Involutions in Coxeter groups

We combinatorially characterize the number cc₂ of conjugacy classes of involutions in any Coxeter group in terms of higher rank odd graphs. This notion naturally generalizes the concept of odd graphs, used previously to count the number of conjugacy classes of reflections. We provide uniform bounds and discuss some extremal cases, where the number cc₂ is smallest or largest possible. Moreover, we provide formulae for cc₂ in free and direct products as well as for some finite and affine types, besides computing cc₂ for all triangle groups, and all affine irreducible Coxeter groups of rank up to eleven.