Coxeter groups, graphs and Ricci curvature

We consider a notion of curvature for graphs introduced in 1998 by Schmuckenschläger which is an analogue of Ricci curvature. First of all we see some general results on the discrete Ricci curvature of any locally finite graph. We then focus on graphs associated with Coxeter groups: Bruhat graphs, weak order graphs and Hasse diagrams of the Bruhat order. In particular we see that the discrete Ricci curvature for the Bruhat order is always 2 and that the discrete Ricci curvature of the weak order graph of a strictly linear Coxeter system (W, S) is (Formula presented.).