In this paper, we study the relationship between two crucial properties in linear dynamical networks of diffusively coupled agents, that is controllability and robustness to noise and structural changes in the network. In particular, for any given network size and diameter, we identify networks that are maximally robust and then analyze their strong structural controllability. We do so by determining the minimum number of leaders to make such networks completely controllable with arbitrary coupling weights between agents. Similarly, we design networks with the same given parameters that are completely controllable independent of coupling weights through a minimum number of leaders, and then also analyze their robustness. We utilize the notion of Kirchhoff index to measure network robustness to noise and structural changes. Our controllability analysis is based on novel graph-theoretic methods that offer insights on the important connection between network robustness and strong structural controllability in such networks.