Hexagonal mesh and torus, as well as honeycomb and certain other pruned torus networks, are known to belong to the class of Cayley groups which are node-symmetric and possess other interesting mathematical properties. In this paper, we use Cayley-graph formulations for the aforementioned networks, along with some of our previous results on subgraphs and coset graphs, to draw conclusions relating to internode distance and network diameter. We also use our results to refine, clarify, and unify a number of previously published properties for these networks and other networks derived from them.