  The NMR relaxation rate and the static spin susceptibility in graphene are
studied within a tight-binding description. At half filling, the NMR relaxation
rate follows a power law as $T^2$ on the particle-hole symmetric side, while
with a finite chemical potential $\mu$ and next-nearest neighbor $t'$, the
$(\mu+3t')^2$ terms dominate at low excess charge $\delta$. The static spin
susceptibility is linearly dependent on temperature $T$ at half filling when
$t'=0$, while with a finite $\mu$ and $t'$, it should be dominated by
$(\mu+3t')$ terms in low energy regime. These unusual phenomena are direct
results of the low energy excitations of graphene, which behave as massless
Dirac fermions. Furthermore, when $\delta$ is high enough, there is a
pronounced crossover which divides the temperature dependence of the NMR
relaxation rate and the static spin susceptibility into two temperature
regimes: the NMR relaxation rate and the static spin susceptibility increase
dramatically as temperature increases in the low temperature regime, and after
the crossover, both decrease as temperature increases at high temperatures.
This crossover is due to the well-known logarithmic Van Hove singularity in the
density of states, and its position dependence of temperature is sensitive to
$\delta$.
