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Vorschau:
Neuer Abschnitt in Beweglichkeit eines Streichholzgraphen bestimmen

Nun möchte ich mit dieser Methode die Beweglichkeit eines Streichholzgraphen bestimmen. Wie ich die Gleichungssystem aufstelle, siehe im Thread Streichholzgraphen 4-regulär und 4/n-regulär (n>4) und 2/5 den Beitrag No.251. Dann fülle ich die Koeffizientenmatrix mit Nullspalten und Nullzeilen zu einer quadratischen Matrix auf. Dann addiere ich beginnend bei a11 von links oben nach rechts unten zu jedem aii eine 1, wenn die Teilmatrix von a11 bis aii (führende Hauptminoren) noch nicht invertierbar ist, und überprüfe die drei Voraussetzungen. Dadurch ändern sich manche der aii+1 in andere akl+1 oder entfallen wieder. Die unterstrichenen 0 und 0 geben solche Positionen an, wo ich ursprünglich 0+1 und 0+1 eingesetzt hatte, diese nach dem Überprüfen der Voraussetzungen aber nicht geblieben und auch nicht entfallen sind, sondern aus je zwei ein einziges aij+1 geworden ist, verschoben innerhalb der gleichen Zeile/Spalte. Damit möchte ich wenigstens etwas verdeutlichen, wie diese Ersetzungen mit der Veränderung der Beweglichkeit einhergehen. Genauere Gedanken habe ich mir darüber nicht gemacht. Die neu hinzukommenden Kante und deren Koordinatendifferenzen sind gleichfarbig hervorgehoben. Ganz rechts in der Tabelle bedeuten Spaltenbezeichnung A = Anzahl der aij+1, F = Anzahl der Freiheitsgrade des Graphen.

GraphGleichungssystemAF
\begin{tikzpicture} \draw [cyan] (0,0) node [below] {P1} -- (1,0) node [below] {P2} ; \end{tikzpicture} \begin{pmatrix} F_{1x}\\F_{1y}\\F_{2x}\\F_{2y}\end{pmatrix} = \begin{pmatrix} \textcolor{cyan}{1}&0&0&0\\ \textcolor{cyan}{0}&0\textbf{+1}&0&0\\ \textcolor{cyan}{-1}&0&0\textbf{+1}&0\\ \textcolor{cyan}{0}&0&0&0\textbf{+1} \end{pmatrix}\begin{pmatrix} f_{12}\\0\\0\\0\end{pmatrix} 3 3
\begin{tikzpicture} \draw (0,0) node [below] {P1} -- (1,0) ; \draw [cyan] (1,0) node [below] {P2} -- (1,1) node [above] {P3} ; \end{tikzpicture} \begin{pmatrix} F_{1x}\\F_{1y}\\F_{2x}\\F_{2y}\\F_{3x}\\F_{3y}\end{pmatrix} = \begin{pmatrix} 1&0&0&0&0&0\\ 0&\textcolor{magenta}{\underline{0}}}&0&\textcolor{magenta}{0\textbf{+1}}&0&0\\ -1&\textcolor{cyan}{0}&0\textbf{+1}&0&0&0\\ 0&\textcolor{cyan}{1}&0&\textcolor{magenta}{\underline{0}}&0&0\\ 0&\textcolor{cyan}{0}&0&0&0\textbf{+1}&0\\ 0&\textcolor{cyan}{-1}&0&0&0&0\textbf{+1} \end{pmatrix}\begin{pmatrix} f_{12}\\f_{23}\\0\\0\\0\\0\end{pmatrix} 4 4
\begin{tikzpicture} \draw (0,0) node [below] {P1} -- (1,0) node [below] {P2} -- (1,1) ; \draw [cyan] (1,1) node [above] {P3} -- (0,1) node [above] {P4} ; \end{tikzpicture} \definecolor{zusatzgruen}{rgb}{0.1,0.8,0.1} \begin{pmatrix} F_{1x}\\F_{1y}\\F_{2x}\\F_{2y}\\F_{3x}\\F_{3y}\\F_{4x}\\F_{4y}\end{pmatrix} = \begin{pmatrix} 1&0&0&0&0&0&0&0\\ 0&\textcolor{magenta}{\underline{0}}&0&\textcolor{magenta}{0\textbf{+1}}&0&0&0&0\\ -1&0&\textcolor{zusatzgruen}{\underline{0}}&0&\textcolor{zusatzgruen}{0\textbf{+1}}&0&0&0\\ 0&1&0&\textcolor{magenta}{\underline{0}}&0&0&0&0\\ 0&0&\textcolor{cyan}{-1}&0&\textcolor{zusatzgruen}{\underline{0}}&0&0&0\\ 0&-1&\textcolor{cyan}{0}&0&0&0\textbf{+1}&0&0\\ 0&0&\textcolor{cyan}{1}&0&0&0&0\textbf{+1}&0\\ 0&0&\textcolor{cyan}{0}&0&0&0&0&0\textbf{+1} \end{pmatrix}\begin{pmatrix} f_{12}\\f_{23}\\f_{34}\\0\\0\\0\\0\\0\end{pmatrix} 5 5
\begin{tikzpicture} \draw (0,0) -- (1,0) node [below] {P2} -- (1,1) node [above] {P3} -- (0,1) ; \draw [cyan] (0,1) node [above] {P4} -- (0,0) node [below] {P1} ; \end{tikzpicture} \definecolor{zusatzgruen}{rgb}{0.1,0.8,0.1} \begin{pmatrix} F_{1x}\\F_{1y}\\F_{2x}\\F_{2y}\\F_{3x}\\F_{3y}\\F_{4x}\\F_{4y}\end{pmatrix} = \begin{pmatrix} 1&0&0&\textcolor{cyan}{0}&0&0&0&0\\ 0&0&0&\textcolor{cyan}{1}&0&0&0&0\\ -1&0&\textcolor{zusatzgruen}{\underline{0}}&0&\textcolor{zusatzgruen}{0\textbf{+1}}&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&-1&0&\textcolor{zusatzgruen}{\underline{0}}&0&0&0\\ 0&-1&0&0&0&0\textbf{+1}&0&0\\ 0&0&1&\textcolor{cyan}{0}&0&0&0\textbf{+1}&0\\ 0&0&0&\textcolor{cyan}{-1}&0&0&0&0\textbf{+1} \end{pmatrix}\begin{pmatrix} f_{12}\\f_{23}\\f_{34}\\f_{41}\\0\\0\\0\\0\end{pmatrix} 4 4
\begin{tikzpicture} \draw (0,0) -- (1,0) node [below] {P2} -- (1,1) -- (0,1) node [above] {P4} -- (0,0) ; \draw [cyan] (0,0) node [below] {P1} -- (1,1) node [above] {P3} ; \end{tikzpicture} \begin{pmatrix} F_{1x}\\F_{1y}\\F_{2x}\\F_{2y}\\F_{3x}\\F_{3y}\\F_{4x}\\F_{4y}\end{pmatrix} = \begin{pmatrix} 1&0&0&0&\textcolor{cyan}{1}&0&0&0\\ 0&0&0&1&\textcolor{cyan}{1}&0&0&0\\ -1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&-1&0&\textcolor{cyan}{-1}&0&0&0\\ 0&-1&0&0&\textcolor{cyan}{-1}&0\textbf{+1}&0&0\\ 0&0&1&0&0&0&0\textbf{+1}&0\\ 0&0&0&-1&0&0&0&0\textbf{+1} \end{pmatrix}\begin{pmatrix} f_{12}\\f_{23}\\f_{34}\\f_{41}\\f_{13}\\0\\0\\0\end{pmatrix} 3 3
\begin{tikzpicture} \draw (0,0) node [below] {P1} -- (1,0) -- (1,1) node [above] {P3} -- (0,1) -- (0,0) -- (1,1) ; \draw [cyan] (1,0) node [below] {P2} -- (0,1) node [above] {P4}; \end{tikzpicture} \begin{pmatrix} F_{1x}\\F_{1y}\\F_{2x}\\F_{2y}\\F_{3x}\\F_{3y}\\F_{4x}\\F_{4y}\end{pmatrix} = \begin{pmatrix} 1&0&0&0&1&0&0&0\\ 0&0&0&1&1&0&0&0\\ -1&0&0&0&0&\textcolor{cyan}{-1}&0&0\\ 0&1&0&0&0&\textcolor{cyan}{1}&0&0\\ 0&0&-1&0&-1&0&0&0\\ 0&-1&0&0&-1&0\textbf{+1}&0&0\\ 0&0&1&0&0&\textcolor{cyan}{1}&0\textbf{+1}&0\\ 0&0&0&-1&0&\textcolor{cyan}{-1}&0&0\textbf{+1} \end{pmatrix}\begin{pmatrix} f_{12}\\f_{23}\\f_{34}\\f_{41}\\f_{13}\\f_{24}\\0\\0\end{pmatrix} 3 3

Es ist absehbar, auf welche Vermutung das hinauslaufen soll. Die Anzahl der aij+1 stimmt mit der Anzahl der Freiheitsgrade überein (Ausnahme siehe nächsten Absatz) und wenn man davon die 3 Freiheitsgrade für den gesamten Graph abzieht, bleibt ein Ergebnis >0 übrig, wenn der Graph in sich beweglich ist und 0 bei einem starren Graph.

Ausnahme: Wenn der Graph mehr als doppelt so viele Kanten wie Punkte enthält (zum Beispiel diese 4-regulären Streichholzgraphen mit 60 Knoten und 121 Kanten), dann müssen Nullzeilen an die Koeffizientenmatrix angefügt werden, um sie quadratisch zu machen. In dem Fall zähle ich die aij+1 aus diesen angefügten Nullzeilen nicht mit.

Für den ehemalig zweitkleinsten 4-regulären Streichholzgraph mit 60 Knoten und 120 Kanten im Artikel Ein 4-regulärer Streichholzgraph mit 114 Kanten unter Punkt "3." erhalte ich als Ergebnis 4 Freiheitsgrade. Abzüglich der 3 Freiheitsgrade für den gesamten Graph bleibt 1 Freiheitsgrad übrig, der Graph ist in sich einfach beweglich. Die Eingabe des Graphen erfolgte mit Streichholzgraph_zum_Artikel.htm und die Berechnung mit dem extra (GAP-)Programm. Nachfolgend einige Bewegungsphasen, in der Mitte der Originalgraph:

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