Radius, diameter, and minimum degree (Q1262322)

The authors prove that the diameter of a connected graph \(G\) with \(n\) vertices and minimum degree \(\delta\geq 2\) is bounded from above by \([3n/(\delta +1)]-1\), and that this bound is asymptotically sharp where \(\delta\) is fixed and \(n\) tends to infinity. They show an analogous result for the radius of \(G\), and also give upper bounds for triangle-free and \(C^4\)-free connected graphs.