### The Degree-Diameter Problem for Sparse Graph Classes

#### Abstract

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. 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 $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine 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.

#### Keywords

degree-diameter problem; treewidth; arboricity; sparse graph; surface graph; apex-minor-free graph