text:
\section{Introduction}

Watts and Strogatz \cite{wast} proposed the clustering coefficient of a graph in order to quantify the corresponding property of networks.
For a vertex $u$ of a graph $G$, let
the {\it clustering coefficient} of $u$ in $G$ be
$$C_u(G)=
\begin{cases}
\frac{m(G[N_G(u)])}{{d_G(u)\choose 2}} &, \mbox{ if $d_G(u)\geq 2$,}\\
0 &, \mbox{ otherwise,}
\end{cases}
$$
where $m(G[N_G(u)])$ is the size of the subgraph of $G$ induced by the neighborhood $N_G(u)$ of $u$ in $G$,
that is, $m(G[N_G(u)])$ equals exactly the number of triangles of $G$ that contain $u$.
Furthermore, let the {\it clustering coefficient} of $G$ be the average
$$C(G)=\frac{1}{n(G)}\sum\limits_{u\in V(G)}C_u(G)$$
of the clustering coefficients of its $n(G)$ vertices.

While the clustering coefficient received a lot of attention within social network analysis \cite{bojapr,scwa,wa1,wa2},
some fundamental mathematical problems related to it are still open.
It is unknown, for instance, which graphs maximize the clustering coefficient among all connected graphs of a given order and size.

Watts \cite{wa1,wa2} suggested the so-called {\it connected caveman} graphs as a possible extremal construction.
For integers $k$ and $\ell$ at least $2$,
these arise from $\ell$ disjoint copies $G_1,\ldots,G_\ell$ of $K_{k+1}-e$ arranged cyclically
by adding, for every $i$ in $[\ell]$,
an edge between
one of the two vertices of degree $k-1$ in $G_i$ and
one of the $k-1$ vertices of degree $k$ in $G_{i+1}$, where the indices are identified modulo $\ell$.
Actually, it is rather obvious that these graphs do not have the largest clustering coefficient
among all connected graphs of given order and size,
because removing the edge between $G_1$ and $G_2$,
and adding a new edge between the two vertices of degree $k-1$ in $G_1$,
increases the clustering coefficient.

Fukami and Takahashi \cite{futa1,futa2} considered {\it clustering coefficient locally maximizing graphs}
whose clustering coefficient cannot be increased by some local operations such as an edge swap.

In the present paper
we determine the maximum clustering coefficients
among all connected regular graphs of a given order,
as well as
among all connected subcubic graphs of a given order.
In both cases, we characterize all extremal graphs.
Furthermore, we determine the maximum increase of the clustering coefficient caused by adding a single edge.

\section{Results}

We introduce a slightly modified version of the connected caveman graphs.
For integers $k$ and $\ell$ with $k\geq 3$ and $\ell\geq 2$,
let $G(k,\ell)$ be the $k$-regular connected graph that arises
from $\ell$ disjoint copies $G_1,\ldots,G_\ell$ of $K_{k+1}-e$ arranged cyclically
by adding, for every $i$ in $[\ell]$,
an edge between a vertex in $G_i$ and a vertex in $G_{i+1}$,
where the indices are identified modulo $\ell$.
Note that $G(k,\ell)$ is uniquely determined up to isomorphism by the requirement of $k$-regularity.

\begin{theorem}\label{theorem1}
Let $k$ and $n$ be integers with $n\geq k+2$ and $k\geq 3$.
If $G$ is a connected $k$-regular graph of order $n$, then
$$C(G)\leq 1-\frac{6}{k(k+1)}$$
with equality if and only if $n/(k+1)$ is an integer and $G$ equals $G(k,n/(k+1))$.
\end{theorem}
{\it Proof:} Let $G$ be a connected $k$-regular graph of order $n$.
For a non-negative integer $i$, let $V_i$ be the set of vertices $u$ of $G$ with $m(G[N_G(u)])={k\choose 2}-i$.
Since $G$ is connected and has order at least $k+2$, no vertex has a complete neighborhood, that is, $V_0$ is empty.
For a set $U$ of vertices of $G$, let $\sigma(U)=\sum\limits_{u\in U}C_u(G)$.
In order to obtain a useful decomposition of $G$, we consider some special graphs.
For $k\leq 4$, one such graph suffices, while for $k\geq 5$, two more are needed.

Let $G_1,\ldots,G_r$ be a maximal collection of disjoint subgraphs of $G$ that are all copies of $K_{k+1}-e$.
Let $A=V(G_1)\cup \cdots \cup V(G_r)$ and $R=V(G)\setminus A$.
Note that every vertex in $A$ has at most one neighbor in $R$.
Suppose that $R$ contains a vertex $u$ from $V_1$.
Since every vertex in $N_G(u)$ has at least two neighbors in the closed neighborhood $N_G[u]$ of $u$,
the subgraph $G_{r+1}$ of $G$ induced by $N_G[u]$
does not intersect $A$
and is a copy of $K_{k+1}-e$.
Now, $G_1,\ldots,G_r,G_{r+1}$ contradicts the maximality of the above collection,
which implies that $R$ does not intersect $V_1$.

Since each $G_i$ contains $k-1$ vertices from $V_1$ and two vertices whose neighborhood induces $K_1\cup K_{k-1}$,
and $|A|=r(k+1)$,
we have
\begin{eqnarray*}
\sigma(A)=\sum\limits_{i\in [r]}\sum\limits_{u\in V(G_i)}C_u(G)
= r\left((k-1)\frac{{k\choose 2}-1}{{k\choose 2}}+2\frac{{k-1\choose 2}}{{k\choose 2}}\right)
= |A|\left(1-\frac{6}{k(k+1)}\right).
\end{eqnarray*}
Since $R$ does not intersect $V_1$, we have
\begin{eqnarray*}
\sigma(R)=\sum\limits_{u\in R}C_u(G)
\leq |R|\frac{{k\choose 2}-2}{{k\choose 2}}
=|R|\left(1-\frac{4}{(k-1)k}\right).
\end{eqnarray*}
First, let $k\leq 4$.
Since $1-\frac{4}{(k-1)k}<1-\frac{6}{k(k+1)}$ in this case, we obtain
\begin{eqnarray*}
C(G) &=& \frac{1}{n(G)}(\sigma(A)+\sigma(R))\\
&\leq & \frac{1}{n(G)}(|A|+|R|)\left(1-\frac{6}{k(k+1)}\right)\\
&=& 1-\frac{6}{k(k+1)},
\end{eqnarray*}
with equality if and only if $A=V(G)$ and $R=\emptyset$, which implies that $k+1$ divides $n$, and $G$ equals $G(k,n/(k+1))$.

Now, let $k\geq 5$.
In this case, we need to refine the partition of $V(G)$ into $A$ and $R$ further.
Let $G_1,\ldots,G_r$, $A$, and $R$ be exactly as above, and recall that $R$ does not intersect $V_1$.
Let $H_1,\ldots,H_s$ be a maximal collection of disjoint subgraphs of $G[R]$
that are all copies of the two possible graphs
that arise from $K_{k+1}$ by removing two edges.
Let $B=V(H_1)\cup \cdots \cup V(H_s)$, and $S=V(G)\setminus (A\cup B)$.
Note that every vertex in $A\cup B$ has at most two neighbors in $S$.
Suppose that $S$ contains a vertex $u$ from $V_2$.
Since $k\geq 5$, every vertex in $N_G(u)$ has at least three neighbors in the closed neighborhood $N_G[u]$ of $u$.
Hence, the subgraph $H_{s+1}$ of $G$ induced by $N_G[u]$
does not intersect $A\cup B$
and is a copy of one of the two possible graphs that arise from $K_{k+1}$ by removing two edges.
Now, $H_1,\ldots,H_s,H_{s+1}$ contradicts the maximality of the above collection,
which implies that $S$ does not intersect $V_1\cup V_2$.

If $H_i$ for some $i$ in $[s]$ is a copy of $K_{k+1}$ minus two non-incident edges,
then $H_i$ contains $k-3$ vertices from $V_2$
and four vertices whose neighborhood induces a graph that arises from $K_1\cup (K_{k-1}-e)$ by adding at most two edges,
which implies
\begin{eqnarray*}
\sigma(V(H_i))
& \leq & (k-3)\frac{{k\choose 2}-2}{{k\choose 2}}+4\frac{{k-1\choose 2}-1+2}{{k\choose 2}}
=\frac{k^3-13 k+28}{(k-1)k}.
\end{eqnarray*}
If $H_i$ for some $i$ in $[s]$ is a copy of $K_{k+1}$ minus two incident edges,
then $H_i$ contains $k-2$ vertices from $V_2$,
two vertices whose neighborhood induces a graph that arises from $K_1\cup K_{k-1}$ by adding at most one edge, and
one vertex whose neighborhood induces a graph that arises from $K_1\cup K_1\cup K_{k-2}$ by adding at most one edge,
which implies
\begin{eqnarray*}
\sigma(V(H_i))
& \leq & (k-2)\frac{{k\choose 2}-2}{{k\choose 2}}+2\frac{{k-1\choose 2}+1}{{k\choose 2}}+\frac{{k-2\choose 2}+1}{{k\choose 2}}
=\frac{k^3-13 k+24}{(k-1)k}.
\end{eqnarray*}
Using $|B|=s(k+1)$, we obtain
\begin{eqnarray*}
\sigma(B) & = & \sum\limits_{i\in [s]}\sigma(V(H_i))
\leq |B|\frac{k^3-13 k+28}{(k-1)k(k+1)}
= |B|\left(1-\frac{12k-28}{(k-1)k(k+1)}\right).
\end{eqnarray*}
Since $S$ does not intersect $V_1\cup V_2$, we have
\begin{eqnarray*}
\sigma(S) & \leq & |S|\frac{{k\choose 2}-3}{{k\choose 2}}
= |S|\left(1-\frac{6}{(k-1)k}\right).
\end{eqnarray*}
Since
$$\max\left\{ 1-\frac{12k-28}{(k-1)k(k+1)},1-\frac{6}{(k-1)k}\right\}<1-\frac{6}{k(k+1)}$$
for $k\geq 5$,
we obtain
\begin{eqnarray*}
C(G) &=& \frac{1}{n(G)}(\sigma(A)+\sigma(B)+\sigma(S))\\
&\leq &\frac{1}{n(G)}(|A|+|B|+|S|)\left(1-\frac{6}{k(k+1)}\right)\\
&=& 1-\frac{6}{k(k+1)},
\end{eqnarray*}
with equality if and only if $A=V(G)$ and $B=S=\emptyset$,
which implies that $k+1$ divides $n$, and $G$ equals $G(k,n/(k+1))$.
This completes the proof. $\Box$

\medskip

\noindent Recall that the {\it diamond} is the unique graph with degree sequence $2,2,3,3$.

\begin{theorem}\label{theorem2}
If $G$ is a connected subcubic graph of order $n$ at least $6$, then
\begin{eqnarray}\label{e1}
C(G)\leq
\begin{cases}
\frac{7}{12}+\frac{12}{12n} & \mbox{, if $n\equiv 0$ mod $4$,}\\
\frac{7}{12}+\frac{13}{12n} & \mbox{, if $n\equiv 1$ mod $4$,}\\
\frac{7}{12}+\frac{14}{12n} & \mbox{, if $n\equiv 2$ mod $4$, and}\\
\frac{7}{12}+\frac{11}{12n} & \mbox{, if $n\equiv 3$ mod $4$.}\\
\end{cases}
\end{eqnarray}
\end{theorem}
{\it Proof:} Let ${\cal G}$ be the set of all connected subcubic graphs of order $n$ at least $6$. We assume that $G$ is chosen within ${\cal G}$ in such a way that
\begin{enumerate}[(i)]
\item its clustering coefficient $C(G)$ is as large as possible,
\item subject to condition (i), the size $m(G)$ of $G$ is as small as possible, and,
\item subject to conditions (i) and (ii), the number of triangles in $G$ that contain at most one vertex of degree $2$ in $G$ is as small as possible.
\end{enumerate}
We establish a series of structural properties of $G$.

\begin{claim}\label{claim1}
Every subgraph $D$ of $G$ that is a diamond is induced and forms an endblock.
\end{claim}
{\it Proof of Claim \ref{claim1}:}
Since $G$ is subcubic, connected, and of order more than $4$, the subgraph $D$ is induced.
If all vertices in $D$ have degree $3$ in $G$,
then contracting $D$ to a single vertex $u$,
adding a new triangle $xyz$,
and adding the new edge $ux$
yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)+\frac{1}{3n}$,
contradicting the choice of $G$.
Note that the two neighbors of $u$ in $G'$ that are distinct from $x$ may be adjacent,
in which case $C(G')>C(G)+\frac{1}{3n}$.
In view of the order,
this implies that $D$ contains exactly one vertex of degree $2$,
and, hence, forms an endblock of $G$.
$\Box$

\begin{claim}\label{claim2}
Every edge of $G$ that lies in some cycle also lies in some triangle.
\end{claim}
{\it Proof of Claim \ref{claim2}:}
If the edge $uv$ of $G$ lies in some cycle but in no triangle,
then removing $uv$ yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)$ and $m(G')<m(G)$,
contradicting the choice of $G$.
Note that, if some triangle of $G$ contains $u$ or $v$, then $C(G')>C(G)$.
$\Box$

\begin{claim}\label{claim3}
Every block of $G$ is $K_2$, $K_3$, or a diamond.
\end{claim}
{\it Proof of Claim \ref{claim3}:}
Suppose, for a contradiction,
that $B$ is a block of $G$ that is neither $K_2$, nor $K_3$, nor a diamond.
Note that every edge of $B$ lies in some cycle,
and, hence, by Claim \ref{claim2},
also lies in some triangle.
Let $uvw$ be a triangle in $B$.
Since $B$ is not $K_3$,
we may assume that $u$ has a neighbor $x$ in $V(B)\setminus \{ v,w\}$.
Since the edge $ux$ of $B$ lies in some triangle,
we may assume that $x$ and $v$ are adjacent.
Since $B$ is not a diamond,
we may assume, by symmetry, that $x$ has a neighbor $y$ in $V(B)\setminus \{ u,v,w\}$.
Since the edge $xy$ of $B$ lies in some triangle,
we obtain that $y$ is adjacent to $u$ or $v$,
contradicting the assumption that $G$ is subcubic.
$\Box$

\medskip

\noindent Let ${\cal D}$ be the set of blocks of $G$ that are diamonds.
For $i\in \{ 2,3\}$, let ${\cal I}_i$ be the set of blocks of $G$ that are triangles that contain exactly $i$ vertices of degree $3$ in $G$.
Let ${\cal I}={\cal I}_2\cup {\cal I}_3$.
Finally, let ${\cal S}$ be the set of vertices of $G$ that do not lie in some triangle and have degree at most $2$ in $G$.

The triangles in ${\cal I}$ are called {\it inner triangles}.
By condition (iii), $G$ has as few inner triangles as possible
given the other conditions.

If $u\in {\cal S}$ has degree $2$, then {\it resolving $u$} means to remove $u$ from $G$, and to connect its two neighbors by a new edge.
If $u\in {\cal S}$ has degree $1$, then {\it resolving $u$} simply means to remove $u$ from $G$.
Note that resolving some vertex from ${\cal S}$ yields a connected subcubic graph $G'$ of order $n-1$ with $C(G')\geq C(G)$.

\begin{claim}\label{claim4}
Either ${\cal D}$ or ${\cal I}$ is empty.
\end{claim}
{\it Proof of Claim \ref{claim4}:}
Suppose, for a contradiction,
that $G$ contains a diamond $D$ and an inner triangle $T$.
By Claim \ref{claim1}, $D$ contains a vertex $u$ of degree $2$ in $G$.
Let $v$ be a neighbor of $u$.
If $T\in {\cal I}_2$, then
contracting $T$ to a single vertex $w$,
removing $u$,
adding a new triangle $xyz$, and
adding the new edge $xw$
yields a graph $G'$ in ${\cal G}$ with $C(G')=C(G)+\frac{1}{3n}$,
contradicting the choice of $G$.
If $T\in {\cal I}_3$, then
contracting $T$ to a single vertex,
removing $u$,
adding a new triangle $xyz$, and
adding the new edge $xv$
yields a graph $G'$ in ${\cal G}$ with $C(G')=C(G)+\frac{1}{3n}$,
contradicting the choice of $G$.
$\Box$

\begin{claim}\label{claim5}
$|{\cal D}|\leq 2$.
\end{claim}
{\it Proof of Claim \ref{claim5}:}
Suppose, for a contradiction,
that $G$ has three blocks $B_1$, $B_2$, and $B_3$ that are diamonds.
By Claim \ref{claim1}, these are all endblocks, and
removing the three vertices, say $v_1$, $v_2$, and $v_3$, of degree $2$ in $G$ from $B_1$, $B_2$, and $B_3$,
adding a new triangle $xyz$, and
adding a new edge between $x$ and one of the two neighbors of $v_1$ in $B_1$
yields a graph $G'$ in ${\cal G}$ with $C(G')=C(G)+\frac{2}{3n}$,
contradicting the choice of $G$.
$\Box$

\begin{claim}\label{claim6}
${\cal S}$ is empty.
\end{claim}
{\it Proof of Claim \ref{claim6}:}
Suppose, for a contradiction,
that ${\cal S}$ contains some vertex $u$.

If ${\cal I}_2$ contains a triangle $T$, then
resolving $u$,
contracting $T$ to a single vertex $v$,
adding a new triangle $xyz$, and
adding the new edge $xv$
yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)+\frac{2}{3n}$,
contradicting the choice of $G$.
If ${\cal I}_3$ contains a triangle $T$, then
resolving $u$,
contracting $T$ to a single vertex $v$,
adding a new triangle $xyz$, and then
replacing one of the edges incident with $v$, say $vw$,
with the two new edges $xv$ and $yw$
yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)+\frac{2}{3n}$,
contradicting the choice of $G$.
Hence, we may assume that $G$ has no inner triangles.

If $G$ has a block $B$ that is a triangle, then $B$ is an endblock, and
resolving $u$,
adding a new vertex $v$, and
adding two new edges between $v$ and the two vertices of degree $2$ in $B$
yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)+\frac{1}{3n}$,
contradicting the choice of $G$.
Hence, we may assume that $G$ has no blocks that are triangles.

If $G$ has two blocks $B_1$ and $B_2$ that are diamonds,
then $B_1$ and $B_2$ are endblocks by Claim \ref{claim1}, and
resolving $u$,
removing the two vertices, say $v_1$ and $v_2$, of degree $2$ in $G$ from $B_1$ and $B_2$, respectively,
adding a new triangle $xyz$, and
adding a new edge between $x$ and one of the two neighbors of $v_1$ in $B_1$
yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)+\frac{1}{n}$,
contradicting the choice of $G$.
Hence, since $G$ is not a tree,
we may assume that $G$ has exactly one endblock $B$ that is a diamond, and all other blocks of $G$ are $K_2$s.
In this case $C(G)=\frac{8}{3n}$.
Since ${\cal G}$ contains a graph $G'$ with two endblocks that are triangles, we obtain $C(G')\geq \frac{14}{3n}>C(G)$,
contradicting the choice of $G$.
$\Box$

\medskip

\noindent Recall that the {\it paw} is the unique graph with degree sequence $1,2,2,3$.

\begin{claim}\label{claim7}
$|{\cal I}|\leq 1$.
\end{claim}
{\it Proof of Claim \ref{claim7}:}
Suppose, for a contradiction,
that $G$ has two inner triangles $T_1$ and $T_2$.

If $T_1,T_2\in {\cal I}_3$, then
contracting both triangles to single vertices $u_1$ and $u_2$,
adding a new paw $P$, and then
replacing one of the edges incident with $u_1$, say $u_1v$,
with the two new edges $xu_1$ and $xv$,
where $x$ is the vertex of degree $1$ in $P$,
yields a graph $G'$ in ${\cal G}$ with $C(G')=C(G)+\frac{1}{3n}$,
contradicting the choice of $G$.

If $T_1\in {\cal I}_2$ and $T_2\in {\cal I}_3$, then
contracting both triangles to single vertices $u_1$ and $u_2$,
adding a new diamond $D$, and
adding the new edge $xu_1$,
where $x$ is a vertex of degree $2$ in $D$,
yields a graph $G'$ in ${\cal G}$ with $C(G')=C(G)$, $m(G')=m(G)$,
and less inner triangles than $G$,
contradicting the choice of $G$.

If $T_1,T_2\in {\cal I}_2$, then
replacing $T_1$ with an edge between the two vertices, say $a$ and $b$, outside of $T_1$ that have neighbors in $T_1$,
adding a new triangles $xyz$, and
adding the new edge $xu_2$,
where $u_2$ is the vertex in $T_2$ of degree $2$ in $G$,
yields a graph $G'$ in ${\cal G}$ with $C(G')=C(G)$, $m(G')=m(G)$,
and less inner triangles than $G$,
contradicting the choice of $G$.
Note that,
in this last construction,
the vertices $a$ and $b$ are non-adjacent in $G$ by Claim \ref{claim3}, and
two triangles that contributed to ${\cal I}_2$ are replaced by one that contributes to ${\cal I}_3$.
$\Box$

\medskip

\noindent For $t(G)=(|{\cal D}|,|{\cal I}_2|,|{\cal I}_3|)$, the above claims imply
$t(G)\in \{
(0,0,0),
(1,0,0),
(2,0,0),
(0,0,1),
(0,1,0)\}.$
For each of these cases, we can determine $C(G)$ exactly.
Let $k$ be the number of vertices of $G$ that do not lie in a triangle.

If $t(G)=(i,0,0)$ for some $i\in\{ 0,1,2\}$,
then $G$ arises from a tree of order $2k+2$
with $k$ vertices of degree $3$ and $k+2$ endvertices,
by replacing $k+2-i$ endvertices with triangles and $i$ endvertices with diamonds,
and we obtain
$n=4k+6+i$, and
$$C(G)
=\frac{1}{n}\left(\frac{7}{3}(k+2-i)+\frac{8}{3}i\right)
=\frac{7}{12}+\frac{14-3i}{12n}.$$
If $t(G)=(0,0,1)$,
then $G$ arises from a tree of order $2k+4$
with $k+1$ vertices of degree $3$ and $k+3$ endvertices,
by replacing all endvertices as well as one internal vertex with triangles,
and we obtain
$n=4k+12$, and
$$C(G)
=\frac{1}{n}\left(\frac{7}{3}(k+3)+1\right)
=\frac{7}{12}+\frac{1}{n}.$$
If $t(G)=(0,1,0)$,
then $G$ arises from a tree of order $2k+3$
with $k$ vertices of degree $3$, one vertex of degree $2$, and $k+2$ endvertices,
by replacing all endvertices as well as the vertex of degree $2$
with triangles,
and we obtain
$n=4k+9$, and
$$C(G)
=\frac{1}{n}\left(\frac{7}{3}(k+2)+\frac{5}{3}\right)
=\frac{7}{12}+\frac{13}{12n}.$$
Considering the different parities of $n$ modulo $4$,
the desired result follows.
$\Box$

\medskip

\noindent Our next goal is the characterization of all extremal graphs for (\ref{e1}).
Therefore, let ${\cal B}_0$ be the set of all connected subcubic graphs $G$ of order at least $6$
such that every block of $G$ is $K_2$, $K_3$, or a diamond, and every block of $G$ that is a diamond is an endblock of $G$.
Let the {\it type} $t(G)$ of a graph $G$ in ${\cal B}_0$ be the $3$-tuple $(d,i_2,i_3)$, where
$d$ is the number of blocks of $G$ that are diamonds, and, for $j$ in $\{ 2,3\}$,
$i_j$ is the number of blocks of $G$ that are triangles that contain exactly $j$ vertices of degree $3$ in $G$.

Let
$${\cal B}
=
\Big\{ G\in {\cal B}_0:t(G)\in
\{
(0,0,0),
(1,0,0),
(0,1,0),
(0,0,1),
(0,1,1),
(0,2,0),
(0,3,0)\}\Big\}.$$

\begin{theorem}\label{theorem3}
If $G$ is a connected subcubic graph of order at least $6$,
then $G$ satisfies (\ref{e1}) with equality if and only if $G\in {\cal B}$.
\end{theorem}
{\it Proof:} First, let $G\in {\cal B}$.
For $t(G)\in \{ (0,0,0),(1,0,0),(0,1,0),(0,0,1)\}$,
we verified at the end of the proof of Theorem \ref{theorem2} that (\ref{e1}) holds with equality.
For $t(G)\in \{ (0,1,1),(0,2,0),(0,3,0)\}$,
very similar simple calculations imply the same.

Now, let $G$ satisfy (\ref{e1}) with equality.
Similarly as above, let ${\cal G}$ be the set of all connected subcubic graphs of order $n(G)$.
We consider the claims from the proof of Theorem \ref{theorem2}.
Clearly, Claim \ref{claim1} still holds.
Suppose, for a contradiction, that Claim \ref{claim2} fails,
that is, $G$ has some edge $uv$ that lies in some cycle but in no triangle.
Iteratively removing from $G$ first the edge $uv$, and then further edges that lie in cycles but not in triangles as long as possible
yields a graph $G'$ in ${\cal G}$ with $C(G')\geq C(G)$ such that every edge of $G'$ that lies in some cycle also lies in some triangle.
Now, the argument from the proof of Claim \ref{claim3} applies, and, hence, $G'\in {\cal B}_0\subseteq {\cal G}$.
By (\ref{e1}) for $G'$, we obtain $C(G')=C(G)$,
which, as observed in the proof of Claim \ref{claim2}, implies that $u$ and $v$ are vertices of $G'$
that do not lie in some triangle and have degree at most $2$ in $G'$.
Arguing as in the proof of Claim \ref{claim6},
we obtain the existence of a graph $G''$ in ${\cal G}$ with $C(G'')>C(G')=C(G)$,
which contradicts (\ref{e1}) for $G''$.
Hence, Claim \ref{claim2} holds.
Considering their respective proofs,
it follows that also Claims \ref{claim3}, \ref{claim4}, \ref{claim5}, and \ref{claim6} hold.
Let the type $t(G)$ of $G$ be $(d,i_2,i_3)$.
By Claim \ref{claim5}, $d\leq 2$.
If $d=2$, then, by Claim \ref{claim4}, $i_2=i_3=0$,
and the calculation at the end of the proof of Theorem \ref{theorem2} implies the contradiction that (\ref{e1}) does not hold with equality.
Hence, $d\in \{ 0,1\}$, and, if $d=1$, then, by Claim \ref{claim4}, $i_2=i_3=0$.
If $i_3\geq 2$, then arguing as in the proof of Claim \ref{claim7}
yields the existence of a graph $G'$ in ${\cal G}$ with $C(G')>C(G)$,
which is a contradiction.
Hence, $i_3\leq 1$.
If $i_2\geq 4$, then arguing as in the proof of Claim \ref{claim7}
yields the existence of a graph $G'$ in ${\cal G}$ with $C(G')=C(G)$ that is of type $(d',i_2',i_3')=(d,i_2-4,i_3+2)$.
Now, as observed above, $i_3'\geq 2$ implies the existence of a graph $G''$ in ${\cal G}$ with $C(G'')>C(G')=C(G)$,
which is a contradiction.
Hence, $i_2\leq 3$.
Finally,
if $i_2\geq 2$ and $i_3=1$, then arguing as in the proof of Claim \ref{claim7}
yields the existence of a graph $G'$ in ${\cal G}$ with $C(G')=C(G)$ that is of type $(d',i_2',i_3')=(d,i_2-2,i_3+1)$.
Again, as observed above, $i_3'\geq 2$ implies a contradiction.
Hence, $i_3=1$ implies $i_2\leq 1$.
Altogether, it follows that $G\in {\cal B}$,
which completes the proof. $\Box$

\medskip

\noindent Our final result shows that adding a single edge can increase the clustering coefficient of a graph from $0$ to almost $1$,
which means that it is a rather sensitive parameter.

\begin{theorem}\label{theorem4}
If $G$ is a graph of order $n$ at least $3$, and $u$ and $v$ are non-adjacent vertices in $G$, then
$$C(G+uv)\leq C(G)+\left(1-\frac{2}{n}+\frac{4}{n(n-1)}\right)$$
with equality if and only of $G$ is $K_{2,n-2}$, and $u$ and $v$ are of degree $n-2$ in $G$.
\end{theorem}
{\it Proof:} Since the statement is trivial if $d_G(u)=0$ or $d_G(v)=0$, we may assume that neither $u$ nor $v$ are isolated in $G$.
Let $G'=G+uv$.

If $w$ is a common neighbor of $u$ and $v$ in $G$, then the degree of $w$ in $G'$ equals the degree of $w$ in $G$
but $G'[N_{G'}(w)]$ contains exactly one edge more than $G[N_G(w)]$, which implies
\begin{align*}
C_w(G') - C_{w}(G) &=\frac{1}{{d_G(w)\choose 2}}\leq 1
\end{align*}
with equality if and only if $d_G(w) = 2$.

If $d_G(u) \geq 2$, then
\begin{align*}
C_u(G') - C_u(G) &= \frac{m(G'[N_{G'}(u)])}{{d_{G'}(u)\choose 2}} -\frac{m(G[N_G(u)])}{{d_G(u)\choose 2}}\\
&= \frac{m(G[N_{G}(u)]) + |N_G(u) \cap N_G(v)|}{{d_G(u)+1\choose 2}} -\frac{m(G[N_G(u)])}{{d_G(u)\choose 2}}\\
&= \frac{2(d_G(u) -1)|N_G(u) \cap N_G(v)|- 4 m(G[N_G(u)])}{(d_{G}(u)+1)d_{G}(u)(d_{G}(u)-1)}\\
&\leq \frac{2(d_G(u) -1)d_G(u)- 4 m(G[N_G(u)])}{(d_{G}(u)+1)d_{G}(u)(d_{G}(u)-1)}\\
&\leq \frac{2}{d_{G}(u)+1}
\end{align*}
with equality if and only if $N_G(u)\subseteq N_G(v)$ and $N_G(u)$ is independent.

If $d_G(u)=1$, then
$$C_u(G') - C_u(G)\leq 1=\frac{2}{d_G(u)+1}$$
with equality if and only if $N_G(u)\subseteq N_G(v)$ and $N_G(u)$ is independent.

We obtain that
\begin{align*}
n(C(G')-C(G)) &= (C_u(G')-C_u(G))+(C_v(G')-C_v(G))+\sum_{w\in N_G(u) \cap N_G(v)}(C_w(G')-C_w(G))\\
& \leq \frac{2}{d_G(u)+1}+\frac{2}{d_G(v)+1}+|N_G(u)\cap N_G(v)|
\end{align*}
with equality if and only if
$N_G(u)=N_G(v)$ is independent and every vertex in $N_G(u)$ has degree $2$ in $G$,
that is, $G$ is $K_{2,n-2}$, and $u$ and $v$ are of degree $n-2$ in $G$.
Since $d:=\min\{ d_G(u),d_G(v)\}\leq n-2$ and the function $f:\mathbb{N}\to\mathbb{R}:d\mapsto \frac{4}{d+1}+d$ is strictly increasing, we conclude
\begin{align*}
n(C(G')-C(G)) & \leq \frac{2}{d_G(u)+1}+\frac{2}{d_G(v)+1}+|N_G(u)\cap N_G(v)|\\
& \leq \frac{4}{d+1}+d\\
& \leq \frac{4}{n-1}+n-2,
\end{align*}
which completes the proof. $\Box$

redpajama_set_name: RedPajamaArXiv


SP&A acts as the principal observatory of the University's national and international academic performance. We produce analytics to inform major strategic developments in the University. Liaising with external bodies and agencies, we coordinate the University's statutory student number returns.
SP&A manages the University's strategic planning and resourcing processes.
SP&A assist departments to forward plan using key data to inform strategy.
Space Management and Timetabling are responsible for the allocation and effective utilisation of space across the University campus, the management of centrally timetabled rooms and the production of personalised timetables.


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These can be made using foil, ceramic or semiconductor gauges.
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You should have seen how much milk they downed afterwards.
We're pretty cute guys and gals. When we get Jack and Conor Maynard to come in and host a show for us, we treat them to an Easter Egg or two. N'AWH! We're so adorable, aren't we?
And because we really, really like them, we dipped them in... Extra hot sauce. Soz, fellas.
These two love a challenge, so we laid six hot sauces in front of them, and asked them to dip their chocolate eggs into them. Don't worry - we armed them with litres of milk, after Jack and Conor nearly cried for some.
The duo started off pretty chilled; by sweating profusely and nearly swearing into the microphones. And that was the pretty mild sauce.
Now, baring in mind Jack and Conor were chatting on the radio, it wasn't too good when we gave them the super, duper hot sauce and they legit couldn't talk for a good ten minutes.


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Delta Pharma Limited has implemented a comprehensive quality system (ISO 9001:2015). Quality Assurance (QA) procedures ensure that the quality of the products is maintained by strict compliance with international pharmacopoeias (BP, USP, etc) specifications on raw materials, packaging materials and finished products. This department enables implementation, assessment, and reporting which is necessary for guaranteeing that the products manufactured by maintaining cGMP and conform to the requirements and expectations of the customers.
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English Bulldog News Forums - English Bulldog Poetry: Contest Winner!
English Bulldog Poetry: Contest Winner!
Here is the winner of the May 2011 4 Paw Member short story contest!
It is a poem written by Libra926, about her beautiful bulldog Vegas.
I don't want to, but I NEED to.
I can't see my Daddy!
I can see..but I'm not sure what I'm seeing.
I can hear..but I'm not sure what I'm hearing.
I can smell..but I'm not sure what I'm smelling.
I can smell. I smell my mommy crying.
I can hear. I hear my mommy praying.
I can see. I see my mommy kneeling.
I can think. I think my mommy is hurting.
Very moving. Absolutely fantastic job!
That is the one that made me and I mean like a Congrats on winning....that is a beautiful poem.
Loved it, that was my pick. Congratualtions!!
I can only imagine what it feels like before, during and after one of these seizures. So often...he's blinded and the was he seeks out comfort is by burying his nose into you and smelling you. On more than one occasion..my husband and I are both by his side helping him thru it. Once the seizure is over..he buries his nose into my hands, only to move and bury his nose into my husband's side. It's very emotional..and that's where I found the words for this poem.
Congrats Libra, it really is a lovely tribute and hands down the best entry, although the others were fun to read too.
Congrads, great poem and love the pic. beautiful bully.
It made me cry, but smile to know that Vegas has so much love.
That is such a pretty poem and how ironic that you would understand exactly how it is to have a seizure from watching him and then put it into words.


Today marks rapper and all round entertainer Falz' birthday and to celebrate that he decided to drop a surprise album entitled "27" for the fans. off the album is this banging hiphop jam dubbed NEXT which features our very own Medikal and Nigerian singer and beatmaker Maleek Berry. cope it below..


text:
Enjoy a treat during K-W Humane Society fundraiser
Bake for the Animals will take place from February 24 to March 1
Feb 23, 2020 10:00 AM By: Ariel Deutschmann
The biggest and sweetest event for the Humane Society is back in the region.
The Kitchener-Waterloo and Stratford-Perth locations will host Bake for the Animals from February 24 to March 1.
To participate, residents can register to set up an online bakery page, selling treats to support the animals in the shelter.
Calla James is the senior manager of marketing and communications for The Humane Society of Kitchener-Waterloo & Stratford-Perth.
She says what began as a one-day cupcake sale has turned into a week long event with multiple goodies for both people and animals.
"We had such an inquiry about other things, 'Can I do cakes? Can I do doughnuts? Can I do pies?' " said James.
"We have so many incredible bakers in the area that people did expand into other things. Pup-cakes that come in are incredibly decorated and well-made that our animals enjoy."
According to James, each person who registers for the event receives access to downloadable party materials online and are mailed a host kit.
Groups or businesses can also participate during the week by planning their own bake sale, apply to be a sponsor, or make a donation.
With many ways to help out this year, Spencer says they are looking to surpass their fundraising goals of $46,000 for Kitchener-Waterloo and $10,000 for Stratford-Perth.
"It's a great online opportunity for people to contribute to," said James about Bake for the Animals, "We're hoping to beat those (goals) this year to help more animals that come through our doors."
To learn more about Bake For The Animals, or to join the event, click here.
About the Author: Ariel Deutschmann
Ariel Deutschmann is a feature writer and reporter who covers community events, businesses, social initiatives, human interest stories and more involving Guelph and Wellington County

redpajama_set_name: RedPajamaCommonCrawl


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text:
The Romulan Gambit – An Enterprise Story
They Went In Search of Peace – but found themselves charting the unknown possibilities of existence.
Chapter 2 : Recovery
Posted on March 20, 2020 May 23, 2020 by mjohnston55
Location: Military Assault Commander Operations, San Francisco
Year : 2156.70
They had been circling each other for three minutes. Neither was prepared to make a move. Judo was as much a mental as a physical discipline and each of them was looking for a way in, a way to take their opponent down, outwitting the other, both physically and mentally. He reached out with his left foot and went for her right, trying to take her off balance, roll her round and flatten her. The problem was, she knew what he was going to do and why.
As he went in, she stepped to one side, turned, dropped her right shoulder, pulled at the lapels of his Judogi and hurled him over her, the perfect Seol Nage shoulder throw. He came down with a thump on the mat, felt his neck and grimaced, shaking his head from side to side and, if he was honest, finding a new emotion. On this occasion, he was irritated with his failure.
She had outmanoeuvred him at every session for the last ten weeks but his response this time was different. There was no resignation or distraction, his adrenaline began to pump faster and he gave himself a precious thirty seconds to focus. He lifted himself off the mat and took his position. The command came – "rei" – and they bowed to each other. He moved toward her. This time, he was in the zone. Everything slowed down. He had more time to think, more time to plan and was ready for some lateral thinking. He moved in on her, grabbed her lapels and went to move his left foot forward. She sensed the move and was ready to repeat the shoulder throw. But it was a feint. This time his left foot darted across behind her left not her right foot. She lost her balance, fell back and he came down over her at right angles and pinned her down – the perfect Yokoshiho Katame, the four direction hold. The instructor waited to see if she could move free and finally called time: "Mate". He let go of her and oddly enough she gave him an encouraging smile, hardly the sign of an implacable Judo opponent.
The instructor reminded them they had not finished, "Sergeant, Private remember: best of three." The atmosphere in the training room changed; everyone was now watching. Water bottles were put to one side, Pads put down. Everyone wanted to see how this would end.
She came in swiftly aiming to grab his lapels and finish the bout quickly. He had other ideas. He moved his feet away from her, frustrating her first attempt. She realised his pace had doubled in these last two bouts and she was now up against a worthy opponent.
They had each other's lapels. An onlooker not familiar with Judo would think they were performing some kind of strange dance as they deftly probed with their feet: looking for a weakness or momentary lack of concentration.
Finally, she saw her chance; came in ready to twist and take him over her right shoulder. As this was his weak side, she was counting on him remaining defensive, not counter attacking. But she was wrong; he ducked down and with his left arm, held her turn and dropped his left shoulder. A combination of momentum and his grip, sent her flying over his left shoulder, rolling and crashing to the ground with a decisive thump. The MACOs present burst into spontaneous applause but the Private, rather than enjoying the moment, became alarmed. He went over and reached out with his hands to pull her up. "You OK?" he said. She took his hands smiled at him, replying "Just fine, but thank you." She sat up, but still holding his hands, rocked backwards using the momentum to throw him over her. Everyone laughed, including the Private, as he lay there flat out on the mat.
"OK, Team, a great day. It's all getting better, some good work today. 0800 hours for small fire arms practice. Well done everyone. And good work, Private."
The Private lifted himself up, went to grab his towel and made for the changing rooms. He had just reached the door of the training room when a voice behind him reached out to him. "Hey, Private: my place at seven. Bring a chilled bottle of white." The Private knew exactly what that meant. "Sure thing," he replied and without turning round, put up his right arm and held his thumb and finger in a gesture which communicated a simple message "A.O.K."
Location: Sky Tower, San Francisco Harbour
He hit the entry chime at 6.58 pm Pacific Daylight Time. The hydraulic hiss came seconds later. He held the bottle in his left hand with a big grin on his face. "Reportin' for duty ma'am." She took him in and just smiled. He wore a sand coloured pair of suede chinos and a burgundy silk top with a fold below the neck line, understated but classy. How things had changed, she thought. Despite the loose fitting top, she knew it covered a body in perfect shape, in fact in better shape than it had ever been. He broke the silence. "Well, am I gonna stand here all night or are you gonna invite me in?"
She reached out, grabbed him by the waist, pulled him in and lifted her face to his. They kissed, not as a prelude to anything but with a warmth, a sincerity and an openness. He still held the bottle in his hand as he moved his hand behind her back and slid it underneath her top and pressed hard. "Woah! That's cold, Private! Remove it at once!" He laughed, "Yes ma'am."
They let go of each other and walked across to the island kitchen unit in the centre of the apartment. She took the wine from him. "I'll put that in the wine chiller." He looked slightly crest-fallen, wondering if it was the wrong choice. "No, it's just fine! It's just we have something to celebrate," she said, interpreting his look and presenting him with a half bottle of champagne to open. As he opened it, she spoke, "You know, today was the first time you haven't compensated for that left arm?" A discreet <Pop> from the champagne, as he opened it with aplomb. "Hey, you're right! When I hit the mat, something happened to me today and I stopped overthinking."
She put two flutes on the island. He carefully poured and handed the first glass to her, as the bubbles began to settle. She took the glass and spoke, "It was strange for me, too. I wanted to win in front of the guys but I enjoyed seeing you get your mojo." Both glasses expertly poured, they now leaned against the island and stood looking at each other. "Congratulations, Soldier! You did good today." They brought the glasses together, clinked and lifted them to their lips, all the while looking across at each other, their eyes smiling. She put her glass down on the island and, rubbing the rim with her finger, looked up at him intensely. "This is more than your first win; it's about recovery." As he looked at her, he pursed his lips and nodded, "You're damn right." He picked up his glass again and looked at it and thought very carefully about what he was going to say next. "It's been a hell of a journey and no one has been more important to getting me here than you." He nodded as he offered his thanks to her. "I'm just glad I have been here for you," she acknowledged. "It really has been my privilege, if that doesn't sound too humble, to watch you turn it round."
He frowned, "Now, you know I don't like all that hero stuff. Part of this has been realising what a deep hole I was in and that I needed to do a lot of climbin' to get out. So I am not too happy with all that privilege stuff." She offered a resigned smile, "That's why I like you so much and why I wanted to help." Changing the subject, "I need to give the Chowder some attention; why don't you pop out on to the balcony? The screen's up and it's a beautiful night."
Charles Tucker the 3rd picked up his glass, walked over to the Observation Deck and slid the glass panel to one side. He took in the breathtaking sight of San Francisco Bay on a beautiful night in the mid 22nd Century. He looked West and took in the twinkling lights of the buildings, the traffic on the water (which he noted was light tonight) and the shuttles crossing the sky. The moon was out and the stars filled the sky. It sure was a beautiful night. He could hear one or two warning horns from the bigger craft on the water; yep, somethings will always be, whatever the technology .
As he stood there, his mind went back to a year ago. When he thought about the date, he realised it was almost a year to the day that they had sent little Elizabeth out into the heavens. Ambassador Soval had given the address at the child's funeral rite and his old friend, Jonathan Archer, had read some lines from Kahlil Gibran on sorrow.
Throughout, T'Pol and Tucker, the surrogate parents, had remained Bonded to ensure he kept his composure. But afterwards, when everyone had filed out, not having any idea what to say to them and they were alone, T'Pol let go.
Trip received it all. Rage, anger, despair: it all flooded into him. Despite his arm being in a sling he had used his good arm to pull her to him, as she bled internally. After what had seemed like an age of agony, the feelings began to dissipate and what she shared was in some respects more disturbing. T'Pol revealed to him she just didn't care anymore; she was completely lost. Finally, Trip verbalised. "My God, this is a mess." She placed her hands around his neck and they stood like that for quite some time. No sharing. Empty. Just a physical connection.
Eventually, she stood back from him and looked up at him, her eyes wide open. "I am leaving for Vulcan." He thought about that for a moment and let out an ironic sigh, turned his head to the side and nodded. These two had feigned and mis-routed each other many times in the last two years. Their relationship was so often about bluff, but he knew this was entirely different. He looked up to the Cargo Bay roof as if his next words were costing him deeply. "T'Pol, erm, you and I, we've been round the block a few times over the years." She nodded, "I agree." "But this? This is not good." he concluded. She simply nodded this time. "I just hope those guys, the priests or whatever they are, can help you." She looked down, deep in thought for several seconds before lifting her head. Raising an eyebrow with a resigned look on her face, "Who can say?" she uttered. Trip looked at her and realised this amazingly bright Science Officer was out of ideas, out of hope. Out of everything.
She turned to leave. The only sound in the Cargo Bay was the rustle of her robes, as she moved toward the exit. Just as she reached it, she turned and looked at him, "I have been able to make one deduction." Trip's eyebrows went up questioning. "I owe you an apology, Trip , I am… " and she hesitated to use such an informal human word, "… sorry." And with that, she turned and left.
Suddenly, he was shaken out of his reverie. "Hey, Private, you OK?" Charles Tucker realised he had drifted rather a long way. "Yeah, I'm good. But you know, reachin' this point – its made me kinda reflective." She looked hard at him, "Anything you want to share? The Chowder's ready, come and sit."
Private Tucker got stuck in. "Wow, Amanda this is the best yet! You've really got the hang of this. You have surpassed yourself." She smiled, "Praise indeed." He put his spoon down for a moment and wondered where to start, "Well, I am bound to wonder how I got from the shipwreck I was a year ago to now." "Of course, " she said, aware of the irony, "it's only… human."
"For one thing, Malcolm's father lending me his beach property in the Indian Ocean for three months? That was a Godsend. Being able to get some diving done and catch lobster without any of these Terra Prime idiots knowing where I was – that was perfect. And whilst I was there I did somethin' I have never done before, I read. It was one of Malcolm's Pop's books that gave me the idea to join the MACOs." "What do you mean?" She enquired. "Well, there was some British chap who was really famous in the First World War back in the Twentieth Century. He was serving out in the Middle East. Became a bit of a hero. Afterwards, he just wanted to be a regular guy in one of their military forces. It made me realise I wanted to do the same." He began eating again.
"I think there was more to it than that." Amanda Cole had become a little more serious. "Another reason why you joined the MACOs, I mean." She hesitated to say more. "Come on then," Tucker encouraged. She took a deep breath, "You needed to get your pride back in yourself." Tucker looked bemused and shook his head, "Whaddaya mean?" he questioned. Amanda ploughed on, "Look Trip, you don't mind if I call you Trip?" "Course not." She drove her point home: "Someone stole it and misused it." Trip stopped again and thought about the implications of what Amanda had said. "My God woman, you're right! Once I stopped thinking about that little munchkin I felt like, I dunno, like I'd suffered some kind of… rape."
She exhaled and relaxed, "Exactly! When I came back to Headquarters from my duty on Colombia, you were three months into the training. Six months on, it was obvious to me you were still in trouble. You had lost something; that spark I fell for to begin with…" Trip smiled and looked at her wistfully, "You're bang on, Amanda".
"God, that made me so damned angry," she intervened. "You'd become less of a man. Which you sure as hell are not and did not deserve." He smiled and looked at her. "Thank you, Amanda. I owe you one heck of a lot." He finished up his Chowder, "Now, beautiful lady, what's the next course?" She teased him. "You're such a Southern Charmer! Wait and see, Private. Wait and see…"
As he finished eating the second course he could not help but smile to himself. Amanda had just cooked him his favourite dish, Pan Fried Catfish, though he was glad she had left out the Hush Puppies, knowing how health conscious he now was. Somehow it was symbolic of everything about their relationship and how easily it came. She knew him because they came from the same area of Florida, even went to the same movie theatres when they were young. It was certainly different from T'Pol. He couldn't and didn't want to disguise his enthusiasm. "I can't believe you did that, how did you know?" She laughed, "Never you mind," reaching for his hand across the table, "You know that adage 'the way to a man's heart…' " She changed the subject. "Shall we rest up for a while, before dessert?" he nodded. "Yep, that would be great."
Trip got up and walked to the observation deck but this time turned to face South, looking down the Bay. She came up behind him and followed his gaze. He was looking at Star Fleet Headquarters. "It's not them is it? It's her." Trip turned to Amanda he knew it was time to talk. "Can we sit, darlin?"
They sat on one of the sofas on the roof terrace; Amanda pouring the last of the wine. "Amanda, what I'm gonna tell you is known only to five people. So this is for you only, promise me?" She nodded. "It's strange, because in some sense, you're part of this story." He paused, drawing breath and summoning up the memory. "During the search for the Xindi I had a lot of trouble sleeping and the Doctor suggested T'Pol helped me with some meditation and what not." Amanda smiled, "I remember – the neuro-pressure thing." "Yup, that's it." "Well, T'Pol and I became very close. She was a great comfort to me and I somehow became the focus of very strong emotions which she began to experience." Amanda laughed "Oh, you mean, like jealousy?" Trip smiled as only he could with that huge half moon look. "Yeah, you got it." "Well, one thing led to another and we were… intimate." Amanda became thoughtful. She knew this was answering a question that had been in her mind for several weeks, "I see." Trip looked at her. "I bet you don't."
"Vulcans may seem very uncomplicated compared to us, but the truth is it's the other way round. When a mate is chosen for a Vulcan in childhood, not all but some, with the help of their families, develop a Bond which is consummated when they physically mate. It's not merely an understanding or real affection beyond their legal obligations; it's more. Way more… " She continued to look puzzled, "I am still not with you." "They share each other's thoughts."
"Ok, but you're not a Vulcan and you haven't gone through this 'ritual'?" He put his glass down and held her hand, looking directly at her and hesitated. "I know… but we have it all … without even trying." She withdrew her hand and got up and walked over to the observation panel and looked out at the view. Everything was now crystal clear. She had puzzled over why Trip had not taken her to bed, now she knew. Whether he wanted to or not, she knew instinctively he couldn't.
"You know, Trip, I have always believed you were special. Not all that 'save the planet' stuff but you, the man: your integrity, your honesty, your honour." He was scratching his head, vaguely embarrassed at his confession to her, "Thank you."
Amanda turned to look at him. In the end, she reasoned to herself, T'Pol was a woman even if she was a Vulcan one and Amanda knew women would react to Trip in similar ways. "Trip, can you remember the last thing she said to you?" He frowned, "Oddly enough, yes. I was thinking of it earlier. She said she owed me an apology." Amanda's eyes darted about, "Do you know why she apologised?" He shook his head. "I do," she said. "It was because she realised, like the child, she had drawn you into something without your consent." Trip looked down for a moment and played the scene again in his mind. It wasn't a 'sorry, I am leaving you', it was an apology for not explaining the potential consequences of becoming intimate. Trip began to shake his head, finally realising what she was trying to tell him in those final moments together and looked up at Amanda. "My God, you're right!"
Amanda looked at Trip, "You have to believe me, Trip, this has nothing to do with my feelings but you need to find her and find a way of breaking this 'bond' between you. Only then can both of you move on… in every sense of the word."
PreviousChapter 3: Discovery
NextAct 1 : Chapter 1 : Romulus
The Romulan Gambit - An Enterprise Story


text:
Home Latest News Alkem Laboratories Gets EIR Report For Its Baddi Facility In Himachal Pradesh
Alkem Laboratories Gets EIR Report For Its Baddi Facility In Himachal Pradesh
Drug firm Alkem Laboratories on Friday said the US health regulator has issued an establishment inspection report (EIR) for its manufacturing facility at Baddi in Himachal Pradesh. The Baddi facility was inspected by the United States Food and Drug Administration (USFDA) from February 17 to 21, 2020, Alkem Laboratories said in a filing to BSE. The USFDA has now issued an EIR for the manufacturing facility, it added. On February 21, the company in a regulatory filing had said USFDA had issued a Form 483 with two observations at the end of the inspection of the facility. It had said it shall put together a detailed response with adequate corrective and preventive measures to address the USFDA observations.
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The Left Lies to Itself About NFL Ratings and Attendance
RUSH: To the phones, Kokomo, Indiana. This is Bill. It's great to have you, sir. Hi.
CALLER: Yes, sir. Great honor. Like you, I'm interested in the NFL ratings and how the politics plays into it, and so I was checking to see what the ratings were doing this year, and the first story I linked to on a Google search said that the ratings decrease this year was due to the protests started by the folks that were upset by Colin Kaepernick's treatment by the NFL. And I thought that was kind of funny because for like the last year, folks have been bending over backwards to try and tell us that the people who were boycotting the NFL because of Colin Kaepernick taking a knee, that had no effect.
RUSH: Wait a minute. I want to be sure I heard this right. You went on a Google search and they threw at you a USA Today story which says the reason the NFL is losing audience is because people are upset, not at Kaepernick, but rather upset at the way he's being treated by the NFL?
CALLER: Yes, sir. I believe it was — the USA Today affiliate. And they went on to list how many Twitter followers he has and 30,000 people this and 300,000 people that. I thought it was really funny, because all I've heard for a year was, well, everybody's watching because their cable is getting cut or they're cutting the cable, that Kaepernick had nothing to do with any of it.
RUSH: You know, that last point, cord cutting, that's happening faster than anybody thought. I don't think I printed this. I saw it over the weekend as I was perusing the tech blogs. So cut me some slack if I don't remember the number right. But it's — oh, this can't be. It's something like 35 or 40 million people have canceled satellite cable over like a six- or eight month period, in order to go streaming and so forth. I'm gonna have to find the story.
But regardless what the number is, people cutting the cord, that is canceling cable or canceling satellite. It is happening fast. It's all Millennials, most of it, and it's happening much faster than anybody thinks. That obviously is gonna have an effect on live TV ratings. There's another aspect of this, that millennials are watching TV but not live. You know, they'll watch, let's say The Last Ship. The Last Ship's a TNT show that airs on Sunday. They'll watch it, but during the week.
They won't watch it live as it is aired or broadcast live. And that's a ratings challenge. You know, find that viewer. Nielsen measures real-time viewing. They're trying to accommodate DVR, people that record programs on satellite or cable and watch 'em later, but streaming, that's a whole other challenge to get those. So it may well be that it's not that fewer people are watching, it's just how, and the ratings people are having trouble catching them.
On Kaepernick, this is classic. I can assure you the NFL's ratings are not down because people are mad at the NFL over how they're treating Kaepernick. The media is treating it that way. The media is mad at the NFL. The media is trying to get you mad at the NFL. The media has made Kaepernick a champion. And they're trying to peddle this story that Kaepernick is a great player and a team would be eminently helped if he would be their quarterback, but the league is filled with racist white supremacist owners and they're not gonna have somebody who's gonna protest the country or the flag.
They're trying to create that narrative, that viewers are actually in Kaepernick's corner. It's just the opposite. And I'm gonna tell you something else now. These pregame protests that take place on the sidelines, they're now beginning to happen in the game. And that ain't going to help. "What do you mean it's happening?" Well, one instance yesterday. The Bennett brothers, Martellus Bennett plays for the Green Bay Packers. His brother Michael Bennett plays for the Seattle Seahawks.
Michael Bennett was in Las Vegas recently, before the season started. He was in a strip club or some other similar type establishment where gunfire broke out. The cops were called. Among the people fleeing the scene after the cops told everybody to stop where they were was Michael Bennett. So the cops, entering a scene like that, gunfire, everybody is worried about being shot, so they tell everybody to stop, hold your ground, put down your guns. If anybody doesn't and starts running away, they're gonna get chased.
It's what happened, and Michael Bennett said they threatened to blow his F-ing head off and said they treated him like some third-rate citizen because of race. And his brother for the Packers, Martellus, who obviously supports him and a lot of NFL players support him. We don't know what really happened. I don't think he had anything to do with the event. It's just that he ran when they told everybody to shut up. Well, yesterday he tried to do the black power salute on the field after a play. I didn't see it. I don't know if he's an offensive lineman, defensive lineman. He made a sack or something.
They're telling us he gave the black power salute, but his fist wasn't closed. His hand up in the air, like John Carlos and the other guy at the Mexico Olympics in 1968, he did the black power salute, and it looked like he was emulating that. Whether he did or not, it was reported that he did. And so it has been reported that a player has continued to protest the racism of the NFL actually on the field. He does sit down and take a knee during the anthem and he has two Seahawks players on either side with their hands on his shoulder offering support, one black, one white.
They're making a spectacle of this, and it's growing, these protests are growing, they're not being cut back. Certain owners like Jerry Jones and Clark Hunt, Cowboys and Chiefs, are asking their players not to do this, to respect the country, to respect the flag because it's a respect for the fans. It's respect for the people who are paying to enter the stadium and watch the game. The owners are asking, "Please respect them, if you don't think you can respect the flag or the country." I don't know if there's been any protest of the Cowboys or Chiefs players.
But the idea that NFL viewership is down because fans are mad at the NFL because Kaepernick doesn't have a job is folly. It's absolute insanity. It's hilariously incorrect. But if they want to lie to themselves — and the left does this a lot — if they want to lie to themselves about what's really going on, if they want to lie to themselves about public opinion, if they want to lie to themselves about what the majority of people in this country think on any given issue, go right ahead.
You're gonna continue to lose elections. You're gonna continue to support candidates who are gonna lose elections. You go right ahead. You purposely want to misread the majority thought on culture, morality, patriotism, whatever it is, if you want to misread it in the sports media, in the Drive-By Media, you go right ahead. Be my guest.
RUSH: Kyle in Berryville, Virginia. Hi.
CALLER: Hi, Rush. Great to talk to you.
RUSH: Thank you, sir. How are you doing?
CALLER: Good. I'm doing good. I've been enjoying your show for a long time. I appreciate what you do.
RUSH: I appreciate that, sir, very much. I really do.
CALLER: So I wanted to give you a little bit of fuel for this NFL issue. I believe also that if you are lucky enough to be a professional athlete, it's only because you live in this country where you have that ability, and it's really a shame that some want to disrespect that. And I'm really glad to see that the attendance and everything has gone down and the people watching TV has gone down because of it. I heard you weigh in on that during your first hour. So I was eating lunch during your second hour, and the TV at the restaurant was on ESPN, of course, and they were also talking about the statistics of the poor attendance in LA. They attributed that, the very first reason, to the stadium being old, broken down, and in need of replacement, repair, whatever to get the attendance back up.
RUSH: Which stadium? The LA coliseum?
CALLER: Correct.
RUSH: Well, there's no doubt it's a rat trap. I mean, there's no doubt that it's old, but that excuse doesn't hold water because the day before, 81,000 people streamed in the place to watch USC-Texas.
CALLER: Exactly. And it was an awesome game, too.
RUSH: It was. A bunch of Texas people were staying at the hotel I was staying in in Beverly Hills. Those people, you know 'em when they get out of their limos. You don't need the big long horns on their hood. And it's not the ten-gallon hat. You know the Texans when they're walking around the hotel. You just know that they're there for — look, the LA Coliseum's been around a long time. It's not the best neighborhood. But USC doesn't have any problem. I've heard people say, "Rush, that's a different thing. You've got students who are there on the cheap. College fervor is much different than a capitalistic enterprise."
I don't care. They still sell tickets. You still have to drive there. You still have to party, buy the beer, the popcorn — well, beer at college games may not be allowed; I don't know. But still, you gotta buy the concessions. You gotta put up with all of it. You gotta sit on uncomfortable bench seats and all that stuff. You sit out in the sun, if that bothers you. It's the same circumstances the Rams or any other pro team has to deal with. You think about this, the NFL — you would think an enterprise this big would want be to honest when people are not watching on TV as much and not going to stadiums, you'd think they'd want to really know what's going. And by the way, did you see pictures of the crowd at the LA Coliseum on Sunday?
CALLER: I did.
RUSH: And the Stub Hub down in Carson where the Chargers were playing, 27,000. That wasn't filled. Last week the 49ers, brand-new state-of-the-art stadium, WiFi better than anywhere in the league, it's not filled. They're empty seats all over the league. The NFL is concerned about studio show now. There aren't any blackouts, every game's on TV, home games and so forth. I would want to know why people aren't watching or attending. I'd want to know the real reason so I could change it. And if they're lying to themselves, if they're thinking people are staying away because they're mad at the way the league is treating Kaepernick?
A sportswriter went to a game as a fan, figured the average cost, and I forget where, $500 to take a family to a football game. Now, given that's leisure dollars and that's a discretionary, you don't have to spend that. That's a lot of money to go to a football game where you might have to watch the flag being protested or you might have to watch the country being protested. If I were the NFL, I'd want to know what the truth is, wouldn't you?
CALLER: Yes, absolutely. It's just amazing to me how fast the excuses come out other than the real reasons we know they are.
RUSH: Well, ESPN, they've got a financial interest. I mean, they're paying over a billion dollars a year for one game a week. Monday night game. And they don't even get the best games on Monday Night Football. Those go to Sunday night on NBC, which doesn't pay as much as ESPN does.
New York Post: NFL Players Now Taking Their Protests Onto the Field
DeadLine: Emmy Awards Ratings Hit Low On Trump-Bashing Night; 'Sunday Night Football' Dips


text:
Taken / Taken 2
OPC: P6CDWS
Taken Liam Neeson stars in this action-packed thriller that will have you on the edge of your seat from start to finish. When his daughter is kidnapped in Paris, a former spy (Neeson) sets out to find her at any cost. Relying on his special skills, he tracks down the ruthless gang that abducted her...
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Taken Liam Neeson stars in this action-packed thriller that will have you on the edge of your seat from start to finish. When his daughter is kidnapped in Paris, a former spy (Neeson) sets out to find her at any cost. Relying on his special skills, he tracks down the ruthless gang that abducted her and launches a one-man war to bring them to justice and rescue his daughter. Taken 2 Liam Neeson returns as Bryan Mills, the retired CIA agent 'with a particular set of skills' who stopped at nothing to save his daughter Kim from Albanian kidnappers. When the Father of one of the kidnappers swears revenge and takes Bryan and his wife hostage during their family vacation in Istanbul, Bryan enlists Kim to help them escape... Taken Special Features: Making of Avant Premiere Inside Action: Side by Side Comparison Segments Real Time Mission Intelligence - Bonus View Taken 2 Special Features: Extended and Theatrical Feature Deleted Scenes Alternate Ending Black Ops Field Manual Sam's Tools of the Trade FX Movie Channel Presents: In Character with Liam Neeson
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According to Ixia, IxLoad-Wireless generates real-world traffic enabling network equipment makers and carriers to fully test the performance of LTE components and networks. IxLoad-Wireless supports multiple Cellular IoT interfaces including S1-Lite, a modified version of the existing S1-MME interface used for the control plane to signal or control messages, and the user plane the data being transferred between users, plus S11u for small data transmission. This allows IxLoad to emulate both the network and IoT user equipment (UE) to test IoT network nodes in isolation or the entire IoT ecosystem.
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I couldn't even stand to write anything about this game. Right now all I could stand is to show you the video. Chris Jones was blocked (fouled?) on a shot that he made just seconds before to put us in the lead.


Okja is directed by Bong Joon-ho and stars Ahn Seo-hyun the leading actress, as Mija.
In an alternate version of 2007 a Super Pig is announced that is meant to feed more people and leave less of an environmental impact than the average pig. 26 farms are given a Super Pig with a 10 year plan to find who can raise the Best Super Pig. 10 years later we meet Mija, a young girl who is raising Okja, one of the 26 Super Pig, with her grandfather.
Going into Okja I was expecting something of a modern retelling of Charlotte's Web, but I was pleasantly surprised to find that it was more than just that.
Okja's central theme is one of friendship. From the moment we first see Okja and Mija interact, it is clear that these two are more than just a farm girl and her cattle, they are friends. When Okja is taken from the farm, Mija does everything she can to get her back.
Unfortunately friendship is the only theme that seems to get closure. Other themes that pop up include those of loyalty, environmentalism, consumerism, and animal cruelty, but each are left with a degree of open-endedness.
The Animal Liberation Front (ALF) are an animal activist organization that are introduced early in the movie drive much of the plot. One of the side effects of this is that Mija spends most of her time getting caught up in the events that the ALF cause. Rather than giving Mija much of a purpose in the overall story, she ends up merely being used as a segue between scenes. Very little is affected by her presence and if she was removed the majority of the plot would remain the same.
While Mija played a small role in the grand scheme of things, she did manage to be the most enjoyable character. Mija displayed her relentless determination to save Okja with a pure passion in a way only a young child could. The other characters were much less likeable. Jake Gyllenhaal played a washed up educational animal show host that acts so overly emotional and mean-spirited that it becomes unbelievable. Tilda Swinton plays the CEO of the company responsible for the Super Pigs who flip flops between ideals so often it leaves the viewer confused on how they should interpret her actions.
One of the most memorable parts of the movie was the soundtrack. The tracks where Mija is chasing after Okja come to mind in particular. The mix of accordion and piano was unexpected but fit the tone perfectly. The sense of urgency that Mija gave off was heightened by the music that felt almost out of place, similar to how Mija must have felt out of place in the big city after spending all her time in the countryside. In general, the music was pleasant and added to the experience.
The visual effects for the movie were great. Every movement from Okja seemed like a real animal's and the mix of CG and practical effects are blended nearly seamlessly.
In the end, Okja is a fun, heartwarming movie that brings up questions of morality and quickly shoots them back down. The visuals and soundtrack were great, but the acting wasn't always consistent.


text:
Rothschild Family Treasures, Some Belonging to Marie Antoinette, Come to Christie's
The lots being offered by Christie's in London next week bear eloquent witness to classic Rothschild splendor
By Mitchell Owens
Jean-Honoré Fragonard, Dans les blés. (Estimate: £700,000–£1,000,000)
When it comes to surnames, Rothschild means something to everyone, namely either international banking or top-flight wines. But the five brilliant, adventuresome Rothschild brothers who emerged from Frankfurt am Main's ghetto in the early 19th century to establish financial footholds in Frankfurt, Vienna, Naples, Paris, and London also had considerable domestic panache.
"It's the only family that has its own decorative style, le gôut Rothschild," says Paul Gallois, head of European furniture at Christie's. On Wednesday, July 4, the firm's London showroom unveils "Masterpieces From a Rothschild Collection," an auction of more than 50 treasures, many of which once belonged to Baron Gustave de Rothschild, a member of the French branch of the family, who died in 1911.
A Louis XVI ormolu, Sèvres porcelain, and marquetry gueridon, circa 1782–83. (Estimate: £400,000–£600,000)
The Rothschilds, in fact, were remarkably consistent when it came to the decoration of their rooms, establishing a notably sumptuous look in the 1850s that was carried through from generation to generation for more than a century, whichever tastemaker, from artist Eugène Lami to decorator Henri Samuel to architect Renzo Mongiardino, they happened to commission. From English country houses to Paris hôtels particuliers to Vienna palaces, the clan embraced architectural grandeur, old-master paintings in gilded frames, costly fabrics in luxurious tones, thousands of rare books bound in Morocco leather, and cross-collecting treasure troves of 17th- and 18th-century furniture, exceptional Judaica, and exquisite objets d'art from multiple periods, often equipped with staggering provenance. It was an incredibly rich, even operatic vocabulary that stunned reigning monarchs—Napoléon III and Queen Victoria, among them—and, as Gallois observes, "it's still alive in a way," thanks to similarly luxurious decor concocted by design stars such as Alidad, Jacques Garcia, and Studio Peregalli.
Commissioned by Queen Marie Antoinette of France: a Louis XVI ormolu-mounted mahogany table à écrire, circa 1780, by Jean-Henri Riesener. (Estimate: £600,000–1,000,000)
Vincent Girier Dufournier
The lots being offered by Christie's next week bear eloquent witness to classic Rothschild splendor, including a magnificent pair of circa-1713 tortoiseshell-veneer cabinets on stands (estimated to bring £1,500,000–£2,500,000) that belonged to Philip V of Spain, a 1567 Léonard Limosin 1567 parcel-gilt polychrome enamel charger depicting the Israelites gathering manna from heaven (£120,000–£180,000), and two Louis XV ormolu-mounted Japanese lacquer encoignures, or corner cabinets, made by Bernard III van Risenburgh (£300,000–£500,000) that came into the family through Baron Maurice de Rothschild.
Gallois calls those Risenburgh cabinets his favorite items. Then again, he admits that he has an affection for a spectacular rock crystal casket, or lidded box, a circa-1600 Venetian creation that Christie's explains was "traditionally used to house costly baby linen blessed by the Pope, and given by him to leading European Catholic families on the birth of a male heir." Estimated to bring as much as £150,000 and arguably presented to its original owner by Pope Paul V, the casket eventually ended up in the hands of the scandal-dogged British connoisseur William Beckford in the early 19th century, and passed into the hands of the dukes of Westminster before Baron Gustave de Rothschild acquired it.
A Venetian rectangular parcel-gilt, gilt-bronze and rock crystal casket, circa 1600. (Estimate: £100,000–150,000).
"The Rothschild name is the very best when it comes to collecting," Gallois observes, and given the evidence—including a delicious Jean-Honoré Fragonard oil painting—who's to argue? Other pieces in the sale belonged to Marie Antoinette of France, George III of England, and Prince Anatoly Demidoff, a Russian collector of great renown. There's also a spectacular circa-1640 Savonnerie carpet described as "by descent" from banker Léon Fould, an identification which suggests the origins of Christie's July 4 offerings. Fould's granddaughter Liliane Fould-Springer was the wife of Baron Élie de Rothschild, a grandson of Baron Gustave, so it's likely that the consignors are the late couple's children.


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Definition - What does Tuck-In Acquisition mean?
A tuck-in acquisition refers to a company that is acquired by a platform company that is usually backed by private equity. The term, tuck-in, refers to the idea that the acquisition is being "tucked in" under the infrastructure of the platform company.
When private equity enters a market, it may do so by acquiring a larger company with management and infrastructure capable of absorbing other smaller players. This larger company serves as the platform, and then tuck-in acquisitions are completed to add to the geographic footprint or add complementary services or technology. Tuck-in acquisitions usually have a strong owner, but limited management depth or administrative resources to continue growing, which makes them ideal for larger platform companies to absorb.
While a tuck-in acquisition may work for a seller who is looking to let go of all administration, the process represents a considerable change. The platform company will look to integrate a tuck-in company as quickly as possible into common accounting systems, treasury and operating procedures. Usually, a quick integration win is to revise the pricing of products or services up to the platform company's rate sheet.
While changes are aimed at creating value in the long run, the owner of the tuck-in company needs to be prepared for the significant change that occurs in the first year, and ensure that he/she manages appropriately how the rest of the employees will react to the integration process.


At Stark & Wayne we are constantly thinking, planning, exploring and working towards the ultimate continuous deployment pipelines for our production systems & our customer's production systems.
Dr Nic spoke at CF Summit 2015 on this topic and his slides are now online.
In the video, we are using a Concourse pipeline to deploy a Redis cluster three times. The goal is to only do a deployment/upgrade/reconfiguration of the production Redis cluster if it the "deployment change" has been successfully applied already.
In the demo I have a "try anything" stage where I deploy a cluster of Redis using the latest assets/configuration coming in from upstream/other dev teams.
If that deployment and acceptance tests are ok, then I move to "pre-production". Here, the goal is to re-create "production" and perform an upgrade to the successful "try anything" configuration/packages.
If "pre-production" is successful (and conceptually it can be destroyed at this time), then we have a candidate for the next "production" deployment.
In the demo this production deployment is triggered automatically; though in many organizations a human might want to push the button. Humans need to do something right?
This demo is also a demonstration of BOSH deployments; though hopefully this is orthoganal to the demonstration of Concourse. Concourse was written by people who use BOSH so it currently comes with extensions (called "resources") that work with BOSH.
If you're interested in using Concourse to automate your own assets and production deployments then contact us to help. We think Concourse is a very exciting next-generation open source continously deployment solution.


Italy-based OCME presents a range of pallet checkers designed to improve palletizer performance and productivity.
OCME pallet checkers can reduce palletizer downtime by detecting pallet faults before they are loaded with product and can improve pallet quality.
Traditional OCME pallet checkers employ a single pass system that uses pressure rollers to detect cracked or broken boards and reject substandard pallets.
The typical OCME system consists of a pallet de-stacker, infeed/ exit conveyors, reject system and an integration system to link it to the palletizer.
Robotic pallet checkers employ a six-axis robot to check every board on both sides of the pallet for defects, and nail heads are pressed into the board to prevent subsequent damage to packs or containers loaded on the pallets.
OCME, represented by HBM Packaging in Oceania, is involved in packaging, filling, handling and palletizing systems for the consumer products industry.