Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders

Abstract

We provide a process to extend any bipartite diametrical graph of diameter 4 to an -graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets and , where , we prove that is a sharp upper bound of and construct an -graph 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 , the graph is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities and , and hence in particular, for , the graph 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 -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 of form a matroid whose basis graph is the hypercube . We prove that any -graph of diameter 4 is bipartite self complementary, thus in particular . Finally, we study some additional properties of concerning the order of its automorphism group, girth, domination number, and when being Eulerian.