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 and 4V^2/3E on the diameter (for connected planar graphs), which are also tight.
@article{arxiv.1006.2493,
title = {Diameter Bounds for Planar Graphs},
author = {Radoslav Fulek and Filip Morić and David Pritchard},
journal= {arXiv preprint arXiv:1006.2493},
year = {2010}
}