Proof of the Gobel–Veldman conjecture on diametrical graphs

Abstract: A graph is called noncircular if it is not a cycle. A nontrivial connected graph G of diameter d is called diametrical if, for every vertex v of G, there exists a unique vertex at distance d from v. In 1986, Gobel and Veldman conjectured that the order of a noncircular diametrical graph of diameter d is at least 4(d-1).  We prove this conjecture.