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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
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     {\flushleft\it Том 19, № 2, стр. 144--144 (2022) \hfill{\rm\small УДК \udcs}}
     \newline
        {\rm\small DOI~10.33048/semi.2022.19.xxx}\hfill{\rm\small MSC\ \ \mscs }



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\title[Logarithmic asymptotics of the number of central vertices]
{Logarithmic asymptotics\\of the number of central vertices\\of
almost all $n$-vertex graphs of diameter $k$}
\author{{T.I. FEDORYAEVA}}%


\thanks{\sc Fedoryaeva,~T.I.,
Logarithmic asymptotics of the number of central vertices of almost
all $n$-vertex graphs of diameter $k$}
\thanks{\copyright \ 2022 Fedoryaeva,~T.I.}
\thanks{\rm The work was carried out within the framework of the state contract
of the Sobolev Institute of Mathematics (project no.
FWNF-2022-0018)}
\thanks{\it Received May, 11, 2022, published May, , 2022.}%


\semrtop \vspace{1cm} \maketitle {\small
\begin{quote}
\noindent{\sc Abstract.} 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.


\medskip

\noindent{\bf Keywords:} graph, diameter, radius, central vertices,
number of central vertices, central ratio, center, spectrum of
center, typical graphs, almost all graphs.
\end{quote}
}

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