18.314 Fall 2007 Problem Set 6. Due Tuesday, December 11. 1. Find the resistance between the vertices A and B in the graph: o---o---B | | | | | | o---o---o (3 x 3 grid) | | | | | | A---o---o where the resistance of each edge of the graph is 1 Ohm. 2. Calculate the number of acyclic orientations of the n x 2 grid: o--o--o--o--o--o--o--o | | | | | | | | | | | | | | | | o--o--o--o--o--o--o--o 3. Two people A and B play the following game. Originally, A has k chips and B has n-k chips. Each turn, they flip a coin and if it falls heads then A gives B a chip and if it falls tails then B gives A a chip. If somebody collects all n chips then this person wins the game. For any n and k, find the probability that A wins. 4. Calculate the number of spanning trees of the 6 x 6 torus graph G. The vertices of this graph are pairs (i,j), where i,j are residues modulo 6; the vertex (i,j) is connected with 4 vertices (i,j+1), (i,j-1), (i+1,j), (i-1,j), where again all numbers are taken modulo 6. 5. Find the number of spanning trees of the complete tripartite graph K_{m,n,k}. This is the graph whose vertices are subdivided into tree parts A={1,...,m}, B={m+1,...,m+n}, C={m+n+1,...,m+n+k}, and two vertices are connected by an edge if and only if they are in different parts. 6. Let G be a graph on the vertices 1,2,...,n; and let H be the complementary graph on the same vertices. In other words, for any 1 <=i < j <= n, (i,j) is an edge in H if and only if (i,j) is not an edge in G, and vise versa. Let G+ and H+ be the graphs obtained from G and H by adding the root vertex 0 connected with all other vertices 1,2,...,n. Let T_G(x) be the sum of x^{deg_0(T)} over all spanning trees T of G+. Similary, let T_H(x) be the sum over spanning trees of H+. Show that T_H(x) = (-1)^{n-1} T_G(-x-n). For example, 0----1 0-----1 G+ = / \ / H+ = / \ / / \/ / \ / 2 3 3-----2 T_G = x^2 + 2 x T_H = x^2 + 4x + 3 = T_G(-3-x).