\include{preamble}
\setcounter{try}{43}




\pagestyle{empty}
\begin{document}
\thispagestyle{empty}
\centerline{\bf Math 365 -- Wednesday 4/10/19 -- 10.1 \& 10.2 Graphs}


\bigskip

\begin{try}(Relations and digraphs)
For each the relations in Exercise 43(a), draw the corresponding directed graph where 
$V = \{0,1,2,3\}$ and 
$$a \to b \qquad \text{ if } \qquad a \sim b.$$
What properties of the directed graphs correspond to the symmetric, reflexive, and transitive properties of the corresponding relations? For the digraphs corresponding to equivalence relations, what do the equivalence classes look like?
\end{try}

\vfill

\begin{try}(Applied graphs)
Pick four of the Examples 2--12 in Section 10.1, and quickly summarize them. What is $V$? What is $E$? And what kind of graph results?
For example, in Example 1, 
\begin{align*}
	V &= \{ \text{ people } \}\\
	E &= \{ a\!-\!b ~|~ a \neq b,\text{ and $a$ and $b$ are acquainted }\}
\end{align*}
The resulting graph is simple. 
\end{try}

\vfill

\begin{try}
Let 
$$G = \TikZ{
\node[bV, label=left:{$a$}] (a) at (0,1) {};
\node[bV, label=below:{$b$}] (b) at (1,2) {};
\node[bV, label=below:{$c$}] (c) at (2,1) {};
\node[bV, label=below:{$d$}] (d) at (3,1) {};
\draw (b) .. controls +(.5,.5) and +(-.5,.5) .. (b);
\draw (a) to (c) (b) to (c); 
\draw (a) to [bend right] (c);
\draw (a) to (b);
\draw (c) to (d);
\draw (d) to [bend right] (c);
}
\qquad
H=
\TikZ{
	\node[label=above:{$u$}, bV] (1) at (-.5,1) {};
	\node[label=above:{$v$}, bV] (2) at (.5,1) {};
	\node[label=right:{$w$}, bV] (3) at (1,0) {};
	\node[label=below:{$x$}, bV] (4) at (0,-1) {};
	\node[label=left:{$y$}, bV] (5) at (-1,0) {};
\draw (1) to (4) to (5) to (1) to (3) to (5) to (2) to (3) to (4);
} 
$$
\begin{enumerate}[(a)]
\item In $G$, what is the neighborhood of $a$? What is the neighborhood of $b$?
\item Calculate the degrees of each vertex in $G$ and $H$.
\item Verify the handshake  theorem on $G$ and $H$. 
\end{enumerate}
\end{try}
\vfill

\pagebreak


\begin{try}~
\begin{enumerate}[(a)]
\item Draw $C_6$, $W_6$ $K_6$, and $K_{5,3}$. 

\medskip

\item Which of the following are bipartite? Justify your answer.
$$\begin{matrix}\begin{tikzpicture}[scale=.75]
\node[bV] (a) at (1,2) {};
\node[bV] (b) at (0,1) {};
\node[bV] (c) at (1,0) {};
\node[bV] (d) at (2,1) {};
\node[bV] (e) at (1,1) {};
\draw(a) to (e) (b) to (e) (c) to (e) (d) to (e);
\end{tikzpicture}\end{matrix}\qquad \qquad \begin{matrix}\begin{tikzpicture}[scale=.75]
\node[bV] (a) at (1,2) {};
\node[bV] (b) at (0,1) {};
\node[bV] (c) at (1,0) {};
\node[bV] (d) at (2,1) {};
\node[bV] (e) at (2,2) {};
\draw(a) to (b) (a) to (d) (a) to (e) (c) to (b) (c) to (e) (c) to (d);
\end{tikzpicture}\end{matrix}\qquad \qquad \begin{matrix}\begin{tikzpicture}[scale=.75]
\node[bV] (a) at (1,2) {};
\node[bV] (b) at (0,1) {};
\node[bV] (c) at (1,0) {};
\node[bV] (d) at (2,1) {};
\node[bV] (e) at (2,2) {};
\node[bV] (f) at (3,1) {};
\draw(a) to (b) (a) to (f) (b) to (e) (b) to (d) (c) to (f) (c) to (d) (d) to (e) (e) to (f);
\end{tikzpicture}\end{matrix}$$


 


\medskip

\item Hypercubes are bipartite. 



\medskip

\begin{enumerate}[(i)]
\item The following is the $4$-cube:
$$\begin{tikzpicture}[xscale=3.1,yscale=1.9]
	 \tikzstyle{vertex}=[draw,circle,black,minimum size=20pt,inner sep=1pt, fill=white]
	 \tikzstyle{vertexA} = [vertex]%, fill=red!24
	 \tikzstyle{edge1} = [draw=white,double=black,line width=2pt, double distance=2pt]
	 \tikzstyle{edge2} =  [draw=white,double=black,line width=2pt, double distance=2pt]
	 \tikzstyle{edge3} =  [draw=white,double=black,line width=2pt, double distance=2pt]
	 \tikzstyle{edge4} =  [draw=white,double=black,line width=2pt, double distance=2pt]
	 \tikzstyle{edge} = [draw,thick,-,black]
	 \node[vertexA] (1000) at (0,0) {$1000$};
	 \node[vertex] (1010) at (0,1) {$1010$};
	 \node[vertex] (1100) at (1,0) {$1100$};
	 \node[vertexA] (1110) at (1,1) {$1110$};
	 \node[vertex] (0000) at (0.23, 0.4) {$0000$};
	 \node[vertexA] (0010) at (0.23,1.4) {$0010$};
	 \node[vertexA] (0100) at (1.23,0.4) {$0100$};
	 \node[vertex] (0110) at (1.23,1.4) {$0110$};
	 \node[vertex] (1001) at (-1,-1) {$1001$};
	 \node[vertexA] (1011) at (-1,2) {$1011$};
	 \node[vertex] (0011) at (-0.66,2.7) {$0011$};
	 \node[vertexA] (0001) at (-0.66,-0.3) {$0001$};
	 \node[vertexA] (1101) at (2,-1) {$1101$};
	 \node[vertex] (0101) at (2.34,-0.3) {$0101$};
	 \node[vertex] (1111) at (2,2) {$1111$};
	 \node[vertexA] (0111) at (2.34,2.7) {$0111$};
	 \draw[edge4] (0000) -- (0001) (0100) -- (0101) (0110)--(0111) (0010)--(0011);
	 \draw[edge3] (0000)--(0010)  (0100)--(0110);	 
	 \draw[edge1] (1000) -- (0000) (0100) -- (1100);
	 \draw[edge2] (1000) -- (1100) (0000) -- (0100);
	 \draw[edge1] (1010) -- (0010) (0110) -- (1110);
	 \draw[edge2] (1010) -- (1110) (0010) -- (0110);
	 \draw[edge3] (1000) -- (1010) (1100)--(1110);
	 \draw[edge3]  (0001)--(0011) (0101)--(0111);
	 \draw[edge1] (1001) -- (0001) (0101) -- (1101);
	 \draw[edge2] (1001) -- (1101) (0001) -- (0101);
	 \draw[edge1] (1011) -- (0011) (0111) -- (1111);
	 \draw[edge2] (1011) -- (1111) (0011) -- (0111);
	 \draw[edge3] (1001) -- (1011) (1101)--(1111);
	 \draw[edge4] (1000) -- (1001) (1100) -- (1101) (1110)--(1111) (1010)--(1011);
	 	 \node[vertexA] at (0,0) {$1000$};
	 	 \node[vertex, fill=white]  at (2,2) {$1111$};
 \end{tikzpicture}$$
Shade in the vertices that have an even number of 0's. Explain why the $4$-cube is bipartite.

\medskip

\item Explain why $Q_n$ is bipartite in general.\\
	{[Hint: Show that a vertex with an even (respectively, odd) number of 0's will never be adjacent to another vertex with an even  (respectively, odd) number of 0's.]}
 
 \end{enumerate} 
 \end{enumerate}
\end{try}
\vfill

\pagebreak

\begin{try}~
\begin{enumerate}[(a)]
\item For each of the following pairs of graphs, first list their degree sequences. Then decide whether they are isomorphic or not. If not, say why. If they are, give a bijection on the vertices that preserves the edges, and draw the unlabeled graph that represents the corresponding isomorphism class of graphs. 
\begin{center}
\def\arraystretch{1.5}
\begin{tabular}{l@{\qquad\qquad}l}
(i)&(ii)\\
$\TikZ{
\foreach \x in {1,2,...,5}{\node[label=left:{$u_{\x}$}, bV] (\x) at (0,\x*0.5) {};}
\draw (1) to (5);}\qquad\qquad \TikZ{
\foreach \x/\y [count=\c from 1] in {(1,1)/1, (3,1)/2, (1,-1)/4, (3,-1)/5, (1,0)/3}{\node[label=above:{$v_{\c}$}, bV] (\c) at \x {};}
\draw (1) to (2) to (3) to (4) to (5);}$&$\TikZ{
\foreach \x in {1,2,...,5}{\node[bV] (\x) at (\x*360/5+18:1) {}; \node[anchor=\x*360/5+180+18] at (\x) {$u_\x$};}
\draw (1) to (2) to (3) to (4) to (5) to (1);
}\quad \TikZ{
\foreach \x in {1,2,...,5}{\node[bV] (\x) at (\x*360/5+18:1) {}; \node[anchor=\x*360/5+180+18] at (\x) {$v_\x$};}
\draw (1) to (3) to (5) to (2) to (4) to (1);
}$\\~\\
(iii)&(iv)\\
$\TikZ{
	\node[label=above:{$u_1$}, bV] (1) at (-1,1) {};
	\node[label=above:{$u_2$}, bV] (2) at (1,1) {};
	\node[label=below:{$u_3$}, bV] (3) at (1,-1) {};
	\node[label=below:{$u_4$}, bV] (4) at (-1,-1) {};
	\node[label=above:{$u_5$}, bV] (5) at (0,0) {};
\draw (1) to (2) to (3) to (1) to (4) to (5) (4) to (3);
}\qquad 
\TikZ{
	\node[label=above:{$v_1$}, bV] (1) at (0,1) {};
	\node[label=above right:{$v_2$}, bV] (2) at (1,0) {};
	\node[label=below:{$v_3$}, bV] (3) at (0.5,-1) {};
	\node[label=below:{$v_4$}, bV] (4) at (-0.5,-1) {};
	\node[label=above left:{$v_5$}, bV] (5) at (-1,0) {};
\draw (1) to (2) to (3) to (4) to (5) to (1) (5) to (2) to (4);
} 
$&$\TikZ{
\foreach \x in {1,2,...,7}{\node[bV] (\x) at (\x*360/7+90-360/7:1.2) {}; \node[anchor=\x*360/7-90-360/7] at (\x) {$u_\x$};}
\draw (1) to (2) to (3) to (4) to (5) to (6) to (7) to (1) (2) to (7) (3) to (5);
}\quad \TikZ{
\foreach \x in {1,2,...,7}{\node[bV] (\x) at (\x*360/7+90-360/7:1.2) {}; \node[anchor=\x*360/7-90-360/7] at (\x) {$v_\x$};}
\draw (1) to (2) to (3) to (4) to (5) to (6) to (7) to (1) (2) to (6) (3) to (7);
}$\\
(v)&(vi)\\
$\TikZ{
	\node[label=above:{$u_1$}, bV] (1) at (0,1) {};
	\node[label=above:{$u_2$}, bV] (2) at (1,1) {};
	\node[label=below:{$u_3$}, bV] (3) at (2,0) {};
	\node[label=below:{$u_4$}, bV] (4) at (1,0) {};
	\node[label=below:{$u_5$}, bV] (5) at (0,0) {};
\draw (1) to (2) to (4) to (5) to (1) to (4) (5) to (2) to (3) to (4);
}\qquad 
\TikZ{
	\node[label=above:{$v_1$}, bV] (1) at (-.5,1) {};
	\node[label=above:{$v_2$}, bV] (2) at (.5,1) {};
	\node[label=right:{$v_3$}, bV] (3) at (1,0) {};
	\node[label=below:{$v_4$}, bV] (4) at (0,-1) {};
	\node[label=left:{$v_5$}, bV] (5) at (-1,0) {};
\draw (1) to (4) to (5) to (1) to (3) to (5) to (2) to (3) to (4);
} 
$&
$\TikZ{
	\node[label=above:{$u_1$}, bV] (1) at (0,1) {};
	\node[label=above:{$u_2$}, bV] (2) at (1,.5) {};
	\node[label=below:{$u_3$}, bV] (3) at (1,-.5) {};
	\node[label=below:{$u_4$}, bV] (4) at (0,-1) {};
	\node[label=below:{$u_5$}, bV] (5) at (-1,-.5) {};
	\node[label=above:{$u_6$}, bV] (6) at (-1,.5) {};
\draw (1) to (2) to (3) to (4) to (5) to (6) to (1) to (4) (6) to (2)  (3) to (5);
}\qquad 
\TikZ{
	\node[label=above:{$v_5$}, bV] (1) at (0,.5) {};
	\node[label=above:{$v_2$}, bV] (2) at (1,1) {};
	\node[label=below:{$v_3$}, bV] (3) at (1,-1) {};
	\node[label=below:{$v_6$}, bV] (4) at (0,-0.5) {};
	\node[label=below:{$v_4$}, bV] (5) at (-1,-1) {};
	\node[label=above:{$v_1$}, bV] (6) at (-1,1) {};
\draw (1) to (2) to (3) to (4) to (5) to (6) to (1) to (4) (6) to (2)  (3) to (5);
}
$
\end{tabular}
\end{center}



\item How many isomorphism classes are there of simple graphs with 4 vertices? Draw them. 



\item 
How many edges does a graph have if its degree sequence is $4, 3, 3, 2, 2$? Draw a graph with this degree sequence. Can you draw a simple graph with this sequence?




\item For which values of $n, m$ are these graphs regular? What is the degree?

\centerline{ (i) $K_n$ \qquad (ii) $C_n$ \qquad (iii) $W_n$ \qquad (iv) $Q_n$ \qquad (v) $K_{m,n}$}



\item How many vertices does a regular graph of degree four with 10 edges have?



\item Show that isomorphism of simple graphs is an equivalence relation.  




\end{enumerate}
\end{try}

\end{document}

