Bipartite diametrical graphs of diameter 4 and extreme orders
Summary: 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 \(2m=|U| \leq |W|\), we prove that \(2^{m}\) is a sharp upper bound of \(|W|\) and construct an \(S\)-graph \(G(2m,2^{m})\) 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 \(m \geq 2\), the graph \(G(2m,2^{m})\) is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities \(2m\) and \(2^{m}\), 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 of \(G(2m,2^{m})\) form a matroid whose basis graph is the hypercube \(Q_{m}\). We prove that any \(S\)-graph of diameter 4 is bipartite self complementary, thus in particular \(G(2m,2^{m})\). Finally, we study some additional properties of \(G(2m,2^{m})\) concerning the order of its automorphism group, girth, domination number, and when being Eulerian.
scientific article; zbMATH DE number 5313270