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{\bf Introduction to Graph Theory: The final test.} \hfill 7 June 2000\\

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Name(\hspace{3in}) No.(\hspace{2in})\\

\begin{itemize}
\item[1.] Obtain {\it diameter} and {\it radius} of 
          an $n \times m$ grid graph as a function of $n$ and $m$. 
          Also answer where are its {\it center} and {\it periphery}.
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\item[2.] A game is played with a die by five people around a circular table.
          If the player with the die throws an odd number, he passes the die
          to the player on his left; if he throws a 2 or 4, he passes it to 
          the player on his right; if he throws a 6, he keeps the die and 
          throws again.
          Write down the corresponding transition matrix and directed graph.
\clearpage
\item[3.] Obtain a path with the maximum flow from $v$ to $w$, 
          assuming all the flow is from leftwards to rightwards.
\end{itemize}

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\begin{itemize}
\item[4.] Answer either of the two below.
   \begin{itemize}
   \item How can you apply a topic that you have learned in this course, 
         e.g., {\it Chinese postman's problem}, to real world problems? 
         Give me examples.
   \item In the qustion-2 above, show that each state is persistent and 
         aperiodic, and also show that the corresponding Markov chain is 
         ergodic.
   \end{itemize}
\end{itemize}

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