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.
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