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COMBINATORICS. PROBLEM SET 13 Seminar problems Problem 13.1. Let G be a labeled tree on n vertices. Compute the number of spanning trees of G using Kirchhoff matrix tree theorem. Problem 13.2. The same for the cycle graph on n vertices (i.e., the graph whose vertices and edges are those of the regular n-gon). Problem 13.3. Compute the number of spanning trees of the complete graph Kn on n vertices (the graph is labeled). Problem 13.4. Compute the number of spanning trees of the complete bipartite graph Kp,q (the graph is labeled). Problem 13.5. Let Qn be the hypercube graph (or the Hamming graph). That is, consider the n-dimensional cube [0, 1]n and form a graph having vertices and edges of [0, 1]n . Compute the number of spanning trees of Qn . (Hint: write the Laplacian matrix L and find all its eigenvectors and eigenvalues counted with multiplicities). Problem 13.6. Let Tn (N ) be the generating function X N #vertices Tn (N ) = all gluings of 2n-gon as in the Harer–Zagier theorem. Show that for even n the polynomial Tn (N ) is odd (i.e., contains only odd powers of N ), and for odd n, Tn (N ) is an even polynomial. Problem 13.7. Show that the polynomial Tn (N ) starts as Tn (N ) = Cn N n+1 + terms of lower degree in N (Cn is the nth Catalan number). Problem 13.8. Consider the Harer-Zagier theorem for n = 4 (i.e., the gluings of 8-gon). Surfaces of which genus can arise as gluings of 8-gon? How many are there gluings of each sort? Draw a couple examples for each possible genus. 1