The Excess Zero Graph of a Coxeter Group

For a Coxeter group $W$ with length function $\mathrm{\ell }$, the excess zero graph ${\mathcal{E}}_0(W)$ has vertex set the non-identity involutions of $W$, with two involutions $x$ and $y$ adjacent whenever $\mathrm{\ell }(xy)=\mathrm{\ell }(x)+\mathrm{\ell }(y)$. Properties of this graph such as connectivity, diameter and valencies of certain vertices of ${\mathcal{E}}_0(W)$ are explored.