In this paper we present a decentralized control strategy for the connectivity maintenance for groups of mobile robots performing some desired task. Exploiting a completely decentralized estimation strategy for the algebraic connectivity of the graph, we prove the connectivity maintenance property in the general case, i.e. in presence of a generic (bounded) additional control term. Then, we address two specific decentralized control applications: rendezvous and formation control. We analytically prove that our control strategy ensures the global connectivity maintenance, while preserving the convergence properties of the rendezvous and formation controllers respectively. Simulations and experimental results are presented as well, in order to show the effectiveness of the proposed control law.