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% produkt.tex
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% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\bgroup
\definecolor{darkgreen}{rgb}{0,0.6,0}
\begin{frame}[t]
\frametitle{Inzidenz- und Laplace-Matrix}
\vspace{-20pt}
\begin{columns}[t,onlytextwidth]
\begin{column}{0.40\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick]

\def\r{2.2}

\coordinate (A) at ({\r*cos(0*72)},{\r*sin(0*72)});
\coordinate (B) at ({\r*cos(1*72)},{\r*sin(1*72)});
\coordinate (C) at ({\r*cos(2*72)},{\r*sin(2*72)});
\coordinate (D) at ({\r*cos(3*72)},{\r*sin(3*72)});
\coordinate (E) at ({\r*cos(4*72)},{\r*sin(4*72)});

\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (C);
\draw[color=white,line width=5pt] (B) -- (D);
\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (D);

\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (A) -- (B);
\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (B) -- (C);
\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (C) -- (D);
\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (D) -- (E);
\draw[->,shorten >= 0.2cm,shorten <= 0.2cm] (E) -- (A);

\draw (A) circle[radius=0.2];
\draw (B) circle[radius=0.2];
\draw (C) circle[radius=0.2];
\draw (D) circle[radius=0.2];
\draw (E) circle[radius=0.2];

\node at (A) {$1$};
\node at (B) {$2$};
\node at (C) {$3$};
\node at (D) {$4$};
\node at (E) {$5$};
\node at (0,0) {$G$};

\node at ($0.5*(A)+0.5*(B)-(0.1,0.1)$) [above right] {$\scriptstyle 1$};
\node at ($0.5*(B)+0.5*(C)+(0.05,-0.07)$) [above left] {$\scriptstyle 2$};
\node at ($0.5*(C)+0.5*(D)+(0.05,0)$) [left] {$\scriptstyle 3$};
\node at ($0.5*(D)+0.5*(E)$) [below] {$\scriptstyle 4$};
\node at ($0.5*(E)+0.5*(A)+(-0.1,0.1)$) [below right] {$\scriptstyle 5$};
\node at ($0.6*(A)+0.4*(C)$) [above] {$\scriptstyle 6$};
\node at ($0.4*(B)+0.6*(D)$) [left] {$\scriptstyle 7$};

\end{tikzpicture}
\end{center}
\vspace{-15pt}
\begin{block}{Berechne}
%\vspace{-20pt}
\begin{align*}
\uncover<4->{L(G)}&\uncover<4->{=}B(G)B(G)^t
\end{align*}
\end{block}
\end{column}
\begin{column}{0.58\textwidth}
\begin{center}
\begin{tikzpicture}[>=latex,thick]

\def\dx{0.84}
\def\dy{0.48}

\node at (0,0) {$\displaystyle
\begin{aligned}
B(G)
&=
\begin{pmatrix*}[r]
-1& 0& 0& 0&+1&-1& 0\\
+1&-1& 0& 0& 0& 0&-1\\
 0&+1&-1& 0& 0&+1& 0\\
 0& 0&+1&-1& 0& 0&+1\\
 0& 0& 0&+1&-1& 0& 0
\end{pmatrix*}
\\[20pt]
\uncover<2->{
L(G)
&=
\begin{pmatrix*}[r]
 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{-1}\\
\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{-1}&\uncover<3->{ 0}\\
\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}&\uncover<3->{ 0}\\
\uncover<3->{ 0}&\uncover<3->{-1}&\uncover<3->{-1}& 3&\uncover<3->{-1}\\
\uncover<3->{-1}&\uncover<3->{ 0}&\uncover<3->{ 0}&\uncover<3->{-1}& 2
\end{pmatrix*}}
\end{aligned}$};
\end{tikzpicture}
\end{center}
\end{column}
\end{columns}
\end{frame}
\egroup