Problem set 1
Due date: 02/19/2016
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. Show that the following graphs all have largest eigenvalue 2 by writing
the corresponding eigenvector to the graphs.
P2. Given the eigenvalues of the adjacency matrices of the graphs G1 and
G2 . Let us define the graph G1 G2 as follows: the vertex set of G1 G2 is
V (G1 )×V (G2 ) and the vertices (u1 , v1 ) ∈ V (G1 G2 ) and (u2 , v2 ) ∈ V (G1 G2 )
are adjacent if and only if u1 = u2 and (v1 , v2 ) ∈ E(G2 ) or v1 = v2 and
(u1 , u2 ) ∈ E(G1 ). What are the eigenvalues of the adjacency matrix of G1 G2 ?
P3. Let G be an arbitrary graph, and let µ1 ≥ µ2 ≥ · · · ≥ µn be the eigenvalues
of its adjacency matrix. Show that µ1 ≥ |µn |.
P4. Let T be a tree on n vertices.
Show that the largest eigenvalue of its
√
adjacency matrix is at most n − 1. For which trees can equality hold true?
(A tree2 is a connected cyclefree graph. A tree on n vertices has exactly n − 1
edges.)
P5. Let G be an arbitrary graph and let χ(G) denote its chromatic number. It
is the smallest number k such that the vertices of G can be colored with k colors
such that no two adjacent vertices have the same color. Let µ(G) denote the
largest eigenvalue of the adjacency matrix of G. Show that χ(G) ≤ µ(G) + 1.
1
The extended syllabus can be found at
http://math.mit.edu/~csikvari/extended_syllabus_algebraic_combinatorics.pdf
2If you haven’t seen this concept before then I suggest you finding all trees on 6 vertices,
there are 6 of them.
1
2
2. Homework problems
√
√ (2)
1. The following graph has eigenvalues ± 5, ± 2 , 0(4) . Find the corresponding eigenvectors, give them as you have seen at the class, namely write
the elements of an eigenvector to a picture of the graph.
2. Let Kn,n,n be the complete tripartite graph with color classes of size n. (So
it has three classes, each of them has size n , and two vertices are adjacent if
and only if they are in different classes.) What are the eigenvalues of Kn,n,n
and what are the corresponding eigenvectors?
3. Let G be a simple graph on n vertices and e(G) edges, and let µ(G) denote
the largest eigenvalue of the adjacency matrix of G.
√
(a) Show that µ(G) ≤ 2e(G).
(b) Improve on your previous argument a bit to show that
√
(
)
1
.
µ(G) ≤ 2e(G) 1 −
n
4. Let G be a simple graph on n vertices, and let G be the complement of
G, i. e., the graph in which two vertices are adjacent if and only if they were
not adjacent in G. Let µ(G) and µ(G) denote the largest eigenvalue of the
adjacency matrix of G and G, respectively. Show that
√
µ(G) + µ(G) ≤ 2n.
5. Determine which simple graphs have largest eigenvalue less than 2.
6. Let G be a d–regular simple graph, and let d = µ1 ≥ µ2 ≥ · · · ≥ µn be the
eigenvalues of its adjacency matrix. Let G be the complement of G, i. e., the
graph in which two vertices are adjacent if and only if they were not adjacent
in G. What are the eigenvalues of the adjacency matrix of G?
3
7. Let G be a simple graph, and let G[k] be the following full "blow-up" graph
of G: we replace every vertex of G with a clique of size k, and we connect every
vertices between two such cliques if and only if the corresponding vertices were
adjacent. (A clique of size k is just a complete subgraph, i. e., k vertices such
that all possible edges going between them.) What are the eigenvalues of the
adjacency matrix of G[k] if the eigenvalues of the adjacency matrix of G are
µ1 ≥ µ2 ≥ · · · ≥ µn ?
8. Given the eigenvalues of the adjacency matrices of the graphs G1 and
G2 . Let us define the graph G1 G2 as follows: the vertex set of G1 G2 is
V (G1 )×V (G2 ) and the vertices (u1 , v1 ) ∈ V (G1 G2 ) and (u2 , v2 ) ∈ V (G1 G2 )
are adjacent if and only if u1 = u2 or (u1 , u2 ) ∈ E(G1 ), and v1 = v2 or
(v1 , v2 ) ∈ E(G2 ), but (u1 , v1 ) ̸= (u2 , v2 ). What are the eigenvalues of the
adjacency matrix of G1 G2 ?
9. Let Tn be a tree on n vertices with largest eigenvalue µ(Tn ). Let Pn be
the path on n vertices, so it is a tree on n vertices with edges (i, i + 1), where
1 ≤ i ≤ n − 1. Show that µ(Tn ) ≥ µ(Pn ).
10. Let G be a triangle-free graph on n vertices with eigenvalues µ1 (G) ≥
µ2 (G) ≥ · · · ≥ µn (G). Show that
√
µ1 (G) + µn (G) ≤ (3 − 2 2)n.
11. (a) Assume that the edges of the graph G has an orientation such that
every vertex has indegree at most D+ , and outdegree at most D− . Show
that the following inequality holds true for the largest eigenvalue µ(G) of the
adjacency matrix of G:
√
µ(G) ≤ 2 D+ D− .
(b) Let T be a tree on at least 3 vertices with maximum degree D. Show that
for the largest eigenvalue µ(T ) of the adjacency matrix of T we have
√
µ(T ) ≤ 2 D − 1.
12. Let G be a graph with e(G) edges and largest eigenvalue µ(G). Show that
e(G)
it contains a matching of size at least µ(G)
2 . (A matching is a set of edges
e1 , . . . , ek such that ei and ej has distinct end vertices if i ̸= j.)