  We give rigidity results for the discrete Bonnet-Myers diameter bound and the
Lichnerowicz eigenvalue estimate. Both inequalities are sharp if and only if
the underlying graph is a hypercube. The proofs use well-known semigroup
methods as well as new direct methods which translate curvature to
combinatorial properties. Our results can be seen as first known discrete
analogues of Cheng's and Obata's rigidity theorems.
