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

For a group $H$ and a non empty subset $\mathrm{\Gamma }\subseteq H$, the commuting graph $G=\mathcal{C}(H,\mathrm{\Gamma })$ is the graph with $\mathrm{\Gamma }$ as the node set and where any $x,y\in \mathrm{\Gamma }$ 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 $\text{SL}(2,\text{C})$, 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 $\mathcal{C}(H,\mathrm{\Gamma })$ for every finite subgroup $H\subset \text{SL}(2,\text{C})$ for different subsets $\mathrm{\Gamma }\subseteq H$, and investigating metric properties of them when $\mathrm{\Gamma }=H$.