Logarithmic asymptotics of the number of central vertices of almost all $n$-vertex graphs of diameter $k$

The asymptotic behavior of the number of central vertices and F. Buckley's central ratio ${\mathbb R}_{c}(G)=|{\mathbb C}(G)|/|V(G)|$ for almost all $n$-vertex graphs $G$ of fixed diameter $k$ is investigated.
The logarithmic asymptotics of the number of central vertices for almost all such $n$-vertex graphs is established: $0$ or $\log_2 n$ ($1$ or $\log_2 n$), respectively, for arising here subclasses of graphs of the even (odd) diameter.
It is proved that for almost all $n$-vertex graphs of diameter $k$, ${\mathbb R}_{c}(G)=1$ for $k=1,2$, and ${\mathbb R}_{c }(G)=1-2/n$ for graphs of diameter $k=3$, while for $k\geq 4$ the value of the central ratio ${\mathbb R}_{c}(G)$ is bounded by the interval $(\frac{\Delta}{6} + r_1(n), 1-\frac{\Delta}{6} - r_1(n))$ except no more than one value (two values) outside the interval for even diameter $k$ (for odd diameter $k$) depending on $k$. Here $\Delta\in (0,1)$ is arbitrary predetermined constant and $r_1(n),r_2(n)$ are positive infinitesimal functions.