Diameter bounds for planar graphs

The inverse degree   of a graph is the sum of the reciprocals of the degrees of its vertices. We prove that in any connected planar graph, the diameter is at most 5/2 times the inverse degree, and that this ratio is tight. To develop a crucial surgery method, we begin by proving the simpler related upper bounds (4(|V|−1)−|E|)/3(4(|V|−1)−|E|)/3 and 4|V|2/3|E|4|V|2/3|E| on the diameter (for connected planar graphs), which are also tight.