It has recently been proposed that the miscibility of nanoparticles with a polymer matrix can be controlled by grafting polymer chains to the nanoparticle surface. As a first step to study this situation, we have used molecular dynamics simulations on a single nanoparticle of radius R ($4\sigma \le$R$\le 16\sigma$, where $\sigma$ is the diameter of a polymer monomer) grafted with chains of length 500 in a polymer melt of chains of length 1000. The grafting density $\Sigma$ was varied between $0.04$-$0.32$ chains/$\sigma^2$. To facilitate equilibration a Monte Carlo double-bridging algorithm is applied - new bonds are formed across a pair of chains, creating two new chains each substantially different from the original. For the long brush chains studied here, the structure of the brush assumes its large particle limit even for $R$ as small as 8$\sigma$, which is consistent with recent experimental findings. We study autophobic dewetting of the melt from the brush as a function of increasing $\Sigma$. Even these long brush and matrix chains of length $6$ and $12$ $N_e$, respectively, (the entanglement length is $N_e \sim 85$) give somewhat ambiguous results for the interfacial width, showing that studies of two or more nanoparticles are necessary to properly understand these miscibility issues. Entanglement between the brush and melt chains were identified using the primitive path analysis. We find that the number of entanglements between the brush and melt chains scale simply with the product of the local monomer densities of brush and melt chains.