18.314     PROBLEM SET 6    (due Thursday December 08, 2005)


Problem 1.  Find an explicit formula for the number of spanning trees
in the product K_3 x K_3 x ... x K_3 of n copies of the complete graph 
K_3 on 3 vertices.

Problem 2.  Calculate the chromatic polynomial for the n x 2 grid,
i.e., for the graph of the form

     .__.__.__.__.__.__.
     |  |  |  |  |  |  | 
     !__!__!__!__!__!__!

(there are 2 vertices in each column and n vertices in each row)

Problem 3.  Calculate the number of acyclic orientations of the 
complete bipartite graph K_{3,3}.

Problem 4.  Prove the Hurwitz's identity 
   2 (n+2)^{n-1} = \sum_{k=0}^n C(n,k) (k+1)^{k-1} (n-k+1)^{n-k-1},    
where C(n,k) is the binomial coefficient.    
(Hint: Try to count spanning trees in the complete graph K_{n+2} that 
contain the edge (1,2)).

Problem 6.  Calculate the total resistance between the vertices A and B
in the Wheatstone bridge:

         A
        / \
       R1  R2
      /     \
     /___R3__\
     \       /
      \     /
       R4  R5
        \ /
         B

where the resistances of the edges are R1, R2, R3, R4, and R5 as shown
above.  (A partial credit will be given for calculating the total 
resistance in the case R1 = R2 = R3 = R4 = R5 = 1).

Problem 7.  Calculate the total resistance of the n-hypercube between
the pair of opposite vertices.  Assume that resistances of all edges
of the hypercube are 1.

Problem 8.  Consider a random walk on n vertices labelled by 1,...,n
such that the probabilities to go from the vertex i to the vertex i+1 
and from i to i-1 are 1/2.  (Expect the cases i = 1, n when we go to 
the only adjacent vertex with probability 1.)
Calculate the probability P(i) that a random walk that starts at i
reaches 1 before it reaches n.  For example, P(1) = 1 and P(n)=0.

Problem 9. (bonus)  Consider a random walk on an infinite set of vertices
labelled by integers such that we can go from i to i+1 or to i-1 with 
the probability 1/2.  Calculate the probability p(n) that a walk with 
2n steps that starts and ends at 0 reaches its maximal value exactly once.
For example, the walk with the steps -1,1,1,1,-1,-1 reaches its maximum
value once, but the walk with the steps 1,1,-1,1,-1,-1  reaches its 
maximum value twice:  

       /\               /\/\
      /  \             /    \ 
    \/            
 hits the maximum    hits the maximal 
 value once          value twice

Can you give a combinatorial proof?