  We develop combinatorics of parabolic double cosets in finite Coxeter groups
as a follow-up of recent articles by Billey-Konvalinka-Petersen-Slofstra-Tenner
and Petersen. (1) We construct a double coset system as a generalization of a
two-sided analogue of a Coxeter complex and present its order structure with
its local dimension function on certain connected components. As applications
of double cosets to Bruhat graphs, we also prove: (2) every parabolic double
coset is regular, (3) invariance of degree on Bruhat graph on lower intervals
as an analogy of the one for Kazhdan-Lusztig polynomials, (4) every noncritical
Bruhat interval satisfies out-Eulerian property.
