On Graphs with Prescribed Center and Periphery

A connected graph \(G\) is unicentered if \(G\) has exactly one central vertex. It is proved that for integers \(r\) and \(d\) with \(1 \leq r < d \leq 2r\), there exists a unicentered graph \(G\) such that rad\((G) = r\) and diam\((G) = d\). Also, it is shown that for any two graphs \(F\) and \(G\) with rad\((F) = n \geq 4\) and a positive integer \(d\) (\(4 \leq d \leq n\)), there exists a connected graph \(H\) with diam\((H) = d\) such that the periphery and the center of \(H\) are isomorphic to \(F\) and \(G\), respectively.