On the Graph Corresponding to Center of Distances of Metric Space

The center of distances $C(Y)$ of a metric space $(Y,\rho )$ is the set of all $t\ge 0$ for which the equation $\rho (y,p)=t$ has a solution for each point $p\in Y$. We say that a simple graph $C{G}_Y$ is the central graph of a metric space $(Y,\rho )$ if $Y$ is the vertex set of $C{G}_Y$ and distinct vertices $x,y\in Y$ are adjacent if and only if $\rho (x,y)\in C(Y)$. We prove the inequality $\mathrm{\mid }E(C{G}_Y)\mathrm{\mid }\ge \lceil \frac{1}{2}\mathrm{\mid }Y\mathrm{\mid }\rceil$ for all finite metric spaces $(Y,\rho )$ with $\mathrm{\mid }Y\mathrm{\mid }\ge 2$ and non-empty $C{G}_Y\mathrm{.}$ It is also proved that the inequality $\mathrm{\mid }E(C{G}_X)\mathrm{\mid }\ge \mathrm{\mid }X\mathrm{\mid }-1$ holds for each finite ultrametric space $(X,d)$. The central graphs of metric spaces $(Y,\rho )$ and ultrametric spaces $(X,d)$ satisfying $\mathrm{\mid }E(C{G}_Y)\mathrm{\mid }=\lceil \frac{1}{2}\mathrm{\mid }Y\mathrm{\mid }\rceil$ and, respectively, $\mathrm{\mid }E(C{G}_X)\mathrm{\mid }=\mathrm{\mid }X\mathrm{\mid }-1$ are described up to graph isomorphism.