We study the dynamics of the Olami-Feder-Christensen (OFC) model of earthquakes, focusing on the behavior of sequences of epicenters regarded as a growing complex network. Besides making a detailed and quantitative study of the effects of the borders (the occurrence of epicenters is dominated by a strong border effect which does not scale with system size), we examine the degree distribution and the degree correlation of the graph. We detect sharp differences between the conservative and nonconservative regimes of the model. Removing border effects, the conservative regime exhibits a Poisson-like degree statistics and is uncorrelated, while the nonconservative has a broad power-law-like distribution of degrees (if the smallest events are ignored), which reproduces the observed behavior of real earthquakes. In this regime the graph has also an unusually strong degree correlation among the vertices with higher degree, which is the result of the existence of temporary attractors for the dynamics: as the system evolves, the epicenters concentrate increasingly on fewer sites, exhibiting strong synchronization, but eventually spread again over the lattice after a series of sufficiently large earthquakes. We propose an analytical description of the dynamics of this growing network, considering a Markov process network with hidden variables, which is able to account for the mentioned properties.