Degree diameter problem on honeycomb networks

The degree diameter problem involves finding the largest graph (in terms of the number of vertices) subject to constraints on the degree and the diameter of the graph. Beyond the degree constraint there is no restriction on the number of edges (apart from keeping the graph simple) so the resulting graph may be thought of as being embedded in the complete graph. In a generalization of this problem, the graph is considered to be embedded in some connected host graph, in this paper the honeycomb network. We consider embedding the graph in the kk-dimensional honeycomb grid and provide upper and lower bounds for the optimal graph. The particular cases of dimensions 2 and 3 are examined in detail.