Spectral bounds of multi-way Cheeger constants via cyclomatic number

TL;DR

This paper introduces new spectral bounds for multi-way Cheeger constants in graphs, utilizing cyclomatic number shifts to achieve bounds with absolute constants and providing simpler proofs for certain cases.

Contribution

It establishes bounds for multi-way Cheeger constants using cyclomatic number shifts, replacing $C(k)$ with absolute constants, and offers new lower bounds based on spectral radius.

Findings

Upper bounds with absolute constants for Cheeger constants

Simplified proof of Miclo's inequalities on trees

Lower bounds in terms of spectral radius

Abstract

As a non-trivial extension of the celebrated Cheeger inequality, the higher-order Cheeger inequalities for graphs due to Lee, Oveis Gharan and Trevisan provide for each $k$ an upper bound for the $k$-way Cheeger constant in forms of $C(k)\sqrt{\lambda_k(G)}$, where $\lambda_k(G)$ is the $k$-th eigenvalue of the graph Laplacian and $C(k)$ is a constant depending only on $k$. In this article, we prove some new bounds for multi-way Cheeger constants. By shifting the index of the eigenvalue via cyclomatic number, we establish upper bound estimates with an absolute constant instead of $C(k)$. This, in particular, gives a more direct proof of Miclo's higher order Cheeger inequalities on trees. We also show a new lower bound of the multi-way Cheeger constants in terms of the spectral radius of the graph. The proofs involve the concept of discrete nodal domains and a probability argument showing generic properties of eigenfunctions.