CS7540 Spectral Algorithms, Spring 2017
Lecture #3
Eigenvalues of Adjacency Matrix and Chromatic Number
Presenter: Anup B. Rao
Jan 17, 2017
DISCLAIMER: These notes are not necessarily an accurate representation of what
I said during the class. They are mostly what I intend to say, and have not been carefully
edited.
We study eigenvalues of adjacency matrix and its relation to combinatorial quantities
associated to graphs. Let G = (V, E, w) be a weighted undirected graph. The adjacency
matrix A is defined as
A(i, j) = w(i,j) ,
where wi,j is the weight of the edge (i, j) ∈ E. Let
µ1 ≥ µ2 ... ≥ µn
be the eigenvalues of A with corresponding eigenvector v1 , ..., vn .
We will assume throughout that the graph is connected.
Some additional notations:
• If S ⊂ V is a set of vertices, then by G(S) we denote the induced subgraph on S.
• Also, let dmax be the maximum degree of a vertex in V, dmax (S) the maximum degree
of a vertex in the induced graph on S and similarly davg be the average degree.
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Top Eigenvalue
Recall that v1 is the eigenvector of A corresponding to the highest eigenvalue µ1 . Note
that if A is a d regular undirected graph, then we’ll have
µ1 = d,
and


1
1  1 
.
v1 = √ 
n  ··· 
1
For an undirected graph (which we’ll focus on for much of this course), this vector can
also be obtained from Rayleigh quotients,
xT Ax
v1 = arg max T .
x x
x
For directed graphs, bounds on u1 is given by the following theorem, which we will
prove in a later lecture.
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Theorem 1.1 (Perron Frobenius, not covered in this lecture). Suppose G is a strongly
connected graph. Then,
1. µ1 ≥ −µn , and
2. µ1 > µ2 , and
3. v1 > 0 pointwise.
Back to undirected graphs, we can show the following facts:
1. For every S ⊂ V, davg (S) ≤ µ1 ≤ dmax .
2. If µ1 = dmax for a connected graph G, then G is a regular graph.
In the undirected case, the availability of the Rayleigh quotient also gives a simple
proof to v1 > 0: Consider the vertex x formed by
def
x(i) = |v1 (i)| ,
then since all entries of A are non-negative, we get
xT Ax ≥ v1T Av1 ,
which gives a larger Rayleigh coefficient unless x = v1 .
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Coloring and Eigenvalue
Definition 2.1 (Chromatic Number). The minimum number of colors required to color
a graph G such that no two adjacent vertices get the same color is called the chromatic
number and denoted by χ(G).
We first show that for any (unweighted) graph G, χ ≤ dmax + 1.
Theorem 2.2 (Wilf). For any unweighted graph G, χ(G) ≤ µ1 + 1.
The proof of this has several steps:
1. Show that the adjacency matrix of any subset of the vertices, S, A(S), satisfies
λmin (A) ≤ λmin (A(S)) ≤ λmax (A(S)) ≤ λmax (A).
2. Show that for any inducted subgrpah, u1 ≥ davg (S).
3. Exhibit an ordering of vertices so that each vertex has at most u1 neighbors preceeding it. This is done by iteratively peeling off the minimum degree vertex.
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Lower Bounding Chromatic Number
The following lower bound was proved by Hoffman. It holds even for weighted graphs,
though the chromatic number is not depended on the weights.
Lemma 3.1. Let


A11 A12 · · · A1k
AT12 A2,2 · · · A2k 


·
· 
A=
 ·

 ·
·
· 
·
· · · Akk
AT1k
be a block partitioned symmetric matrix. Then,
(k − 1)λmin (A) + λmax (A) ≤
i=k
X
λmax (Aii ).
i=1
Proof. We start with the k = 2 case: Let the top eigenvector be:
x1
x=
,
x2
and define the vector
"
y=
kx2 k
x
kx1 k 1
kx1 k
− kx2 k x2
#
,
aka. normalize the two blocks of x so that it’s orthogonal to the all 1s vector.
It can be checked that:
y T Ay + xT Ax ≥ λmax (A) + λmin (A),
and
y T Ay + xT Ax =
xT1 A11 x1 xT2 A22 x2
+
≤ λmax A11 + λmax A22 .
kx1 k2
kx2 k2
The general case can be obtained by applying this inductively on the blocks.
Theorem 3.2 (Hoffman).
χ(G) ≥ 1 +
µ1
.
−µn
Proof. Partition the blocks according to the color classes, the above lemma will imply:
(k − 1)λmin (A) + λmax (A) ≤ 0,
which upon rearranging gives the condition.
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