# Expanders - how to find them, and what to find in them

## TL;DR

This paper explores the definition, properties, and examples of expander graphs, introduces methods to identify large expanding subgraphs, and investigates their structural features like separators, cycles, and minors.

## Contribution

It provides a comprehensive overview of expander graph definitions, presents new techniques for finding large expanding subgraphs, and analyzes their structural properties.

## Key findings

- Existence of large expanding subgraphs in general graphs
- Properties such as small separators and cycle structures in expanders
- Methods for embedding large minors in expanders

## Abstract

A graph $G=(V,E)$ is called an expander if every vertex subset $U$ of size up to $|V|/2$ has an external neighborhood whose size is comparable to $|U|$. Expanders have been a subject of intensive research for more than three decades and have become one of the central notions of modern graph theory. We first discuss the above definition of an expander and its alternatives. Then we present examples of families of expanding graphs and state basic properties of expanders. Next, we introduce a way to argue that a given graph contains a large expanding subgraph. Finally we research properties of expanding graphs, such as existence of small separators, of cycles (including cycle lengths), and embedding of large minors.