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\title{General Mathematics and Computational Science II}
\author{Midterm Exam}
\date{March 14, 2014}
\maketitle

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\begin{enumerate}

\item Show that if $v$ is a vertex of odd valency in a finite graph,
then there exists a path connecting $v$ to another vertex $u$ of odd
valency.  
\hfill (10)

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\item Show that a tree with more than one vertex is bipartite.
\hfill (10)

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\item The structure of chemical molecules can be considered as a graph.
Show that the graph of pentanol, an alcohol with the chemical formula
$\ce{C_5 H_{11} OH}$, is a tree.

\emph{Note:} Chemists know that Carbon (C), Oxygen (O), and Hydrogen
(H) atoms have valencies $4$, $2$, and $1$, respectively.  Subscripts
in the chemical formula denote the number of each atom in the
molecule.
\hfill (10)

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\item Identify the symmetries and corresponding frieze groups for the
following five ornaments.  Further, clearly draw all centers and lines
of symmetry into the picture.

\begin{center}
\includegraphics[width=8cm]{friezepatterns-umble.pdf}
\end{center}
{\small(From R.N. Umble, \emph{Transformational Plane Geometry}, \\ \url{http://www.millersville.edu/~rumble/Math.355/Book/TPG-Spring2012.pdf}.)}
\hfill (10)

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\item 
\begin{enumerate}
\item Show that $V = \{1,3,5,7\}$ endowed with multiplication modulo
$8$ is a group.

\item Show that $V$ is isomorphic to the dihedral group
\[
  D_2 = \langle a, b \mid a^2=b^2=e, ba=a^{-1} b \rangle \,.
\]

\item Can you think of a realization of this group as the symmetry group
of some geometric shape?

\end{enumerate}
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\item Let $G$ be the group generated by $\Phi_A$ and $\Phi_B$, two
rotations about different centers of rotation $A$ and $B$.  Show that
$G$ contains a translation.  \hfill(10)
\end{enumerate}
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