The maximum anchored k-core problem plays an important role in marketing, network architecture, and social media; the problem allows network designers and influencers to find the most pivotal vertices which increase the size of the network. In this thesis, we investigate two mixed integer programming (MIP) formulations for the maximum anchored k-core problem: (i) a naive model and (ii) a strong model. We examine the MIP formulations analytically and computationally. We also compare the computational performance of the MIP models with two existing heuristic algorithms: Residual Core Maximization (RCM) and Onion-Layer based Anchored k-core (OLAK). Furthermore, we propose valid inequalities and fixing procedures to improve the computational performance of the MIP models. Finally, we conduct experiments on a set of benchmark instances. Our computational experiments show the superiority of the strong model against the naive model, and the heuristics.