On Graphs with Prescribed Center and Periphery

Abstract

A connected graph $G$ is unicentered if $G$ has exactly one central vertex. It is proved that for integers $r$ and $d$ with $1\le r<d\le 2r$, 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$(F)=n\ge 4$ and a positive integer $d$ ($4\le d\le n$), 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.