In which we show how to find the eigenvalues and eigenvectors of Cayley graphs of Abelian groups, we find tight examples for various results that we proved in earlier lectures, and, along the way, we develop the general theory of harmonic analysis which includes the Fourier transform of periodic functions of a real variable, the discrete Fourier transform of periodic functions of an integer variable, and the Walsh transform of Boolean functions.
Earlier, we prove the Cheeger inequalities
and the fact that Fiedler’s algorithm, when given an eigenvector of , finds a cut such that . We will show that all such results are tight, up to constants, by proving that
- The dimension- hypercube has and , giving an infinite family of graphs for which , showing that the first Cheeger inequality is exactly tight.
- The -cycle has , and , giving an infinite family of graphs for which , showing that the second Cheeger inequality is tight up to a constant.
- There is an eigenvector of the 2nd eigenvalue of the hypercube , such that Fiedler’s algorithm, given such a vector, outputs a cut of expansion , showing that the analysis of the Fiedler’s algorithm is tight up to a constant.
In this lecture we will develop some theoretical machinery to find the eigenvalues and eigenvectors of Cayley graphs of finite Abelian groups, a class of graphs that includes the cycle and the hypercube, among several other interesting examples. This theory will also be useful later, as a starting point to talk about constructions of expanders.
For readers familiar with the Fourier analysis of Boolean functions, or the discrete Fourier analysis of functions , or the standard Fourier analysis of periodic real functions, this theory will give a more general, and hopefully interesting, way to look at what they already know.
We will use additive notation for groups, so, if is a group, its unit will be denoted by , its group operation by , and the inverse of element by . Unless, noted otherwise, however, the definitions and results apply to non-abelian groups as well.
Definition 1 (Character) Let be a group (we will also use to refer to the set of group elements). A function is a character of if
- is a group homomorphism of into the multiplicative group .
- for every ,
Though this definition might seem to not bear the slightest connection to our goals, the reader should hang on because we will see next time that finding the eigenvectors and eigenvalues of the cycle is immediate once we know the characters of the group , and finding the eigenvectors and eigenvalues of the hypercube is immediate once we know the characters of the group .
Remark 1 (About the Boundedness Condition) If is a finite group, and is any element, then
and so if is a group homomorphism then
and so is a root of unity and, in particular, . This means that, for finite groups, the second condition in the definition of character is redundant. In certain infinite groups, however, the second condition does not follow from the first, for example defined as is a group homomorphism of into but it is not a character.
Just by looking at the definition, it might look like a finite group might have an infinite number of characters; the above remark, however, shows that a character of a finite group must map into -th roots of unity, of which there are only , showing a finite upper bound to the number of characters. Indeed, a much stronger upper bound holds, as we will prove next, after some preliminaries.
Proof: Let be such that . Note that
where we used the fact that the mapping is a permutation. (We emphasize that even though we are using additive notation, the argument applies to non-abelian groups.) So we have
and since we assumed , it must be .
If is finite, given two functions , define the inner product
We will prove Lemma 3 shortly, but before doing so we note that, for a finite group , the set of functions is a -dimensional vector space, and that Lemma 3 implies that characters are orthogonal with respect to an inner product, and so they are linearly independent. In particular, we have established the following fact:
Corollary 4 If is a finite group, then it has at most characters.
It remains to prove Lemma 3, which follows from the next two statements, whose proof is immediate from the definitions.
Fact 5 If are characters of a group , then the mapping is also a character.
Fact 6 If is a character of a group , then the mapping is also a character, and, for every , we have .
To complete the proof of Lemma 3, observe that:
- the function is a character;
- the assumption of the lemma is that there is an such that , and so, for the same element ,
- thus is a character that is not identically equal to 1, and so
Notice that, along the way, we have also proved the following fact:
Fact 7 If is a group, then the set of characters of is also a group, with respect to the group operation of pointwise multiplication. The unit of the group is the character mapping every element to 1, and the inverse of a character is the pointwise conjugate of the character.
The group of characters is called the Pontryagin dual of , and it is denoted by .
We now come to the punchline of this discussion.
Proof: We give a constructive proof. We know that every finite abelian group is isomorphic to a product of cyclic groups
so it will be enough to prove that
- the cyclic group has characters;
- if and are finite abelian groups with and characters, respectively, then their product has characters.
For the first claim, consider, for every , the function
Each such function is clearly a character ( maps to 1, is the multiplicative inverse of , and, recalling that for every integer , we also have ), and the values of are different for different values of , so we get distinct characters. This shows that has at least characters, and we already established that it can have at most characters.
For the second claim, note that if is a character of and is a character of , then it is easy to verify that the mapping is a character of . Furthermore, if and are two distinct pairs of characters, then the mappings and are two distinct characters of , because we either have an such that , in which case , or we have a such that , in which case . This shows that has at least characters, and we have already established that it can have at most that many
This means that the characters of a finite abelian group form an orthogonal basis for the set of all functions , so that any such function can be written as a linear combination
For every character , , and so the characters are actually a scaled-up orthonormal basis, and the coefficients can be computed as
Example 1 (The Boolean Cube) Consider the case , that is the group elements are , and the operation is bitwise xor. Then there is a character for every bit-vector , which is the function
Every boolean function can thus be written as
which is the boolean Fourier transform.
Example 2 (The Cyclic Group) To work out another example, consider the case . Then every function can be written as
which is the discrete Fourier transform.
2. A Look Beyond
Why is the term ”Fourier transform” used in this context? We will sketch an answer to this question, although what we say from this point on is not needed for our goal of finding the eigenvalues and eigenvectors of the cycle and the hypercube.
The point is that it is possible to set up a definitional framework that unifies both what we did in the previous section with finite Abelian groups, and the Fourier series and Fourier transforms of real and complex functions.
In the discussion of the previous section, we started to restrict ourselves to finite groups when we defined an inner product among functions .
If is an infinite abelian group, we can still define an inner product among functions , but we will need to define a measure over and restrict ourselves in the choice of functions. A measure over (a sigma-algebra of subsets of) is a Haar measure if, for every measurable subset and element we have , where . For example, if is finite, is a Haar measure. If , then is also a Haar measure (it is ok for a measure to be infinite for some sets), and if then the Lebesgue measure is a Haar measure. When a Haar measure exists, it is more or less unique up to multiplicative scaling. All locally compact topological abelian groups have a Haar measure, a very large class of abelian groups, that include all finite ones, , , and so on.
Once we have a Haar measure over , and we have defined an integral for functions , we say that a function is an element of if
For example, if is finite, then all functions are in , and a function is in if the series converges.
If , we can define their inner product
and use Cauchy-Schwarz to see that .
Now we can repeat the proof of Lemma 3 that for two different characters, and the only step of the proof that we need to verify for infinite groups is an analog of Lemma 2, that is we need to prove that if is a character that is not always equal to 1, then
and the same proof as in Lemma 2 works, with the key step being that, for every group element ,
because of the property of being a Haar measure.
We don’t have an analogous result to Theorem 8 showing that and are isomorphic, however it is possible to show that itself has a Haar measure , that the dual of is isomorphic to , and that if is continuous, then it can be written as the “linear combination”
In the finite case, the examples that we developed before correspond to setting and .
Example 3 (Fourier Series) The set of characters of the group with the operation of addition modulo 1 is isomorphic to , because for every integer we can define the function
and it can be shown that there are no other characters. We thus have the Fourier series for continuous functions ,
3. Cayley Graphs and Their Spectrum
Let be a finite group. We will use additive notation, although the following definition applies to non-commutative groups as well. A subset is symmetric if .
Definition 9 For a group and a symmetric subset , the Cayley graph is the graph whose vertex set is , and such that is an edge if and only if . Note that the graph is undirected and -regular.
We can also define Cayley weighted graphs: if is a function such that for every , then we can define the weighted graph in which the edge has weight . We will usually work with unweighted graphs, although we will sometimes allow parallel edges (corresponding to positive integer weights).
Example 4 (Cycle) The -vertex cycle can be constructed as the Cayley graph .
Example 5 (Hypercube) The -dimensional hypercube can be constructed as the Cayley graph
where the group is the set with the operation of bit-wise xor, and the set is the set of bit-vectors with exactly one .
If we construct a Cayley graph from a finite abelian group, then the eigenvectors are the characters of the groups, and the eigenvalues have a very simple description.
Lemma 10 Let be a finite abelian group, be a character of , be a symmetric set. Let be the adjacency matrix of the Cayley graph . Consider the vector such that .
Then is an eigenvector of , with eigenvalue
Proof: Consider the -th entry of :
The eigenvalues of the form , where is a character, enumerate all the eigenvalues of the graph, as can be deduced from the following observations:
- Every character is an eigenvector;
- The characters are linearly independent (as functions and, equivalently, as vectors in );
- There are as many characters as group elements, and so as many characters as nodes in the corresponding Cayley graphs.
It is remarkable that, for a Cayley graph, a system of eigenvectors can be determined based solely on the underlying group, independently of the set .
4. The Cycle
The -cycle is the Cayley graph . Recall that, for every , the group has a character .
This means that for every we have the eigenvalue
where we used the facts that , that , and .
For we have the eigenvalue . For we have the second largest eigenvalue . If is an eigenvalue of the adjacency matrix, then is an eigenvalue of the normalized Laplacian. From the above calculations, we have that the second smallest Laplacian eigenvalue is .
The expansion of the cycle is , and so the cycle is an example in which the second Cheeger inequality is tight.
5. The Hypercube
The group with bitwise xor has characters; for every there is a character defined as
Let us denote the set by , where we let denote the bit-vector that has a in the -th position, and zeroes everywhere else. This means that, for every bit-vector , the hypercube has the eigenvalue
where we denote by the weight of , that is, the number of ones in .
Corresponding to , we have the eigenvalue .
For each of the vectors with exactly one , we have the second largest eigenvalue. The second smallest Laplacian eigenvalue is .
Let us compute the expansion of the hypercube. Consider “dimension cuts” of the form . The set contains half of the vertices, and the number of edges that cross the cut is also equal to half the number of vertices (because the edges form a perfect matching), so we have and so .
These calculations show that the first Cheeger inequality is tight for the hypercube.
Finally, we consider the tightness of the approximation analysis of Fiedler’s algorithm.
We have seen that, in the -dimensional hypercube, the second eigenvalue has multiplicity , and that its eigenvectors are vectors such that . Consider now the vector ; this is still clearly an eigenvector of the second eigenvalue. The entries of the vector are
Suppose now that we apply Fiedler’s algorithm using as our vector. This is equivalent to considering all the cuts in the hypercube in which we pick a threshold and define .
Some calculations with binomial coefficients show that the best such “threshold cut” is the “majority cut” in which we pick , and that the expansion of is
This gives an example of a graph and of a choice of eigenvector for the second eigenvalue that, given as input toFiedler’s algorithm, result in the output of a cut such that . Recall that we proved , which is thus tight, up to constants.