Diameter graphs of polygons and the proof of a conjecture of Graham

We show that for an n-gon with unit diameter to have maximum area, its diameter graph must contain a cycle, and we derive an isodiametric theorem for such n-gons in terms of the length of the cycle. We then apply this theorem to prove Grahamʹs 1975 conjecture that the diameter graph of a maximal 2m-gon ( m ⩾ 3 ) must be a cycle of length 2 m − 1 with one additional edge attached to it.