Problem set 2
Due date: 03/11/2016
Before you start working on the homework problems read the extended
syllabus1 carefully and check which problems you need to solve.
1.
Practice problems
P1. (a) Let T be a tree on n vertices, and mk (G) be the number of k–matchings.
(A k-matching is a set of edges e1 , . . . , ek such that they cover 2k vertices
together, i. e., they have distinct end vertices. Note that m0 (G) = 1.) Show
that for the characteristic polynomial of the tree T we have
⌊n/2⌋
det(xI − A) =
∑
(−1)k mk (T )xn−2k .
k=0
(b) Compute the characteristic polynomials of the following two trees.
P2. Show that if G is a strongly regular with parameters (n, d, a, b) then its
complement G is also strongly regular. What are the parameters of G?
P3. Let G be a strongly regular graph with parameters (n, d, a, b). Show that
if (n, d, a, b) ̸= (4k + 1, 2k, k − 1, k) for some k then all eigenvalues of G are
integer.
P4. (a) Show that if H is an induced subgraph of G then the smallest eigenvalue
of H is at least as large as the smallest eigenvalue of G. Is it true that if H
is any subgraph of G then the smallest eigenvalue of H is at least as large as
the smallest eigenvalue of G? (Induced subgraph means that you can delete
vertices (and all edges incident to the deleted vertices), but you cannot delete
an edge without deleting at least one of its end vertex.)
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The extended syllabus can be found at
http://math.mit.edu/~csikvari/extended_syllabus_algebraic_combinatorics.pdf
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(b) Let A be an n × n real symmetric matrix with eigenvalues λ1 ≥ · · · ≥ λn .
Show that
xT Ax
λk = max min
U :dim U =k x∈U ||x||2
x̸=0
and
λk =
min
U :dim U =n−k+1
max
x∈U
x̸=0
xT Ax
.
||x||2
(c) Let G be a graph with eigenvalues λ1 ≥ · · · ≥ λn . Let H be a graph obtained by deleting the vertex v from G. Let ρ1 ≥ · · · ≥ ρn−1 be the eigenvalues
of H. Show that
λ1 ≥ ρ1 ≥ λ2 ≥ ρ2 ≥ · · · ≥ ρn−1 ≥ λn .
2.
Homework problems
1. Let G be a strongly regular graph with parameters (n, d, 0, 2).
(a) How many vertices does G have in terms of d?
(b) Show that there exists an r such that d = r2 + 1.
2. Let G be the following graph: V (G) = {(i, j) | 1 ≤ i ≤ n, 1 ≤ j ≤ n}, and
E(G) = {((i, j), (k, l) | (i, j) ̸= (k, l) and i = k or j = l}.
(a) Show that G is strongly regular. What are the parameters? Compute the
eigenvalues of G.
(b) Show that G is a Cayley-graph Cay(Γ, S) for some Γ and S. Determine
the eigenvalues using the formula for the eigenvalues of a Cayley-graph.
3. Let L(G, x) be the characteristic polynomial of the Laplacian matrix of G.
Let G be the complement of G. Show that the number of spanning trees of G
is n12 L(G, n).
4. Let G be a graph obtained from the complete graph Kn by deleting the
edges of a complete graph on k vertices (k < n). How many spanning trees
does G have?
5. Let p be a prime of the form 4k+1. The Paley-graph Pp is defined as follows:
the vertex set of Pp is Fp , the finite field on p elements, an let (a, b) ∈ E(Pp ) if
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a − b is a quadratic residue modulo p, i. e., there exists some c ̸= 0 such that
a − b = c2 in Fp .
(a) Show that Pp is a strongly regular graph and determine its parameters.
(Hint: an elementary way to do it is to use the observation that Pp and its
complement Pp are isomorphic.)
(b) How many spanning trees does the Paley-graph Pp have?
6. Let G = (A, B, E) be a biregular bipartite graph such that |A| = n1 , |B| =
n2 , every vertex in A has degree d1 , every vertex in B has degree d2 . Let
µ1 ≥ µ2 ≥ · · · ≥ µn be the eigenvalues of the adjacency matrix of the graph
G. For U ⊆ A and V ⊆ B, let e(U, V ) the number of edges going between the
set U and V . Show that
e(U, V ) − d1 |U ||V | ≤ µ2 |U |1/2 |V |1/2 .
n2 7. (a) Let G be a strongly regular graph with parameters (n, d, a, b). Assume
that the different eigenvalues of G are d > ϑ1 > ϑ2 . Show that
(d − ϑ1 )(d − ϑ2 ) = nb.
(b) Let G be a strongly regular graph with parameters (n, d, a, b). Assume
that G is not the empty or the complete graph, and n = p is a prime. Show
that G is a conference graph, i. e., it has parameters (p, p−1
, p−5
, p−1
).
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8. Let G be a 3–regular simple graph with second largest eigenvalue at most 2.
Give a concrete upper bound on the number of vertices (and naturally prove
it).
9. Let G be a connected graph. Let H be a 2-lift of G: V (H) = V (G) ×
{0, 1}, and for every (u, v) ∈ E(G), exactly one of the following two pairs
are edges of H: ((u, 0), (v, 0)) and ((u, 1), (v, 1)) ∈ E(H), or ((u, 0), (v, 1))
and ((u, 1), (v, 0)) ∈ E(H). If (u, v) ∈
/ E(G), then none of ((u, 0), (v, 0)),
((u, 1), (v, 1)), ((u, 0), (v, 1)) and ((u, 1), (v, 0)) are edges in H. Let τ (G) and
τ (H) denote the number of spanning trees of G and H, respectively. Show
that τ (G) is a divisor of τ (H): there is an integer k such that kτ (G) = τ (H).
10. How many trees are there on the vertices 1, 2, . . . , n which contain the
edges (1, 2), (3, 4),..., (2k − 1, 2k)? (Naturally 2k ≤ n.) Clarification: the trees
have to contain all vertices, i. e., they have to be the spanning trees of Kn
such that they contains the given edges.
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11. Let G be 3–regular simple connected vertex-transitive graph whose smallest
eigenvalue is strictly bigger than −2. Show that G is the complete graph on
4 vertices. (Vertex-transitive means that the automorphism group of G acts
transitively on the vertices of G. Intuitively this means that G looks like the
same way from every vertex.)
12. Let G be a simple graph. For a set S ⊆ V (G), let G[S] denote the
induced subgraph on the vertex set S. For any graph H let L(H, x) denote
the characteristic polynomial of the Laplacian matrix of H. Show that
∑
L(G[S], x)L(G[V \ S], y) = L(G, x + y).
S⊆V (G)
(For S = ∅ we have L(G[∅], x) = 1.)