Coverage control is the problem of navigating a robot swarm to collaboratively monitor features or a phenomenon of interest not known a priori. The problem is challenging in decentralized settings with robots that have limited communication and sensing capabilities. We propose a learnable Perception-Action-Communication (LPAC) architecture for the problem, wherein a convolution neural network (CNN) processes localized perception; a graph neural network (GNN) facilitates robot communications; finally, a shallow multi-layer perceptron (MLP) computes robot actions. The GNN enables collaboration in the robot swarm by computing what information to communicate with nearby robots and how to incorporate received information. Evaluations show that the LPAC models -- trained using imitation learning -- outperform standard decentralized and centralized coverage control algorithms. The learned policy generalizes to environments different from the training dataset, transfers to larger environments with more robots, and is robust to noisy position estimates. The results indicate the suitability of LPAC architectures for decentralized navigation in robot swarms to achieve collaborative behavior.