# Commuting graphs on Coxeter groups, Dynkin diagrams and finite subgroups of $SL(2,\mathbb{C})$

## TL;DR

This paper demonstrates that any simple graph can be realized as a commuting graph of a Coxeter group, linking Dynkin diagrams, finite subgroups of SL(2,C), and the McKay correspondence.

## Contribution

It proves the realizability of all simple graphs as commuting graphs of Coxeter groups and connects ADE Dynkin diagrams and McKay correspondence through this framework.

## Key findings

- Any simple graph can be obtained as a commuting graph of a Coxeter group.
- Dynkin diagrams of ADE type are recoverable as commuting graphs.
- Results relate finite subgroups of SL(2,C) to Coxeter groups and McKay correspondence.

## Abstract

For a group $H$ and a non empty subset $\Gamma\subseteq H$, the commuting graph $G=\mathcal{C}(H,\Gamma)$ is the graph with $\Gamma$ as the node set and where any $x,y \in \Gamma$ are joined by an edge if $x$ and $y$ commute in $H$. We prove that any simple graph can be obtained as a commuting graph of a Coxeter group, solving the realizability problem in this setup. In particular we can recover every Dynkin diagram of ADE type as a commuting graph. Thanks to the relation between the ADE classification and finite subgroups of $\SL(2,\C)$, we are able to rephrase results from the {\em McKay correspondence} in terms of generators of the corresponding Coxeter groups. We finish the paper studying commuting graphs $\mathcal{C}(H,\Gamma)$ for every finite subgroup $H\subset\SL(2,\C)$ for different subsets $\Gamma\subseteq H$, and investigating metric properties of them when $\Gamma=H$.