Title: Commuting graphs on Coxeter groups, Dynkin diagrams and finite subgroups of $SL(2,\mathbb{C})$,
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 $\Gamma=H$.