Characterizing symmetric diametrical graphs of order 12 and diameter 4

Abstract

A diametrical graph G is said to be symmetric if d(u,v)+d(v,u¯)=d(G) for all u,v∈V(G), where u¯ is the buddy of u. If moreover, G is bipartite, then it is called an S-graph. It would be shown that the Cartesian product K2×C6 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.