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 vertex set and where any \(x,y \in \Gamma \) are joined by an edge if x and y commute in H. In this paper, we solve the realizability problem for Coxeter groups by proving that any simple graph can be obtained as a commuting graph of such a group. In particular, we can recover the Dynkin diagrams of ADE type as commuting graphs. We further investigate the commuting graphs \(\mathcal {C}(H,\Gamma )\) for every finite subgroup \(H\subset {{\,\textrm{SL}\,}}(2,\mathbb {C})\) and different subsets \(\Gamma \subseteq H\) . We also study certain distance properties of these graphs when \(\Gamma =H\) .
