Center and its spectrum of almost all $n$-vertex graphs of given diameter

We study typical (valid for almost all graphs of a class under consideration) properties of the center and its spectrum (the set of centers cardinalities) for $n$-vertex graphs of fixed diameter $k$. The spectrum of the center of all and almost all $n$-vertex connected graphs is found. The structure of the center of almost all $n$-vertex graphs of given diameter $k$ is established. For $k= 1,2$ any vertex is central, while for $k\geq 3$ we identified two types of central vertices, which are necessary and sufficient to obtain the centers of almost all such graphs; in addition, centers of constructed typical graphs are found explicitly.
It is proved that the center of almost all $n$-vertex graphs of diameter $k$ has cardinality $n-2$ for $k=3$, and for $k\geq 4$ the spectrum of the center is bounded by an interval of consecutive integers except no more than one value (two values) outside the interval for even diameter $k$ (for odd diameter $k$) depending on $k$. For each center cardinality value outside this interval, we calculated an asymptotic fraction of the number of the graphs with such a center. The realizability of the found cardinalities spectrum as the spectrum of the center of typical $n$-vertex graphs of diameter $k$ is established.