  We provide local recognition results for the reflection graphs on spherical
Coxeter groups. In particular, we study the case $F_4$ which is locally
recognizable under additional constraints only. It is then demonstrated in the
cases $A_n$ and $F_4$ how these graph theoretical recognition results can be
used to characterize the corresponding Coxeter groups in terms of their
reflection centralizers. Finally, we outline the connection to the local
recognition of Chevalley groups which is particularly important in the
classification of the finite simple groups.
