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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(vertical bar V vertical bar - 1) - vertical bar E vertical bar)/3 and 4 vertical bar V vertical bar(2)/3 vertical bar E vertical bar on the diameter (for connected planar graphs), which are also tight. (C) 2010 Elsevier B.V. All rights reserved.
Abhishek Methuku, Balázs Keszegh