The degree-diameter problem for sparse graph classes

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree Δ and diameter k. For fixed k, the answer is Ө(Δ^k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Ө(Δ^[k-1]), and for graphs of bounded arboricity the answer is Ө(Δ^[k/2]), in both cases for fixed k. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.