# An overview of $(\kappa,\tau)$-regular sets and their applications

## TL;DR

This paper provides an overview of ($, au$)-regular sets in graphs, their spectral properties, and their connections to classical combinatorial structures like matchings and Hamilton cycles.

## Contribution

It summarizes main results on ($, au$)-regular sets, characterizes graphs with classical structures, and discusses their spectral properties and algorithms for determination.

## Key findings

- Characterization of graphs with ($, au$)-regular sets
- Equivalence of ($, au$)-regular sets determination to classical structures
- Spectral properties of ($, au$)-regular sets

## Abstract

A ($\kappa$,$\tau$)-regular set is a vertex subset S inducing a $\kappa$-regular subgraph such that every vertex out of S has $\tau$ neighbors in S. This article is an expository overview of the main results obtained for graphs with ($\kappa$,$\tau$)-regular sets. The graphs with classical combinatorial structures, like perfect matchings, Hamilton cycles, efficient dominating sets, etc, are characterized by ($\kappa$,$\tau$)-regular sets whose determination is equivalent to the determination of those classical combinatorial structures. The characterization of graphs with these combinatorial structures are presented. The determination of ($\kappa$,$\tau$)-regular sets in a finite number of steps is deduced and the main spectral properties of these sets are described.