The excess zero graph of a Coxeter group

Abstract For a Coxeter group 𝑊 with length function ℓ, the excess zero graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">E</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>W</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{E}_{0}(W) has vertex set the non-identity involutions of 𝑊, with two involutions 𝑥 and 𝑦 adjacent whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="normal">ℓ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>⁢</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="normal">ℓ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:mi mathvariant="normal">ℓ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> \ell(xy)=\ell(x)+\ell(y) . Properties of this graph such as connectivity, diameter and valencies of certain vertices of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi mathvariant="script">E</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>W</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathcal{E}_{0}(W) are explored.