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     \hspace{80mm}\raisebox{1ex}{ISSN 1813-3304 }

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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]
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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 asymptotic of the number of central vertices]
{Logarithmic asymptotic\\of the number of central vertices\\of
almost all $n$-vertex graphs of diameter $k$}
\author{{T.I. FEDORYAEVA}}%
\address{Tatiana Ivanovna Fedoryaeva   % Контактные данные всех авторов и место работы указываются только на английском языке
\newline
\newline\hphantom{iii} Sobolev Institute of Mathematics,
\newline\hphantom{iii} pr. Koptyuga, 4,
\newline\hphantom{iii} 630090, Novosibirsk, Russia}%
\email{fti@math.nsc.ru}%

\thanks{\sc Fedoryaeva, T.I.,
Logarithmic asymptotic 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}
}


\section*{Introduction}
\hspace*{\parindent} We study finite labeled ordinary graphs. For a
connected graph $G$, the {\it distance} $\rho_G(u,v)$ between its
vertices $u,v\in V(G)$ is defined as the length of the shortest path
connecting these vertices. In this case, $e_G(v)=\max_{u\in V}
\rho_G(v,u)$ is the {\it eccentricity} of the vertex $v$ of the
graph $G$, $d(G)= \max_{v\in V} e_G(v)$ is the {\it diameter} of the
graph $G$, and $r(G)=\min_{v\in V} e_G(v)$ is the {\it radius} of
the graph $G$. A vertex is called {\it central} if its eccentricity
is equal to the radius of the graph. The {\it graph center}
${\mathbb C}(G)$ is the set of all central vertices of the graph
$G$.

The concept of the graph center is related to its numerous practical
problems arising in varied fields and it is stipulated by the
measurement of proximity centrality in the analysis of various kinds
of networks and connections. Often in graphs corresponding to
communication networks, the diameter is interpreted as the time of
data transfer or the path length between the most distant nodes, the
radius is as the reachability time from the most distant nodes to
the central node, serving as the main distribution center. In this
case, the presence of several such centers is allowed (see, for
example, \cite{rap05}). Finding of the graph center turns out to be
useful for problems of optimal placement of publicly important
institutions and enterprises (hospitals, fire stations, post offices
and other emergency points), when it is required to minimize the
farthest distances to these institutions \cite{buc90}. So, the
location of the hospital at the central vertices of the graph that
arises here reduces the maximum distance that ambulances have to
cover. The concept of centrality is also applied in social sciences
and is actively used in the analysis of social networks
\cite{wasser}, for example, when the most influential persons of
considered network are identified. In biology, it is relevant when
building models of the spread of diseases, in chemistry when
analyzing molecular bonds, etc.

A number of classical results about the graph center are well known
\cite{buc&mil,buc,buc90,emel90,xar69,kop77}. So, the realizability
of an arbitrary graph as a subgraph generated by the center of a
suitable graph is established. Namely, it is proved that for any
graph $H$ there exists a connected graph $G$ such that its subgraph
generated by the center ${\mathbb C}(G)$ is isomorphic to $H$. This
fact was established by G.N.~Kopylov and E.A.~Timofeev \cite{kop77},
its simple justification was also given by S.T.~Hedetniemi (see
\cite{buc&mil}). F.~Buckley investigated the so-called {\it central
ratio} ${\mathbb R}_{c}(G)=|{\mathbb C}(G)|/|V(G)|$ of connected
graph $G$, for which it is obvious that the inequality $0<{\mathbb
R}_{c}(G)\leq 1$ holds. For any rational $q$ in the interval
$(0,1]$, he proved the existence of a graph $G$ such that ${\mathbb
R}_{c}(G)=q$ \cite{buc}.

In \cite{fti21-2} the {\it center spectrum} ${\mathbb
Sp}_{c}({\mathcal K})$ of an arbitrary class of graphs ${\mathcal
K}$ is defined as the set of cardinalities of graphs centers from
this class ${\mathcal K}$. The center spectrum of $n$-vertex graphs
of radius $r$ is established by J.~Hu and S.~Zhan \cite{hu}. The
center spectrum of all and almost all $n$-vertex connected graphs is
found in \cite{fti21-2}. In this article \cite{fti21-2}, the author
obtained a number of structural results on the center and spectrum
of its cardinalities for almost all $n$-vertex graphs of fixed
diameter $k$. For $k=1,2$ any vertex is central, while for $k\geq 3$
two types of central vertices are identified, which are necessary
and sufficient to obtain the centers of almost all such graphs.
While studying the possible  center spectrum of almost all
$n$-vertex graphs of fixed diameter $k$, an unexpected result was
obtained in \cite{fti21-2}. It turned out that the center of almost
all such graphs has cardinality $n$ for $k=1,2$, and $n-2$ for
$k=3$, while for $k\geq 4$ in the distribution of values of center
cardinality for the probability space of $n$-vertex graphs of
diameter $k$ (for almost all graphs) there is no unique value of the
center cardinality (although the radius is uniquely defined
\cite{fti21-1}). Moreover, there are jumps of such cardinality
values depending on the diameter value. Namely, the center spectrum
is bounded by an interval of consecutive integers and additionally
contains at most one value (two values) outside this interval for an
even diameter $k$ (for an odd diameter $k$) depending on the value
of $k$ (for more details see Theorem 2 in preliminary information).
Note that the boundaries of the interval depend on a predetermined
arbitrary integer $p$ and shrink when choosing a greater value $p$.
This leads to the need to consider the case when the parameter $p$
depends on $n$.

Using this approach, in this paper we asymptotically study the
behavior of the number of central vertices and the central ratio for
almost all $n$-vertex graphs of fixed diameter $k$.

The necessary preliminary information is contained in Section 1.
There general properties of classes of typical graphs are also given
(Proposition \ref{l300nn}). In this section it also defined a family
of embedded classes ${\mathcal F}_{n,k,p}$, $p\geq 1$ ($p$ is an
integer constant independent of $n$) of typical $n$-vertex graphs of
fixed diameter $k \geq 3$, which have a number of metric properties
and were constructed by the author in \cite{fti21-1}.

Section 2 is of a technical nature, the purpose of which is to
obtain estimates for some expressions with classical binomial
coefficients (Lemma \ref{l4pn} and Corollary \ref{s1pn}). Wherein
properties of binomial coefficients $\binom{r}{\alpha}$ of real
arguments $r,\,\alpha\in {\mathbb R}$, previously proved by the
author in \cite{fti22-1}, are used. Namely, the analogue of the
unimodality property for classical binomial coefficients (Theorem 3)
and the asymptotics of such generalized binomial coefficients of a
special form (Proposition~2).

In Section 3 we consider a more general case of the class of
$n$-vertex graphs ${\mathcal F}_{n,k,p}$ of diameter $k\geq 3$, when
$p=p(n)$ is a function depending on $n$ and taking positive integer
values. For selected class of functions $p(n)$ it is established
asymptotically exact value of the number $|{\mathcal F}_{n,k,p(n)}|$
(Theorem \ref{t1sn}) and it is proved that ${\mathcal F}_{n,k,p(
n)}$ is the class of typical $n$-vertex graphs of diameter $k$
(Corollary \ref{s2sn}). In particular, this implies that for almost
all $n$-vertex graphs of diameter $k$, neighbourhoods of any two
vertices that do not belong to a fixed diametral path contain at
least $p(n)$ common vertices (Corollary \ref{s2222sn}).

In Section 4, for almost all $n$-vertex graphs of diameter $k$, we
find lower and upper estimates for the number of central vertices,
which depend on $n$ and refine the previously obtained interval
boundaries from the  center spectrum of such graphs (Theorem
\ref{t2sn}).

As a corollary, the asymptotics of the number $\log_2 |{\mathbb
C}(G)|$ is established for almost all $n$-vertex graphs $G$ of a
fixed diameter. It is proved that the logarithmic asymptotics of the
number of central vertices is $0$ or $\log_2 n$ ($1$ or $\log_2 n$)
for the corresponding subclasses of graphs of the even (odd)
diameter (see Corollary \ref{s100sn}).

Also, as a corollary, estimates of the central ratio ${\mathbb
R}_{c}(G)$ are found for almost all $n$-vertex graphs $G$ of
diameter $k$. It is proved that for almost all such graphs,
${\mathbb R}_{c}(G)=1$ 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
$r_i(n)=o(1)$ are positive functions, $ i=1,2$ (see Corollary
\ref{s280nn} for more details).

All obtained typical properties of the center of $n$-vertex graphs
of fixed diameter $k\geq 2$ remain typical for connected graphs of
diameter at least $k$, as well as for graphs (not necessarily
connected) containing a shortest path of length at least $k $ (see,
in particular, Corollary \ref{s80nn}).

\section{Preliminary information}
\hspace*{\parindent} The article uses the generally accepted
concepts and notations of graph theory \cite{emel90, xar69}, as well
as the standard concepts of combinatorial analysis \cite{grah94} and
the theory of functions of real variable \cite{fixt}. We consider
only finite ordinary (i.e.,\/ without loops and multiple edges)
graphs $G=(V, E)$ with set of vertices $V=\{1,2,\ldots,n\}$, $n\in
{\mathbb N}$. Denote by $B_i^G(v)=\{u\in V\,|\, \rho_G(v,u)\leq i
\}$ a {\it ball of radius $i$ centered at a vertex} $v\in V$ in the
metric space of the graph $G$ with the metric $\rho_G$,
$S_i^G(v)=\{u\in V\,|\, \rho_G(v,u)=i \}$ is a {\it sphere of radius
$i$ centered at a vertex} $v\in V$, $\lfloor x \rfloor$ is the
largest integer less or equal to real nonnegative number $x$,
$\bigl[\bigl[\, x,\,y \,\bigr]\bigr]$ is the integer interval
$[x,\,y]\cap {\mathbb Z}$ between the real numbers $x,y\in{\mathbb
R}$, $(n)_k=n(n-1)\cdots(n-k+1)$, and wherein we define $(n)_k=0$
for $n<k$ and $(n)_0=(0)_0=1$. A vertex of degree $1$ is called {\it
pendant}, sphere $S_1^G(v)$ of radius $1$ centered at $v$ is called
the {\it neighbourhood of the vertex $v$}, a shortest path of length
$d(G)$ is the {\it diametral path} of the graph $G$, and under by a
{\it pair of diametral vertices} we mean an unordered sample of two
vertices from the set $V$, the distance between which is equal to
the diameter.

To estimate the measure of the number of graphs with a certain
property, the concept of {\it almost all} is often used; in this
approach, the studied property is considered for graphs with a large
number of vertices. Let ${\mathcal J}_n$ be the class of labeled
$n$-vertex graphs with the fixed set of vertices
$V=\{1,2,\ldots,n\}$, $n\in {\mathbb N}$. Consider some property
${\mathcal P}$, by which each graph may or may not possess. Through
${\mathcal J}^{\mathcal P}_n$ denote the set of all graphs from
${\mathcal J}_n$ that possess the property ${\mathcal P}$. {\it
Almost all graphs possess the property} ${\mathcal P}$ if
$\lim_{n\rightarrow \infty}\frac{|{\mathcal J}^{\mathcal
P}_n|}{|{\mathcal J}_n|}=1$, \/i.e.\/ $|{\mathcal J}^{\mathcal P}_n|
\sim |{\mathcal J}_n|$, and {\it there are almost no graphs with the
property} ${\mathcal P}$, if $\lim_{n\rightarrow
\infty}\frac{|{\mathcal J}^{\mathcal P}_n|}{|{\mathcal J}_n|}=0.$ In
a similar way, we can talk about the property ${\mathcal P}$, which
is possessed by almost all $n$-vertex graphs of the class ${\mathcal
K}$ under study, or when there are almost no graphs in the class
${\mathcal K}$ that have property ${\mathcal P}$.

In the study and selection of almost all graphs in the class of
graphs under consideration it is often useful to define not
characteristic properties themselves for the notion of almost all,
but directly select a subclass of typical graphs itself (in
\cite{fti15, fti16} a more general concept of a class of typical
combinatorial objects and an abstract typical combinatorial object
for a given class of objects admitting  the concept of dimension is
formulated). Further we will also use this formal concept for graphs
(when the dimension of a graph is understood as the number of its
vertices). Let $\Omega $ be an arbitrary class of graphs such that
$\Omega_n\neq\varnothing$ for all large enough $n$, where
$\Omega_n=\Omega\,\cap\, {\mathcal J}_n$. A subclass
$\Omega^{\ast}\subseteq \Omega$ is the {\it class of typical graphs
of the class} $\Omega$ if $\lim_{n\rightarrow
\infty}\,\frac{|\Omega_n^{\ast}|}{|\Omega_n|}=1.$ A property of
graphs of the class under consideration is \/{\it typical}\/ if
almost all graphs of this class have this property. Let us formulate
simple properties of classes of typical graphs related to
set-theoretic operations.
\begin{proposition}[properties of typical graphs]\label{l300nn}
Let ${\mathcal X}$, ${\mathcal Y}$ be  subclasses of a graph class
${\mathcal K}$. Then

{\rm (i)} if ${\mathcal X}$ is the class of typical graphs of class
${\mathcal K}$ and ${\mathcal X}\subseteq{\mathcal Y}$, then
${\mathcal Y}$ is also the class of typical graphs of class
${\mathcal K};$

{\rm (ii)} if ${\mathcal X}$ is the class of typical graphs of class
${\mathcal K}$ and ${\mathcal Y}\subseteq {\mathcal X}$, in
addition, $|{\mathcal Y_n}|=o(|{\mathcal K_n}|)$ as $n\rightarrow
\infty$, then ${\mathcal X}\setminus{\mathcal Y}$ is also the class
of typical graphs of class ${\mathcal K};$

{\rm (iii)} if ${\mathcal X}$ is the class of typical graphs of
class ${\mathcal K}$ and ${\mathcal X}\subseteq {\mathcal Y}$, then
${\mathcal X}$ is the class of typical graphs of class ${\mathcal
Y};$

{\rm (iv)} if ${\mathcal X}$, ${\mathcal Y}$ are the classes of
typical graphs of class ${\mathcal K}$, then ${\mathcal X}\cap
{\mathcal Y}$ is also the class of typical graphs of class
${\mathcal K}$, moreover, subclasses ${\mathcal X}\setminus
{\mathcal Y}$ and ${\mathcal Y}\setminus {\mathcal X}$ have
asymptotically zero fraction in
 ${\mathcal K}$, i.e. $|{\mathcal X_n}\setminus
{\mathcal Y_n}|=o(|{\mathcal K_n}|)$ and $|{\mathcal Y_n}\setminus
{\mathcal X_n}|=o(|{\mathcal K_n}|)$ as $n\rightarrow \infty\,;$

{\rm (v)} if ${\mathcal X}$, ${\mathcal X'}$ are disjoint classes of
typical graphs of classes ${\mathcal K}$ and ${\mathcal K'}$
respectively, then ${\mathcal X}\cup{\mathcal X'}$ is the class of
typical graphs of class ${\mathcal K}\cup{\mathcal K'};$

{\rm (vi)} if ${\mathcal X}\cup {\mathcal Y}$ is the class of
typical graphs of class ${\mathcal K}$ and ${\mathcal X}\cap
{\mathcal Y}=\varnothing$, in addition, subclass ${\mathcal X}$ has
an asymptotic fraction in ${\mathcal K}$ not equal to {\rm 1,} then
${\mathcal Y}$ is the class of typical graphs of class ${\mathcal
K}\setminus {\mathcal X}.$
\end{proposition}
\begin{proof}
Statements (i)-(iii) follow directly from the definitions, statement
(iv) is obtained from property (ii) and equality ${\mathcal X}\cap
{\mathcal Y}={\mathcal X}\setminus ({\mathcal X}\setminus{\mathcal
Y})$.

Prove assertion (v). Since $\frac{|{\mathcal K_n}|}{|{\mathcal
K_n}|+|{\mathcal K'_n}|}$ is a bounded sequence, as $n\rightarrow
\infty$ we obtain
$$\frac{|{\mathcal K_n}|}{|{\mathcal
K_n}|+|{\mathcal K'_n}|} \Bigl(  \frac{|{\mathcal X_n}|}{|{\mathcal
K_n}|} \,-\, \frac{|{\mathcal X'_n}|}{|{\mathcal
K'_n}|}\Bigr)=o(1).$$ It remains to note that $|{\mathcal X_n}\cup
{\mathcal X'_n}| = |{\mathcal X_n}|+|{\mathcal X'_n}|$ and as
$n\rightarrow \infty$
$$\frac{|{\mathcal X_n}\cup {\mathcal X'_n}|}{|{\mathcal K_n}\cup
{\mathcal K'_n}|}\geq\frac{|{\mathcal X_n}|+ |{\mathcal
X'_n}|}{|{\mathcal K_n}|+|{\mathcal K'_n}|}=\frac{|{\mathcal
K_n}|}{|{\mathcal K_n}|+|{\mathcal K'_n}|} \Bigl(  \frac{|{\mathcal
X_n}|}{|{\mathcal K_n}|} \,-\, \frac{|{\mathcal X'_n}|}{|{\mathcal
K'_n}|}\Bigr) + \frac{|{\mathcal X'_n}|}{|{\mathcal
K'_n}|}\,\longrightarrow 1.$$

Prove statement (vi). Since ${\mathcal X}\cup {\mathcal Y}$ is the
class of typical graphs of class ${\mathcal K}$, we have $|{\mathcal
K_n}\setminus({\mathcal X_n}\cup {\mathcal Y_n})|=o(|{\mathcal
K_n}|).$ By virtue of the condition on the fraction of the subclass
${\mathcal X}$, there is the following limit
$$\lim_{n\rightarrow \infty} (1-\frac{|{\mathcal X_n}|}{|{\mathcal
K_n}|})\neq 0.$$ \noindent Besides, ${\mathcal Y_n}={\mathcal
K_n}\setminus ({\mathcal X_n}\cup {\mathcal K_n}\setminus ({\mathcal
X_n}\cup {\mathcal Y_n}))$ and $|{\mathcal K_n}\setminus {\mathcal
X_n}| = |{\mathcal K_n}|- |{\mathcal X_n}|.$ Thus, as $n\rightarrow
\infty$ we conclude
$$\frac{|{\mathcal Y_n}|}{|{\mathcal K_n}\setminus {\mathcal
X_n}|}=\Bigl(1-\frac{|{\mathcal X_n}|}{|{\mathcal K_n}|}-
\frac{|{\mathcal K_n}\setminus({\mathcal X_n}\cup {\mathcal
Y_n})|}{|{\mathcal K_n}|}\Bigr)\Bigl(1-\frac{|{\mathcal
X_n}|}{|{\mathcal K_n}|}\Bigr)^{-1}\,\longrightarrow 1.
$$
\end{proof}
Note that the condition on the fraction of the subclass ${\mathcal
X}$ in assertion (vi) of Proposition \ref{l300nn} is essential. For
example, let ${\mathcal K_n}={\mathcal X_n}\cup \{H_n,G_n\}$,
$\lim_{n\rightarrow\infty} |{\mathcal X_n}|=\infty$ and ${ \mathcal
Y_n}=\{G_n\}$, where $H_n,G_n\in{\mathcal J}_{n}\setminus {\mathcal
X_n}$ are different graphs. Then $|{\mathcal K_n}|\sim |{\mathcal
X_n}\cup {\mathcal Y_n}|$, i.e. ${\mathcal X}\cup {\mathcal Y}$ is
the class of typical graphs of class ${\mathcal K}$. Meanwhile, it
is obvious that
$$\lim_{n\rightarrow \infty} \frac{|{\mathcal Y_n}|}{|{\mathcal K_n}
\setminus {\mathcal X_n}|}=\frac{1}{2}.$$ Therefore ${\mathcal Y}$
is not the class of typical graphs of class ${\mathcal K}\setminus
{\mathcal X}.$

Let ${\mathcal J}_{n,\,d=k}$, ${\mathcal J}_{n,\,d\geq k}$,
${\mathcal J}^{\ast}_{n,\,d\geq k}$ be the following classes of
labeled $n$-vertex graphs: graphs of diameter $k$; connected graphs
of diameter at least $k$ and graphs (not necessarily connected)
having a shortest path of length at least $k$, respectively.
Obviously, the following inclusions hold ${\mathcal
J}_{n,\,d=k}\subseteq {\mathcal J}_{n,\,d\geq k} \subseteq {\mathcal
J}^{\ast}_ {n,\,d\geq k}.$ In \cite{fti21-1} for every $k\geq 3$ a
family of nested classes ${\mathcal F}_{n,k,p}$,\, $p\geq 1$ of
typical $n$-vertex graphs of fixed diameter $k$ is constructed,
which are also typical for classes ${\mathcal J}_{n,\,d\geq k}$,
${\mathcal J}_{n,\,d\geq k}^{\ast}$. Give the definition of the
class of graphs ${\mathcal F}_{n,k,p}$. To do this, we first
consider special graphs of diameter $3$ and their properties. Let
$x,y\in V$ and ${\mathcal F}_{n,3,p}(x,y)$ be the class of all
graphs $F\in {\mathcal J}_{n}$, having the following properties:

a) vertices $x,y$ are not pendant in $F$;

b) $\rho_F(z,x)=\rho_F(z,y)=2$ for some vertex $z\in V$;

c) $d(F)=3$, the graph $F$ has the unique pair of diametral vertices
$x,y$ and does not contain coinciding balls of radius 1 with centers
in different vertices;

d) the following properties of spheres hold:
$$|S_1^F(u)\cap S_1^F(v)|\geq p \;\; \forall u,v\in V\setminus
\{x,y\} \mbox{ and } u\neq v,$$
$$|S_1^F(u)\cap S_1^F(v)|\geq p \;\; \forall v\in V\setminus
\{x,y\}\;\; \forall u\in \{x,y\}.$$

Now, we define graphs of the class ${\mathcal F}_{n,k,p}$ as
follows. Let $u=(u_0,u_1,\ldots,u_{k-2})$ be an arbitrary ordered
sequence of different vertices from the set $V$. Fix a pair of
neighboring elements $u_s$ and $u_{s+1}$, $0\leq s \leq k-3$. On the
set $V\setminus \{u_0,\ldots,u_{s-1},u_{s+2},\ldots,u_{k-2}\}$ of
$n-k+3$ vertices, define an arbitrary graph $F$ from the class
${\mathcal F}_{n-k+3,3,p}(u_s,u_{s+1})$. Finally, join  by edges the
vertices $u_i,u_{i+1}$ for $i\neq s$ and $0\leq i<k-2$. Denote the
resulting graph by $G(u,s,F)$. Let ${\mathcal F}_{n,k,p}$ be the
class of all constructed graphs $G(u,s,F)$ under the condition
$0\leq s\leq \lfloor \frac{k-3} {2}\rfloor$. Further, we need the
following estimates of the number of graphs in this class.

\renewcommand{\thelemma}{\arabic{lemma} {\rm\cite{fti21-1}}}
\begin{lemma}\label{l5a}
Let $k\geq 3$, $p\geq 1$ and $x,y\in V$ be different vertices. Then
$$|{\mathcal F}_{n,k,p}|=\frac{1}{2} (k-2)(n)_{k-1}\, |{\mathcal
F}_{n-k+3,3,p}(x,y)|.$$
\end{lemma}
\renewcommand{\thelemma}{\arabic{lemma}}
\noindent The following theorem contains estimates that give the
asymptotically exact value $2^{\binom{n}{2}}\,\xi_{n,k}$ of the
number of graphs in each of classes ${\mathcal F}_{n,k,p}$,
${\mathcal J}_{n,\,d=k}$, ${\mathcal J}_{n,\,d\geq k}$ and
${\mathcal J}_{n,\,d\geq k} ^{\ast}$ for any fixed $k\geq 3$ and
$p\geq 1$ \cite{fti21-1}.

\renewcommand{\thetheorem}{\arabic{theorem} {\rm\cite{fti21-1}}}
\begin{theorem}\label{t1b}
Let $k\geq 3$, $0<\varepsilon<1$ and $p\geq 1$ be independent of
$n$. Then there exists a constant $c>0$ independent of $n$ such that
for every $n\in {\mathbb N}$ the following inequalities hold

\begin{flushleft}
$ 2^{\binom{n}{2}}\xi_{n,k}\Bigr(1-c\Bigr(\frac
{5+\varepsilon}{6}\Bigl)^{n-k+1}\Bigl)\leq |{\mathcal
F}_{n,k,p}|\leq |{\mathcal J}_{n,\,d=k}|$
\end{flushleft}

\begin{flushright}
$\leq |{\mathcal J}_{n,\,d\geq k}|\leq |{\mathcal J}_{n,\,d\geq
k}^{\ast}|\leq 2^{\binom{n}{2}}\xi_{n,k}\Bigr(1+c\Bigr(\frac
{5+\varepsilon}{6}\Bigl)^{n-k+1}\Bigl),$
\end{flushright}

$$\mbox{where }\xi_{n,k}=q_k\,\,(n)_{k-1}\,
\Bigr(\frac{3}{2^{k-1}}\Bigl)^{n-k+1},\,\,\,\,
q_k=\frac{1}{2}\,(k-2)\,2^{-\binom{k-1}{2}}.$$
\end{theorem}
\renewcommand{\thetheorem}{\arabic{theorem}}
In \cite {fti21-2} the  center spectrum ${\mathbb Sp}_{c}({\mathcal
J}_{n,\,d=k})$ is investigated asymptotically, the following
statement is proved.

\renewcommand{\thetheorem}{\arabic{theorem} {\rm\cite{fti21-2}}}
\begin{theorem}\label{t1k}
Let $k\geq 1$ and $p\geq 1$ be fixed integer constants. Then \\[-10pt]

{\rm (i)} \,\,\,$|{\mathbb C}(G)|=n$ for almost all $n$-vertex
graphs $G$ of diameter $k=1,2;$\\[-10pt]

{\rm (ii)} \,\,$|{\mathbb C}(G)|=n-2$ for almost all\,
$n$-vertex graphs $G$ of diameter $3;$\\[-10pt]

{\rm (iii)} $|{\mathbb C}(G)|\in \bigl[\bigl[\,1+p,\, n-5-p
\,\bigr]\bigr]$ for almost all\, $n$-vertex graphs $G$
of diameter $4;$\\[-10pt]

{\rm (iv)} $|{\mathbb C}(G)|\in \bigl[\bigl[\,2+p,\, n-5-p\,
\bigr]\bigr]\cup\{n-4\}$ for almost all\, $n$-vertex graphs $G$ of
diameter $5${\rm;} moreover, the fraction of such graphs with
$(n-4)$-vertex center is
asymptotically equal to $\frac{1}{3};$\\[-10pt]

{\rm (v)} \,$|{\mathbb C}(G)|\in \{1\} \cup \bigl[\bigl[\,1+p,\,
n-k-1-p\, \bigr]\bigr]$ for almost all\, $n$-vertex graphs $G$ of
even fixed diameter $k\geq 6${\rm;} moreover, the fraction of such
graphs with trivial center is
asymptotically equal to $\frac{k-4}{k-2};$\\[-10pt]

{\rm (vi)} $|{\mathbb C}(G)|\in \{2\}\cup \bigl[\bigl[\,2+p,\,
n-k-p\, \bigr]\bigr]\cup\{n-k+1\}$ for almost all\, $n$-vertex
graphs $G$ of odd fixed diameter $k\geq 7${\rm;} moreover, the
fraction of such graphs with $2$-vertex and $(n-k+1)$-vertex center
is asymptotically equal to $\frac{k-5}{k-2}$ and $\frac{1}{k-2}$
respectively.
\end{theorem}
\renewcommand{\thetheorem}{\arabic{theorem}}

Introduce some asymptotic notations. For numerical functions $f(n)$,
$g(n):{\mathbb N}\rightarrow {\mathbb R}$ we write $f(n)
\,{\scriptstyle\lesssim}\,g(n) $ (respectively $f(n)
\,{\scriptstyle\gtrsim}\, g(n)$) if there exists $N\in {\mathbb N}$
such that for every $n\geq N$ the inequality $f(n)\leq g(n)$
($f(n)\geq g(n)$) holds. To denote the {\it asymptotic equality} of
the numerical functions $f(n)$ and $g(n)$ as $n\rightarrow\infty$,
we use the notation $f(n)\sim g(n)$, which by definition means that
$f(n) = g(n)(1+r(n))$ for all large enough $n$, where $r(n) = o(1)$,
or, equivalently, $\lim_{n\rightarrow\infty}\frac{f(n)}{g(n)}=1$
(for functions positive in some neighborhood of $\infty$). Let
$\Omega$ be the class of graphs under consideration and an numerical
characteristic ${\mathcal X} : \Omega \rightarrow {\mathbb R}$
assigns to each graph $G\in \Omega$ some real number ${\mathcal X}
(G)\in {\mathbb R}$. For almost all $n$-vertex graphs $G$ of class
$\Omega$, the numerical characteristic ${\mathcal X}(G)$ {\it is
asymptotically equal to} the numerical function $f(n)$ (symbolically
${\mathcal X }(G)\sim f(n)$), if for some functions
$f_1,\,f_2:{\mathbb N}\rightarrow {\mathbb R}$ almost all $n$-vertex
graphs $G$ of class $\Omega$ satisfy the relation $f_1(n)\leq
{\mathcal X}(G) \leq f_2(n)$, wherein the numerical functions
$f_1(n)$, $f_2(n)$, $f(n)$ asymptotically coincide as $n\rightarrow
\infty $.

\section{Estimates of some expressions\\with binomial coefficients}
\hspace*{\parindent} In this section, we obtain estimates for some
expressions with usual binomial coefficients. Wherein, we need the
properties of binomial coefficients
\scalebox{1.1}{$\binom{r}{\alpha}$} of real arguments $r,\,\alpha\in
{\mathbb R}$, proved by the author in \cite{fti22-1}. Recall that
the binomial coefficient of real arguments (as a generalization of
the classical binomial coefficient $\binom{n}{m}$ for non-negative
integers $n$ and $m$) is defined as follows (see, for example,
\cite{fowler}):

\begin{eqnarray}\label{303fpn}
\binom{r}{\alpha}=\frac{\Gamma(1+r)}{\Gamma(1+\alpha)\Gamma(1+r-\alpha)}\,,
\end{eqnarray}
here $\Gamma(\alpha)$ is the Euler gamma function, which can be
defined in the {\it Euler-Gauss form} (see, for example,
\cite{fixt}) as

\begin{equation*}\label{300fpn}
\Gamma(\alpha)=\lim_{n\rightarrow\infty} \frac{(n-1)!\,n^
{\alpha}}{\alpha(\alpha +1)(\alpha +2)\cdots(\alpha
+n-1)},\,\alpha\in{\mathbb R}\setminus \{0,-1,-2,\ldots\}.
\end{equation*}
\noindent Note that if $r\in(-1,+\infty)$ and $\alpha\in (-1,r+1)$,
the binomial coefficient $\binom{r}{\alpha}$ is defined by
(\ref{303fpn}) correctly.

In \cite{fti22-1} a number of properties of real binomial
coefficients is established, in particular, the following statements
are proved.
\renewcommand{\thetheorem}{\arabic{theorem} {\rm\cite{fti22-1}}}
\begin{theorem}\label{304lpn}
Let $r\in(-1,+\infty)$ and $\alpha\in (-1,r+1)$. Then
$\scalebox{0.9}{$\binom{r}{\alpha}$}>0$ and binomial coefficient
$\phi(\alpha)\!=\binom{r}{\alpha}$ is strictly increasing on the
interval \,$(-1,\frac{r}{2}]$\, and strictly decreasing on the
interval \, $[\frac{r}{2},r+1)$.
\end{theorem}
\renewcommand{\thetheorem}{\arabic{theorem}}
\renewcommand{\theproposition}{\arabic{proposition} {\rm\cite{fti22-1}}}
\begin{proposition}\label{309lpn}
Let $r$ takes real values, $\alpha\in {\mathbb R}$ does not depend
on $r$ and $0<\alpha <1$. Then the following asymptotic equality is
valid as r tends to infinity
\begin{equation}\label{310lpn}
\binom{r}{r\alpha}\sim \sqrt{\frac{1}{2\pi \alpha
(1-\alpha)r}}\,\,\Bigl(\frac{1}{\alpha}\Bigr)^{\alpha
r}\,\Bigl(\frac{1}{1-\alpha}\Bigr)^{(1-\alpha) r}.
\end{equation}
\end{proposition}
\renewcommand{\theproposition}{\arabic{proposition}}
\renewcommand{\thecorollary}{\arabic{corollary} {\rm\cite{fti22-1}}}
\begin{corollary}\label{312spn}
Let $r$ takes non-negative integer values, $\alpha\in {\mathbb R}$
does not depend on $r$ and $0<\alpha <1$. Then the asymptotic
equality {\rm(\ref{310lpn})} is valid as r tends to infinity.
\end{corollary}
\renewcommand{\thecorollary}{\arabic{corollary}}

Recall that the sequence of numbers $a_1,a_2,\ldots,a_n$ is called
{\it unimodal} if there exists $s$ such that $a_1<a_2<\ldots<a_s\geq
a_{s+1}>a_{s+2}> \ldots>a_n$ (see, for example, \cite{yablon}).
\begin{lemma}\label{l3pn}
Let $n\geq q\geq 1$. Then the sequence $\binom{n}{s} q^{-s}$,
$s=0,1,\ldots, n$ is unimodal, moreover, for
$s=\lfloor\frac{n+1}{q+1}\rfloor$ its largest value is reached and
such $s$ is the largest.
\end{lemma}
\begin{proof}
Introduce sequences $\alpha_s= \binom{n}{s} q^{-s}$, $0\leq s\leq
n$, and $\beta_s= \frac{\alpha_{s+1}}{\alpha_s}$ if $0\leq s\leq
n-1$. It is directly established that
$$\beta_s= \frac{n-s}{(s+1)q}\,,$$
$$\frac{\beta_{s+1}}{\beta_s} = \frac{n-(s+1)}{n-s} \cdot
\frac{s+1}{s+2}\,< 1 \mbox{ if }0\leq s\leq n-2,$$ i.e. $\beta_s$ is
a strictly decreasing sequence. Moreover, $\beta_0\geq 1$ due to the
condition $n\geq q$. Therefore, for some $s^{\ast}$ the sequence
$\alpha_s$ strictly increases for $0\leq s \leq s^{\ast}$,
$\,\alpha_{s^{\ast}}\leq \alpha_{s^{\ast}+1}$ and then strictly
decreases for $s^{\ast}+1 \leq s\leq n$. Note that $s^{\ast}$ is the
largest $s$ satisfying the inequality $\beta_s\geq 1$. Hence
$s^{\ast}=\lfloor\frac{n-q}{q+1}\rfloor$ and
$s^{\ast}+1=\lfloor\frac{n+1}{q+1}\rfloor$.
\end{proof}
\begin{lemma}\label{l4pn}
Let $p(n)=\lfloor\frac{n}{q+1}\Delta\rfloor+1$, where $q\geq 1$ and
$0<\Delta<1$ do not depend on $n$. Then there is
$\varepsilon_{\Delta}$ such that $\varepsilon_{\Delta}$ does not
depend on $n$, $0<\varepsilon_{\Delta}<1$ and the following relation
is fulfilled as $n\rightarrow \infty$
$$\Bigl(\frac{q}{q+1}\Bigr)^n\,\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}q^{-s}
=\varepsilon_{\Delta}^n\,O(\sqrt{n}).$$
\end{lemma}
\begin{proof}
By Proposition 2
%\ref{309lpn}
we obtain
\begin{equation}\label{f51pn}
\binom{n}{\frac{n}{q+1}\Delta}\sim \sqrt{\frac{(q+1)^2}{2\pi \Delta
(q+1-\Delta)n}}\,(q+1)^n {\Delta}^{-\frac{\Delta}{q+1} n}\,
(q+1-\Delta)^{-\frac{q+1-\Delta}{q+1}n}.
\end{equation}
Therefore
\begin{equation*}\label{f52pn}
\binom{n}{\frac{n}{q+1}\Delta}= (q+1)^n
{\Delta}^{-\frac{\Delta}{q+1} n}\,
(q+1-\Delta)^{-\frac{q+1-\Delta}{q+1}n}\,O\Bigl(\frac{1}{\sqrt{n}}\Bigr).
\end{equation*}
Hence,
\begin{eqnarray}\label{f53pn}
n\,\Bigl(\frac{q}{q+1}\Bigr)^n\binom{n}{\frac{n}{q+1}\Delta}\,
q^{-\frac{n}{q+1}\Delta} =\varepsilon_{\Delta}^n\,O(\sqrt{n}),
\mbox{ where}
\\\label{f54pn} \varepsilon_{\Delta}=q\,
{\Delta}^{-\frac{\Delta}{q+1}}\,
(q+1-\Delta)^{-\frac{q+1-\Delta}{q+1}}\,\,q^{-\frac{\Delta}{q+1}}.
\end{eqnarray}
Since $0<\Delta<1$ and $q\geq 1$, we have $\varepsilon_{\Delta}>0$.
Consider the functions $$f(x)=x^{\frac{x}{q+1}}\,
(q+1-x)^{\frac{q+1-x}{q+1}}\,q^{\frac{x}{q+1}} \mbox{ and } g(x)=
\ln f(x)$$ on the interval $(0,q+1).$ Let us study intervals of
monotonicity of the function $f(x)$. It is directly calculated
$$g'(x)=\frac{1}{q+1} \ln \frac{qx}{q+1-x}\,.$$
Note that $g'(x)<0$ on $(0,1)$ and $g'(1)=0$. Therefore, the
function $g(x)$ strictly decreases on $(0,1]$. The function $f(x)$
behaves similarly on the interval $(0,1]$. Therefore, the relation
$f(\Delta)>f(1)=q$ is valid. Now, using (\ref{f54pn}), we obtain
$\varepsilon_{\Delta}=\frac{q}{f(\Delta)}<1.$

Further, due to the inequality $p(n)-1 \leq
\frac{n}{q+1}\Delta\leq\frac{n}{2}$ by Theorem 3
%\ref{304lpn}
we have
\begin{equation}\label{f55pn}
\binom{n}{p(n)-1}\leq \binom{n}{\frac{n}{q+1}\Delta}.
\end{equation}

Now, using the relations $p(n)-1\leq\lfloor\frac{n+1}{q+1}\rfloor$,
$\lfloor x\rfloor\geq x-1$, (\ref{f53pn}), (\ref{f55pn}) and Lemma
\ref{l3pn}, as $n\rightarrow \infty$ we conclude
\begin{eqnarray*}
\Bigl(\frac{q}{q+1}\Bigr)^n\,\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}q^{-s}&
\leq &
p(n)\Bigl(\frac{q}{q+1}\Bigr)^n\,\binom{n}{p(n)-1}q^{-(p(n)-1)}\\
&=&
\Bigl(\frac{q}{q+1}\Bigr)^n\,\binom{n}{\frac{n}{q+1}\Delta}q^{-\frac{n}{q+1}\Delta}\,O(n)\\
&=& \varepsilon_{\Delta}^n\,O(\sqrt{n}).
\end{eqnarray*}
\end{proof}
\begin{remark}
The statement of Lemma {\rm\ref{l4pn}} is not valid for $\Delta=1$.
\end{remark}
Indeed, let $\Delta=1$. First consider the case $q > 1$. Using
Theorem 3 and the inequality $\frac{n}{q+1}\leq p(n)\lesssim
\frac{n}{2}$, we obtain
\begin{eqnarray}\label{f5500pn}
\binom{n}{p(n)-1} &=&\nonumber
\binom{n}{p(n)}\,\frac{p(n)}{n-p(n)+1}\\
&=& \binom{n}{\frac{n}{q+1}}\,\frac{\lfloor\frac{n}{q+1}\rfloor
+1}{n-\lfloor\frac{n}{q+1}\rfloor}\,\,\Omega(1)\\
&=&\nonumber \binom{n}{\frac{n}{q+1}}\,\frac{\Omega(n)}{nq+q+1}\,.
\end{eqnarray}
From the asymptotic equality (\ref{f51pn}) (which is also valid for
$\Delta=1$) we have
\begin{equation}\label{f51111pn}
\binom{n}{\frac{n}{q+1}}= \frac{(q+1)^n }{\sqrt{n}}\,\,
q^{-\frac{q}{q+1}n}\,\Omega(1).
\end{equation}
Now, from (\ref{f5500pn}),(\ref{f51111pn}) it is easy to see that
\begin{eqnarray*}
\Bigl(\frac{q}{q+1}\Bigr)^n\,\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}q^{-s}
&=&
\Bigl(\frac{q}{q+1}\Bigr)^n\,\binom{n}{p(n)-1}q^{-(p(n)-1)}\,\Omega(1)\\
&=& \Bigl(\frac{q}{q+1}\Bigr)^n\,\frac{(q+1)^n }{\sqrt{n}}\,\,
q^{-\frac{q}{q+1}n}\,q^{-\frac{n}{q+1}}\frac{\Omega(n)}{nq+q+1}\\
&=& \frac{\Omega(\sqrt{n})}{nq+q+1}.
\end{eqnarray*}
Therefore, if the assertion of Lemma \ref{l4pn} holds, then
$(nq+q+1)\,\varepsilon_{\Delta}^n = \Omega(1)$ and we get a
contradiction.

Now consider the case $q=1$. It is known that as $n\rightarrow
\infty$
$$\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\sim \frac{2^n}{\sqrt{\pi
n/2}}$$ (see, for example, \cite{yablon}). Therefore
$$2^{-n}\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}=
2^{-n}\binom{n}{\lfloor\frac{n}{2}\rfloor}\,\Omega(1)=\frac{\Omega(1)}{\sqrt{n}}$$
and similarly, as in the case of $q > 1$, we obtain a contradiction.
\begin{corollary}\label{s1pn}
Let $p(n)=\lfloor\frac{n}{q+1}\Delta\rfloor+1$, where is an integer
$q\geq 1$ and $0<\Delta<1$ do not depend on $n$. Then there is
$\varepsilon_{\Delta}$ such that $0<\varepsilon_{\Delta}<1$,
$\varepsilon_{\Delta}$ does not depend on $n$ and as $n\rightarrow
\infty$
$$
\sum\limits_{i=1}^{q}\Bigl(\frac{i}{i+1}\Bigr)^n\,
\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}i^{-s}=\Bigl(\frac{q+
\varepsilon_{\Delta}}{q+1}\Bigr)^n\,O(\sqrt{n}).$$
\end{corollary}
\begin{proof}
Since $p(n)\leq\lfloor\frac{n}{i+1}\Delta\rfloor+1$ for every $i\leq
q$, by Lemma \ref{l4pn} there is $\varepsilon$ such that
$0<\varepsilon<1$, $\varepsilon$ does not depend on $n$ and as
$n\rightarrow \infty$
$$\sum\limits_{i=1}^{q}\Bigl(\frac{i}{i+1}\Bigr)^n\,
\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}i^{-s}
%\,{\scriptstyle\lesssim}\,
\leq \sum\limits_{i=1}^q\Bigl(\frac{i}{i+1}\Bigr)^n\,
\sum\limits_{s=0}^{\lfloor\frac{n}{i+1}\Delta\rfloor}\binom{n}{s}i^{-s}=\varepsilon^n\,O(\sqrt{n}).$$
We can assume that $\frac{q}{q+1}<\varepsilon <1$. Remains to note
that $\varepsilon_{\Delta}=\varepsilon (q+1)-q$ will be required.
\end{proof}

\section{Class of graphs ${\mathcal F}_{n,k,p(n)}$}
\hspace*{\parindent} This section deals with the more general case
of the graphs class ${\mathcal F}_{n,k,p}$ (constructed for every
$k\geq 3$ and $p\geq 1$ in \cite{fti21-1}) when $p=p (n)$ is a
function depending on $n$ and taking positive integer values. Let us
estimate the number of graphs of the class ${\mathcal F}_{n,k,p(n)}$
for a class of functions $p(n)$ distinguished in this section.
Further, we will use the estimates obtained in \cite{fti15,fti21-1}
for the number of graphs of the following subclasses of labeled
$n$-vertex graphs. Let $x,y,u,v$ be different elements of $V$,
$p\geq 1$, $0\leq s<p$\, and
$$a_n=|{\mathcal A}_n(x,y)|,\mbox{ where } {\mathcal A}_n(x,y)=\{G\in
{\mathcal J}_n \,|\, B_1^G(x)\cap B_1^G(y)=\varnothing\},
$$
$${\mathcal B}_{n}(x,y,u,v;s)=\{G\in {\mathcal J}_n \,|\, B_1^G(x)\cap
B_1^G(y)=\varnothing \mbox{ and } |S_1^G(u)\cap S_1^G(v)|=s\},$$
$${\mathcal C}_{n}(x,y,u;s)=\{G\in {\mathcal J}_n \,|\,
B_1^G(x)\cap B_1^G(y)=\varnothing \mbox{ and } |S_1^G(x)\cap
S_1^G(u)|=s\},$$
$$\beta_{n,p}=|{\mathcal
B}_{n,p}(x,y,u,v)|,\mbox{ where } {\mathcal
B}_{n,p}(x,y,u,v)=\bigcup\limits_{0\leq s<p} {\mathcal
B}_{n}(x,y,u,v;s),$$
$$\gamma_{n,p}=|{\mathcal C}_{n,p}(x,y,u)|, \mbox{ where } {\mathcal C}_{n,p}(x,y,u)=
\bigcup\limits_{0\leq s<p} {\mathcal C}_{n}(x,y,u;s).$$
\renewcommand{\thelemma}{\arabic{lemma} {\rm\cite{fti15,fti21-1}}}
\begin{lemma}\label{l1a}
Let $x,y,u,v$ be different elements in $V$ and $p\geq 1$. Then
for every $n\geq p$ the following relations hold\\[1pt]

{\rm (i)} $a_n=2^{\binom{n}{2}}\,\frac{8}{9}
\,\Bigl(\frac{3}{4}\Bigr)^n;$\\[1pt]

{\rm (ii)} $\beta_{n,p}\leq a_n b_p(n)\Bigr(\frac{3}{4}\Bigl)^{n},$
where $b_p(n)=128\sum_{s=0}^{p-1}\binom{n}{s}\; 3^{-(3+s)};$\\[1pt]

{\rm (iii)} $\gamma_{n,p}\leq a_n
c_p(n)\Bigr(\frac{5}{6}\Bigl)^{n},$ where
$c_p(n)=72\sum_{s=0}^{p-1}\binom{n}{s}\; 5^{-(1+s)}.$
\end{lemma}
\renewcommand{\thelemma}{\arabic{lemma}}
\renewcommand{\thelemma}{\arabic{lemma} {\rm\cite{fti21-1}}}
\begin{lemma}\label{l3a}
Let $x,y$ be different vertices in $V$ and $p\geq 1$, $\lambda >0$,
$0<\varepsilon<1$ be arbitrary constants independent of $n$. Then
$$|{\mathcal F}_{n,3,p}(x,y)| \,{\scriptstyle\gtrsim}\, a_n \Bigr(1-
\lambda\,\Bigr( \frac{5+\varepsilon}{6}\Bigl)^{n-2}\Bigl).$$
\end{lemma}
\renewcommand{\thelemma}{\arabic{lemma}}

The next lemma extends Lemma 5
%\ref{l3a}
on a lower estimate of the number of graphs of class ${\mathcal
F}_{n,3,p}(x,y)$, where $p$ is a fixed integer, to the case $p=p(
n)$ for the considered class of functions.
\begin{lemma}\label{l5pn}
Let $x,y$ be different vertices in $V$,
$p(n)=\lfloor\frac{n}{6}\Delta\rfloor+1$ and $0<\Delta<1$, $\lambda
>0$ are arbitrary constants independent of $n$. Then there exists
$\varepsilon_{\Delta}$ such that $0<\varepsilon_{\Delta}<1$,
$\varepsilon_{\Delta}$ does not depend on $n$, and for every
$\varepsilon$ satisfying the inequality
$\varepsilon_{\Delta}<\varepsilon<1$, the following relation is
fulfilled
\begin{equation*}
|{\mathcal F}_{n,3,p(n)}(x,y)|\,{\scriptstyle\gtrsim}\, a_n \Bigr(1-
\lambda\,\Bigr( \frac{5+\varepsilon}{6}\Bigl)^{n-2}\Bigl).
\end{equation*}
\end{lemma}
\begin{proof}
Using Lemma 4,
%\ref{l1a},
Corollary \ref{s1pn} and the inequality
$n\,{\scriptstyle\gtrsim}\,p(n)$, we obtain the existence of
$\varepsilon_{\Delta}$ such that $0<\varepsilon_{\Delta}<1$,
$\varepsilon_{\Delta}$ does not depend on $n$, and for every
$\varepsilon$ satisfying the inequality
$\varepsilon_{\Delta}<\varepsilon<1$, as $n\rightarrow \infty$ we
have
\begin{eqnarray*}
n^2\beta_{n,p(n)}+2n\gamma_{n,p(n)} &=& a_n O(n^2)
\Bigr(\Bigr(\frac{3}{4}\Bigl)^n b_{p(n)}(n) +
\Bigr(\frac{5}{6}\Bigl)^n c_{p(n)}(n)\Bigl)
\\
&=& a_n O(n^2) \sum\limits_{i=1}^{5}\Bigl(\frac{i}{i+1}\Bigr)^n\,
\sum\limits_{s=0}^{p(n)-1}\binom{n}{s}i^{-s}
\\
&=& a_n
\Bigr(\frac{5+\varepsilon_{\Delta}}{6}\Bigl)^{n}\,O(n^{\frac{5}{2}})
\\
&\,\lesssim\,&
a_n\frac{\lambda}{2}\,\Bigr(\frac{5+\varepsilon}{6}\Bigl)^{n-2}.
\end{eqnarray*}
It is not difficult to see that for every $n$ the following
inclusion holds (see the proof of Lemma 4 and relation (2) in
\cite{fti21-1} similarly):
$${\mathcal F}_{n,3,1}(x,y)\setminus ({\mathcal B}_{n,p(n)}(x,y)\cup
{\mathcal C}_{n,p(n)}(x,y)\cup {\mathcal C}_{n,p(n)}(y,x))\subseteq
{\mathcal F}_{n,3,p(n)}(x,y), \mbox{ where}$$
$${\mathcal B}_{n,p(n)}(x,y)=\bigcup_{\substack{u,v\in V\setminus
\{x,y\}
\\ u\neq v}}{\mathcal B}_{n,p(n)}(x,y,u,v),$$
$${\mathcal C}_{n,p(n)}(x,y)=\bigcup_{u\in V\setminus \{x,y\}}{\mathcal
C}_{n,p(n)}(x,y,u).$$
Thus, reckoning Lemma 5,
%\ref{l3a},
conclude
\begin{eqnarray*}
|{\mathcal F}_{n,3,p(n)}| &\,\gtrsim\,& |{\mathcal
F}_{n,3,1}(x,y)|-n^2|{\mathcal
B}_{n,p(n)}(x,y,u,v)| - 2n|{\mathcal C}_{n,p(n)}(x,y,u)|\\
&\,\gtrsim\,& |{\mathcal F}_{n,3,1}(x,y)|- a_n
\frac{\lambda}{2}\,\Bigr(\frac{5+\varepsilon}{6}\Bigl)^{n-2}\\
&\,\gtrsim\,& a_n \Bigr(1-\frac{\lambda}{2}\,\Bigr(
\frac{5+\varepsilon}{6}\Bigl)^{n-2}\Bigl) - \; a_n
\frac{\lambda}{2}\,\Bigr(\frac{5+\varepsilon}{6}\Bigl)^{n-2}.
\end{eqnarray*}
\end{proof}
\begin{remark}\label{r2pn}
In Lemma {\rm \ref{l5pn}}, the value $\varepsilon_{\Delta}$ depends
only on $\Delta$.
\end{remark}
\noindent Further, for an arbitrary $\Delta$, $0<\Delta<1$, the
notation $\varepsilon_{\Delta}$ will be used for the constant found
in Lemma \ref{l5pn}.
\begin{lemma}[lower estimate of $|{\mathcal F}_{n,k,p(n)}|$]\label{l3sn}
Let $k\geq 3$, $0<\Delta<1$,
$\varepsilon_{\Delta}<\varepsilon<\nobreak 1$,
$p(n)=\Bigl\lfloor\frac{n}{6}\Delta\Bigr\rfloor+1$ and $k$,
$\Delta$, $\varepsilon$ do not depend on $n$. Then there exists a
constant $c>0$ independent of $n$ and $k$ such that for every $n\in
{\mathbb N}$ the following inequality holds
$$|{\mathcal F}_{n,k,p(n)}|\geq
2^{\binom{n}{2}}\xi_{n,k} \Bigl(1-c\,\Bigl(\frac{5+
\varepsilon}{6}\Bigr)^{n-k+1}\Bigr).$$
\end{lemma}
\begin{proof}
Similarly, taking into account Lemmas 1,
%\ref{l5a},
4(i)
%\ref{l1a}(i)
and Lemma \ref{l5pn} for $\lambda=1$, to the proof of the lower
estimate $|{\mathcal F}_{n,k,p}|$ for fixed $p$ (see Lemma 8 in
\cite{fti21 -1}).
\end{proof}
\begin{theorem}[asymptotics of $|{\mathcal F}_{n,k,p(n)}|$]\label{t1sn}
Let $k\geq 3$, $0<\Delta<1$,
$\varepsilon_{\Delta}<\varepsilon<\nobreak 1$,
$p(n)=\Bigl\lfloor\frac{n}{6}\Delta\Bigr\rfloor+1$ and $k$,
$\Delta$, $\varepsilon$ do not depend on $n$. Then there exists a
constant $c>0$ independent of $n$ such that for every $n\in {\mathbb
N}$ the following inequalities are valid

\begin{flushleft}
$ 2^{\binom{n}{2}}\xi_{n,k}\Bigr(1-c\Bigr(\frac
{5+\varepsilon}{6}\Bigl)^{n-k+1}\Bigl)\leq |{\mathcal
F}_{n,k,p(n)}|\leq |{\mathcal J}_{n,\,d=k}|$
\end{flushleft}

\begin{flushright}
$\leq |{\mathcal J}_{n,\,d\geq k}|\leq |{\mathcal J}_{n,\,d\geq
k}^{\ast}|\leq 2^{\binom{n}{2}}\xi_{n,k}\Bigr(1+c\Bigr(\frac
{5+\varepsilon}{6}\Bigl)^{n-k+1}\Bigl).$
\end{flushright}
\end{theorem}
\begin{proof}
It follows directly from Lemma \ref{l3sn}, Theorem 1
%\ref{t1b}
and the inclusion ${\mathcal F}_{n,k,p(n)}\subseteq {\mathcal
F}_{n,k,1}$.
\end{proof}
\begin{corollary}\label{s2sn}
Let $k\geq 3$, $0<\Delta<1$ do not depend on $n$ and
$p(n)=\Bigl\lfloor\frac{n}{6}\Delta\Bigr\rfloor+1$. Then as
$n\rightarrow \infty$
\begin{eqnarray*}
|{\mathcal F}_{n,k,p(n)}|\sim |{\mathcal J}_{n,\,d=k}|\sim
|{\mathcal J}_{n,\,d\geq k}| \sim |{\mathcal J}_{n,\,d\geq
k}^{\ast}|\sim 2^{\binom{n}{2}}\xi_{n,k}.
\end{eqnarray*}
\end{corollary}

The following statement follows directly from the definition of the
class ${\mathcal F}_{n,k,p(n)}$ and Corollary \ref{s2sn}.
\begin{corollary}\label{s2222sn}
Let $k\geq 3$, $0<\Delta<1$ be independent of $n$. Then almost all
$n$-vertex graphs of diameter $k$ contain a diametral path such that
neighbourhoods of any two vertices from the set outside this path
have at least $\Bigl\lfloor\frac{n}{6}\Delta\Bigr\rfloor+1$ common
vertices.
\end{corollary}

\section{The spectrum of the center of almost all graphs from ${\mathcal J}_{n,\,d=k}$}
\hspace*{\parindent} Let us establish estimates for the number of
central vertices of almost all $n$-vertex graphs $G$ of fixed
diameter $k$, which depend on $n$ and give the logarithmic
asymptotics of this number $|{\mathbb C}(G)|$. First, turn to the
center spectrum of the class ${\mathcal F}_{n,k,p(n)}$.
\begin{theorem}[spectrum ${\mathbb Sp}_{c}({\mathcal F}_{n,k,p(n)})$]\label{l2z}
Let $k\geq 3$ and $p=p(n)$ be a function depending on $n$ and taking
positive integer values.
Then for every $n\geq 2p(n)+k+4$ the following equalities hold\\[-10pt]

{\rm (i)} \,\,\,\,${\mathbb Sp}_{c}({\mathcal F}_{n,k=3,p(n)})=\{n-2\};$\\[-10pt]

{\rm (ii)} \,\,${\mathbb Sp}_{c}({\mathcal
F}_{n,k=4,p(n)})=\bigl[\bigl[\,1+p(n),\, n-5-p(n)\, \bigr]\bigr];$\\[-10pt]

{\rm (iii)} \,${\mathbb Sp}_{c}({\mathcal F}_{n,k=5,p(n)})=
\bigl[\bigl[\,2+p(n),\, n-5-p(n)\, \bigr]\bigr]\cup\{n-4\};$\\[-10pt]

{\rm (iv)} \,${\mathbb Sp}_{c}({\mathcal F}_{n,k,p(n)})=\{1\} \cup
\bigl[\bigl[\,1+p(n),\, n-k-1-p(n)\, \bigr]\bigr]$ for even $k\geq 6;$\\[-10pt]

{\rm (v)} \,\,${\mathbb Sp}_{c}({\mathcal F}_{n,k,p(n)})=\{2\}\cup
\bigl[\bigl[\,2+p(n),\, n-k-p(n)\, \bigr]\bigr]\cup\{n-k+1\}$ for
odd $k\geq 7.$
\end{theorem}
\begin{proof}
It follows from the description of the center spectrum ${\mathbb
Sp}_{c}({\mathcal F}_{n,k,p})$ for fixed integer values $k\geq 3$
and $p\geq 1$ (Theorem 5 \cite{fti21-2}).
\end{proof}
\begin{theorem}[spectrum of the center of almost all graphs
from ${\mathcal J}_{n,\,d=k}$]\label{t2sn}
Let $k\geq 1$ and
$0<\Delta<1$ be independent of $n$. Then
\\[-9pt]

{\rm (i)} $|{\mathbb C}(G)|=n$ for almost all\,
$n$-vertex graphs $G$ of diameter $k=1,2;$\\[-9pt]

{\rm (ii)} $|{\mathbb C}(G)|=n-2$ for almost all\,
$n$-vertex graphs $G$ of diameter $3;$\\[-9pt]

{\rm (iii)} $|{\mathbb C}(G)|\in \bigl[\bigl[\,2+
\lfloor\frac{n}{6}\Delta\rfloor,\, n-\lfloor\frac{n}{6}\Delta\rfloor
-6 \,\bigr]\bigr]$
for almost all\, $n$-vertex graphs $G$ of diameter $4;$\\[-9pt]

{\rm (iv)} $|{\mathbb C}(G)|\in
\bigl[\bigl[\,3+\lfloor\frac{n}{6}\Delta\rfloor,\,
n-\lfloor\frac{n}{6}\Delta\rfloor -6 \,\bigr]\bigr]\cup\{n-4\}$ for
almost all\, $n$-vertex graphs $G$ of diameter $5${\rm;} moreover,
the fraction of such graphs with $(n-4)$-vertex center
asymptotically equals $\frac{1}{3};$\\[-9pt]

{\rm (v)} $|{\mathbb C}(G)|\in \{1\} \cup
\bigl[\bigl[\,2+\lfloor\frac{n}{6}\Delta\rfloor,\, n-
\lfloor\frac{n}{6}\Delta\rfloor -k-2\,\bigr]\bigr]$ for almost all
$n$-vertex graphs $G$ of even fixed diameter $k\geq 6${\rm;}
moreover, the fraction of such graphs with trivial center
asymptotically equals $\frac{k-4}{k-2}$;\\[ -9pt]

{\rm (vi)} $|{\mathbb C}(G)|\in \{2\}\cup
\bigl[\bigl[\,3+\lfloor\frac{n}{6}\Delta\rfloor,\,
n-\lfloor\frac{n}{6}\Delta\rfloor -k-1 \,
\bigr]\bigr]\cup\,\{n-k+1\}$ for almost all $n$-vertex graphs $G$ of
odd fixed diameter $k\geq 7${\rm;} moreover, the fraction of such
graphs with $2$-vertex and $(n-k+1)$-vertex center asymptotically
equals $\frac{k-5}{k-2}$ and $\frac{1}{k-2}$ respectively.
\end{theorem}
\begin{proof}
The cases $k=1,2,3$ are considered in Theorem 2.
%\ref{t1k}.
Let now $k\geq 4$ and
$p(n)=\Bigl\lfloor\frac{n}{6}\Delta\Bigr\rfloor+1$. Then
$n\,{\scriptstyle\gtrsim}\, 2p(n)+k+4$.

Consider, for example, the case $k=4$. By Theorem \ref{l2z}, for all
large enough $n$ we have ${\mathbb Sp}_{c}({\mathcal
F}_{n,k=4,p(n)})=\bigl[\bigl[\,1+p(n),\, n-5-p(n)\, \bigr]\bigr]$.
Therefore, taking into account Corollary \ref{s2sn}, as
$n\rightarrow \infty$ we obtain
$$\frac{|\{ G\in {\mathcal J}_{n,\,d=4} \,\, |\,\,|{\mathbb C}(G)|\in
\bigl[\bigl[\,1+p(n),\, n-5-p(n)\, \bigr]\bigr] \}|}{|{\mathcal
J}_{n,\,d=4}|} \geq  \frac{|{\mathcal F}_{n,k=4,p(n)}|}{|{\mathcal
J}_{n,\,d=4}|}\,\longrightarrow 1.$$

Similarly, from Corollary \ref{s2sn} and Theorem \ref{l2z} we obtain
the possible cardinalities of the center of almost all $n$-vertex
graphs of fixed diameter $k\geq 5$ for all other indicated cases
(iv)-(vi). Asymptotic value of the corresponding fractions of the
graphs is established in Theorem 2.
%\ref{t1k}.
\end{proof}
Theorem \ref{t2sn} implies a number of properties of the centers of
almost all graphs of fixed diameter $k$. For example, there are
almost no graphs of diameter $k=2,4$ and odd diameter $k$ with a
trivial center, while for any even $k\geq 6$ this is not true.
Similarly, there are almost no graphs with a 2-vertex center of
diameter $k=1,3,5$ and even diameter $k$, however, for every odd
$k\geq 7$ this does not hold. Unexpected is the jump of the center
cardinality outside the interval of consecutive integer values both
from above from $n-\lfloor\frac{n}{6}\Delta\rfloor -k-1$ to $n-k+1$
for odd $k\geq 5$, and from below from
$2+\lfloor\frac{n}{6}\Delta\rfloor$ to $1$ for even $k\geq 6$ and
from $3+\lfloor\frac{n}{6}\Delta\rfloor$ to $2$ for odd $k\geq 7$.

Theorem \ref{t2sn} also allows to find the logarithmic asymptotics
of the number of central vertices of almost all $n$-vertex graphs of
fixed diameter.
\begin{corollary}[logarithmic asymptotics]\label{s100sn}
Let $k\geq 1$ be an integer. The following asymptotic equality is
valid $$\log_2|{\mathbb C}(G)|\sim \log_2(n)$$ for almost all
\,$n$-vertex graphs $G$ of each of the following classes{\,\rm:}
graphs of fixed diameter $k\leq 5;$ graphs of even fixed diameter
$k\geq 6$ with nontrivial center{\rm;} graphs of odd fixed diameter
$k\geq 7$ whose the center cardinality is not equal to $2.$
\end{corollary}
\begin{proof}
Let $k\geq 6$ and $0<\Delta<1$ be a fixed constant. By statements
(v),(vi) of Theorem \ref{t2sn} and Proposition \ref{l300nn}(v) for
almost all $n$-vertex graphs $G$ of even fixed diameter $k\geq 6$
with a nontrivial center we have $|{\mathbb C}(G)|\in
\bigl[\bigl[\,2+\lfloor\frac{n}{6}\Delta\rfloor,\, n-
\lfloor\frac{n}{6}\Delta\rfloor -k-2\,\bigr]\bigr]$, and for almost
all $n$-vertex graphs $G$ of odd fixed diameter $k\geq 7$ whose the
center cardinality is not equal to $2$, we obtain $|{\mathbb
C}(G)|\in \bigl[\bigl[\,3+\lfloor\frac{n}{6}\Delta\rfloor,\,
n-\lfloor\frac{n}{6}\Delta\rfloor -k-1 \,
\bigr]\bigr]\cup\,\{n-k+1\}$. It remains to note that if $P(n)$ is a
polynomial over ${\mathbb R}$ of non-zero degree $m$ with positive
leading coefficient, then $\log_2 P(n) \sim m \log_2 n$ as $
n\rightarrow \infty$.

The case $k\leq 5$ is considered similarly using statements (i)-(iv)
of Theorem \ref{t2sn}.
\end{proof}
\begin{remark}\label{r2}
The fraction of $n$-vertex graphs of fixed even diameter $k\geq 6$
with trivial center {\rm(}of odd diameter $k\geq 7$ with $2$-vertex
center{\rm)} is asymptotically equal to $\frac{k -4}{k-2}$
{\rm(}$\frac{k-5}{k-2}${\rm)}. Moreover, typical graphs of these
classes are constructed in {\rm \cite{fti21-2}}.
\end{remark}

\noindent Thus, for almost all $n$-vertex graphs of a fixed
diameter, the logarithmic asymptotics of the number of central
vertices is $0$ or $\log_2 n$ ($1$ or $\log_2 n$) for the
corresponding subclasses of graphs of the even (odd) diameter.

Theorem \ref{t2sn} also implies estimates of the central ratio
${\mathbb R}_{c}(G)$ for almost all $n$-vertex graphs $G$ of fixed
diameter $k$.

\begin{corollary}[estimates of ${\mathbb R}_{c}(G)$]\label{s280nn}
Let $k\geq 1$ and $0<\Delta<1$ do not depend on $n$. Then
\\[-9pt]

{\rm (i)} ${\mathbb R}_{c}(G)=1$ for almost all\,
$n$-vertex graphs $G$ of diameter $k=1,2;$\\[-9pt]

{\rm (ii)} ${\mathbb R}_{c}(G)=1-2/n$ for almost all\,
$n$-vertex graphs $G$ of diameter $3;$\\[-9pt]

{\rm (iii)} $\frac{\Delta}{6} + r_1(n) \leq {\mathbb R}_{c}(G) \leq
1-\frac{\Delta}{6} - r_2(n)$ for almost all $n$-vertex graphs $G$ of
each of the following classes{\,\rm:} graphs of diameter $4${\rm;}
graphs of diameter $5$ whose center cardinality is not equal to
$n-4${\rm;} graphs of even fixed diameter $k\geq 6$ with nontrivial
center{\rm;} graphs of odd fixed diameter $k\geq 7$ whose center
cardinality is not equal to $2$ and $n-k+1$. Here $r_1(n)$, $r_1(n)$
are positive functions and $r_1(n)=o(n)$, $r_2(n)=o(n)$ as
$n\rightarrow\infty$.
\end{corollary}
\begin{proof}
Similar to the proof of Corollary \ref{s100sn} using Theorem
\ref{t2sn}, Proposition \ref{l300nn} and the inequality $\lfloor
x\rfloor\geq x-1$.
\end{proof}
\begin{remark}\label{r3}
The fraction of $n$-vertex graphs of fixed even diameter $k\geq 6$
with trivial center {\rm(}of odd diameter $k\geq 5$ with $2$-vertex
and $(n-k+1)$-vertex center{\rm)} asymptotically equals
$\frac{k-4}{k-2}$ {\rm(}$\frac{k-5}{k-2}$ and $\frac{1}{k-2}$
respectively{\rm)}. Moreover, typical graphs of these classes are
constructed in {\rm \cite{fti21-2}}.
\end{remark}

Note that from Corollary \ref{s280nn}, by virtue of Proposition
\ref{l300nn}(v), we also find possible values of the central ratio
${\mathbb R}_{c}(G)$ for almost all graphs $G$ of the whole class
${\mathcal J}_{n,\,d=k}$.

The obtained properties of the centers are also valid, by virtue of
Corollary \ref{s2sn}, for graphs of classes ${\mathcal J}_{n,\,d\geq
k}$ and ${\mathcal J}_{n,\,d\geq k}^{\ast}$.
\begin{corollary}\label{s80nn}
For every fixed $k\geq 2$, almost all $n$-vertex graphs of each of
the following classes ${\mathcal J}_{n,\,d\geq k}$, ${\mathcal
J}^{\ast }_{n,\,d\geq k}$ are connected, have diameter $k$, and the
properties stated in Theorem  {\rm\ref{t2sn}} and its corollaries
hold for the center.
\end{corollary}

In conclusion, the author thanks the referee for careful reading of
the article and useful suggestions.

\bigskip
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