The routing capabilities of an interconnection network are strictly related to its bandwidth and latency characteristics, which are in turn quantifiable through the graph-theoretic concepts of expansion and diameter. This paper studies expansion and diameter of a family of subgraphs of the random geometric graph, which closely model the topology induced by the device discovery phase of Bluetooth-based ad hoc networks. The main feature modeled by any such graph, denoted as BTr(n),c(n), is the small number c(n) of links that each of the n devices (vertices) may establish with those located within its communication range r(n). First, tight bounds are proved on the expansion of BTr(n),c(n) for the whole set of functions r(n) and c(n) for which connectivity has been established in previous works. Then, by leveraging on the expansion result, tight (up to a logarithmic additive term) upper and lower bounds on the diameter of BTr(n),c(n) are derived.