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MATH 552. Advanced Graph Theory
Problem Set∗
Spring 2010
Problems
Q1 From the spectral bound on the graph chromatic number of a ring graph, show that
χ(Rn ) > 2.
Q2 Compute λ2 and v2 for a star graph on n vertices. Verify, whether Fiedler’s nodal domain
theorem remains valid when the choice of the vertex set is changed to Wk = {j ∈ V :
vk (j) > 0} in Theorem 5.5.1 of your notes.
Q3 Compute λ2 for a dumbell graph. i.e. two complete graphs (of possibly different sizes)
connected by a single edge. Sketch the nodal domains in v2 as you decrease the number of
nodes in one lump and increase in the other. Keep |V | constant as you study this variation.
Q4 Consider the evolution of a linear differential equation on the graph G = (V, E) by,
ẋ(t) = −LG x(t),
where x(t) ∈ R|V | . Show that the steady state solution of this equation depends on whether
G is connected. Assume that the initial conditions are nonzero.
Q5 We say that a d-regular graph is an expander if its adjacency matrix eigenvalues satisfy
|αi | ≤ d for i ≥ 2 and small . Show that an expander on n vertices is a good approximation of a complete graph Kn . Recall the definition of graph approximation from Section
6.4.
Q6 Give a complete proof of Lemma 3.3.3. Fill in the gaps of the proof sketched in the notes.
∗ Module on Spectral Graph theory by Dr Abubakr Muhammad. LUMS School of Science & Engineering,
Lahore, Pakistan.
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