Maximal planar graphs of diameter two
Karen Seyffarth · Journal of Graph Theory · 1989
Abstract A maximal planar graph is a simple planar graph in which every face is a triangle. We show here that such graphs with maximum degree Δ and diameter two have no more than 3/2Δ + 1 vertices. We also show that there exist maximal planar graphs with diameter two and exactly [3/2Δ + 1] vertices.