Ultrametrics and Complete Multipartite Graphs

Let $(X, d)$ be a semimetric space and let $G$ be a graph. We say that $G$ is the diametrical graph of $(X, d)$ if $X$ is the vertex set of $G$ and the adjacency of vertices $x$ and $y$ is equivalent to the equality \diam X = d(x, y). It is shown that a semimetric space $(X, d)$ with diameter $d^*$ is ultrametric iff the diametrical graph of $(X, d_{\varepsilon})$ with $d_{\varepsilon}(x, y) = \min\{d(x, y), \varepsilon\}$ is complete multipartite for every $\varepsilon \in (0, d^*)$. A refinement of the last result is given for totally bounded ultrametric spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. The bounded ultrametric spaces, which are weakly similar to unbounded ones, are also characterized via complete multipartite graphs

Bipartite Diametrical Graphs of Diameter 4 and Extreme Orders

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 2=||≤||, we prove that 2 is a sharp upper bound of || and construct an -graph (2,2) 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 ≥2, the graph (2,2) is the unique (up to isomorphism) bipartite diametrical graph of diameter 4 and partite sets of cardinalities 2 and 2, and hence in particular, for =3, the graph (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 -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 (2,2) form a matroid whose basis graph is the hypercube . We prove that any -graph of diameter 4 is bipartite self complementary, thus in particular (2,2). Finally, we study some additional properties of (2,2) concerning the order of its automorphism group, girth, domination number, and when being Eulerian