Let W be a Weyl or affine Weyl group and let Wc be the set of fully commutative elements in W. We associate each w ∈ Wc to a digraph G(w). By using G(w), we give a graph-theoretic description for Lusztig's a-function on Wc and describe explicitly all the distinguished involutions of W. The results verify two conjectures in our case: one was proposed by myself in [15, Conjecture 8.10] and the other was by Lusztig in [2].