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Ricci curvature, Bruhat graphs and Coxeter groups

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{ä}ger \cite{shmuck} and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a finite Coxeter group and a classical result obtained by Dyer in \cite{Dyer}. As an application we obtain a bound for the spectral gap of the Bruhat graph of any finite Coxeter group and an isoperimetric inequality for them. Our proofs are case-free.