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{\bf MATH314  \hspace{1cm} HW 1\hfill \name}
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Due \textbf{Jan 21} before class, \textbf{answer without justification will receive 0 points}.
The typing the HW in \LaTeX{} is optional. 
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\question{}{
Give an example of a graph that models something (abstract or physical) that was \emph{not} discussed in class.
}


\textbf{Solution:}

\question{1.4}{
 Let $S = \{-6, -3, 0, 3, 6\}$. Draw the graph $G$ whose vertex set is $S$ and such that
 $ij \in E (G)$ for $i,j \in S$ if $i + j \in S$ or $|i - j| \in S$.
}



\textbf{Solution:}


\question{1.6 (No drawing)}{
 Let $P = (u=v_0,v_1,\ldots,v_k=v)$, be a $u-v$ geodesic in a connected graph $G$. Prove that $d(u, v_i) = i$ for each integer $i$ with $1 \leq i \leq k$.
}


\textbf{Solution:}


\question{1.26}{
Suppose that the vertex set of a graph $G$ is a (finite) set of integers. Two vertices $x$ and $y$ are adjacent if $x + y$ is odd. To which well-known class of graphs is $G$ a member?
}

\textbf{Solution:}


\question{}{
Give an example of a graph on four vertices which is isomorphic with its complement. Do the same for five vertices.
}


\textbf{Solution:}


\question{}{
Find in the book what is Petersen graph. Draw it.
Determine if it is bipartite. 
Find the order of the largest empty induced subgraph. 
}


\textbf{Solution:}






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