of determining if a graph admits a cluster structure. We say that a graph is $(k, \phi)$-clusterable if it can be partitioned into at most $k$ parts such that each part has conductance at least $\phi$. We present an algorithm that accepts all graphs that are $(2, \phi)$-clusterable with probability at least $\frac{2}3$ and rejects all graphs that are $\epsilon$-far from $(2, \phi^*)$-clusterable for $\phi^* \le \mu \phi^2 \epsilon^2$ with probability at least $\frac{2}3$ where $\mu > 0$ is a parameter that affects the query complexity. This improves upon the work of Czumaj, Peng, and Sohler by removing a $\log n$ factor from the denominator of the bound on $\phi^*$ for the case of $k=2$. Our work was concurrent with the work of Chiplunkar et al.\@ who achieved the same improvement for all values of $k$. Our approach for the case $k=2$ relies on the geometric structure of the eigenvectors of the graph Laplacian and results in an algorithm with query complexity $O(n^{1/2+O(1)\mu} \cdot \text{poly}(1/\epsilon, 1/\phi,\log n))$.