Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the Rhombic Dodecahedron. Then we show that for any m ≥ 2, the graph G 2m, 2 is the unique up to isomorphism bipartite diametrical graph of diameter 4 and partite sets of cardinalities 2m and 2, and hence in particular, for m 3, the graph G 6, 8 which is just the Rhombic Dodecahedron is the unique up to isomorphism bipartite diametrical graph of such a diameter and cardinalities of partite sets. Thus we complete a characterization of S-graphs of diameter 4 and cardinality of the smaller partite set not exceeding 6. We prove that the neighborhoods of vertices of the larger partite set ofG 2m, 2 form a matroid whose basis graph is the hypercubeQm. We prove that any S-graph of diameter 4 is bipartite self complementary, thus in particularG 2m, 2 . Finally, we study some additional properties ofG 2m, 2 concerning the order of its automorphism group, girth, domination number, and when being Eulerian.