S-antipodal graphs (Q1366920)

\textit{D. H. Smith} [Q. J. Math., Oxford II. Ser. 22, 551-557 (1971; Zbl 0222.05111)] has introduced the antipodal graph of a graph \(G\) as the graph \(A(G)\) with the same vertex set as that of \(G\) and two vertices adjacent if their distance in \(G\) is \(\text{diam}(G)\). Deleting all the isolated vertices of \(A(G)\) one obtains the S-antipodal graph \(A^\ast (G)\) studied in this paper. Main results: Every graph without isolated vertices is the S-antipodal graph of a hamiltonian graph of diameter 2. Every eulerian graph is the S-antipodal graph of an eulerian graph. A graph \(G\) is the S-antipodal graph of a tree iff \(G\) is complete or complete multipartite.