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 $delta$.
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