# A Cheeger inequality for graphs based on a reflection principle

## TL;DR

This paper introduces a novel Neumann Laplace operator for graphs based on a reflection principle and establishes a Cheeger inequality relating its first eigenvalue to graph boundary properties.

## Contribution

The work defines a new Neumann Laplacian on graphs using reflection and proves a Cheeger inequality for its first eigenvalue, extending spectral graph theory.

## Key findings

- First eigenvalue satisfies Cheeger inequality
- New Neumann Laplacian based on reflection principle
- Connects boundary properties to spectral gap

## Abstract

Given a graph with a designated set of boundary vertices, we define a new notion of a Neumann Laplace operator on a graph using a reflection principle. We show that the first eigenvalue of this Neumann graph Laplacian satisfies a Cheeger inequality.