Characterizing symmetric diametrical graphs of order 12 and diameter 4 (Q1608165)

Summary: A diametrical graph \(G\) is said to be symmetric if \(d (u,v)+d (v,\bar u)=d (G)\) for all \(u,v\in V (G)\), where \(\bar u\) is the buddy of \(u\). If, moreover, \(G\) is bipartite, then it is called an \(S\)-graph. We show that the Cartesian product \(K_{2}\times C_{6}\) is not only the unique \(S\)-graph of order \(12\) and diameter \(4\), but also the unique symmetric diametrical graph of order \(12\) and diameter \(4\).