Quasirandom Cayley graphs

We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, R\"odl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.


Introduction
A fundamental result of Chung,Graham,and Wilson [6], building on earlier work of Thomason [19,20], states that for a sequence of graphs of density p, where p is a fixed positive constant, a number of seemingly distinct notions of quasirandomness are equivalent. The following theorem details some of these equivalences. Theorem 1.1 (Chung-Graham-Wilson). For any fixed 0 < p < 1 and any sequence of graphs (Γ n ) n∈N with |V (Γ n )| = n, the following properties are equivalent: P 1 : for all subsets S, T ⊆ V (Γ n ), e(S, T ) = p|S||T | + o(pn 2 ); P 2 : e(Γ n ) ≥ (1 + o(1))p n 2 , λ 1 (Γ n ) = (1 + o(1))pn and |λ 2 (Γ n )| = o(pn), where λ i (Γ n ) is the ith largest eigenvalue, in absolute value, of the adjacency matrix of Γ n ; P 3 : for all graphs H, the number of labeled copies of H in Γ n is (1 + o(1))p e(H) n v(H) .
Here, the adjacency matrix A(Γ) of an n-vertex graph Γ is the n × n matrix (a st ) s,t∈ [n] where a st = 1 if s and t are adjacent in Γ and 0 otherwise.
In this paper, we will be concerned with studying the extent to which this theorem, and particularly the equivalence between the first two properties, extends to sparse graphs, that is, graphs where the density p tends to zero. We will focus our attention on regular graphs. We then say that an n-vertex d-regular graph Γ = (V, E) is ε-uniform if, for all S, T ⊆ V , e(S, T ) − d n |S||T | ≤ εdn.
The equivalence between properties P 1 and P 2 of Theorem 1.1 now implies that if p is fixed and Γ n is a sequence of graphs with |V (Γ n )| = n and Γ n regular of degree d n = pn, then the sequence Γ n is o(1)-uniform if and only if |λ 2 (Γ n )| = o(d n ).
One direction of this equivalence follows from the famous expander mixing lemma. This says that if Γ = (V, E) is an (n, d, λ )-graph, that is, an n-vertex d-regular graph such that all eigenvalues of the adjacency matrix A(Γ), except the largest, are bounded above in absolute value by λ , then Note that this statement does not rely on knowing that d is large, applying just as well when it is constant as when it is on the order of n. This observation is of fundamental importance in the study of expander graphs (see [10]). Answering a question of Chung and Graham [5], Krivelevich and Sudakov [13] showed that the converse does not hold. That is, there are sequences of sparse graphs for which the discrepancy condition P 1 of Theorem 1.1 does not imply the eigenvalue condition P 2 . However, their example is not regular (and a later example due to Bollobás and Nikiforov [3] is not connected), so we give an alternative example below.
Example 1.2. Take three disjoint vertex sets U, V , and W , with |U| = |V | = t and |W | = n − 2t. Choose a d-regular graph Γ on vertex set U ∪V ∪W such that every vertex in U has no neighbours in U and exactly λ neighbours in V , and vice versa for V , while each vertex in W has the same number of neighbours in U and in V . By taking the vector y = (y 1 , . . . , y n ) with y j = 1 if j ∈ U, −1 if j ∈ V , and 0 otherwise, we see that −λ is an eigenvalue of the adjacency matrix A(Γ). More concretely, suppose d is even, let t = λ = d/2, and let the bipartite graph between U and V be complete. We split the remaining set of vertices W into two subsets W 0 and W 1 , where |W 1 | = d 2 /4 and each vertex in W 1 is joined to both endpoints of a unique associated edge in Γ[U,V ], while no vertex in W 0 is joined to U ∪V . In W , we place a random graph such that every vertex in W 0 has degree d and every vertex in W 1 has degree d − 2. The resulting graph is d-regular. Moreover, provided d → ∞ and d = o(n), a simple calculation (similar to the proof of Proposition 12 in [3]) implies that G is o(1)-uniform, despite having −d/2 as an eigenvalue.
Given such examples, a recent result of Kohayakawa, Rödl, and Schacht [12] comes as something of a surprise. Suppose that G is a finite group and S is a subset of G. The (directed) Cayley graph Cay(G, S) is formed by taking the elements of G as the vertex set and {(sg, g) : g ∈ G, s ∈ S} as the set of (directed) edges. If S is symmetric, that is, S = S −1 , the graph can be viewed as undirected. Unless stated otherwise, we will always use the term Cayley graph to refer to such an undirected graph with a symmetric generating set. We note that many of the standard examples of quasirandom graphs, including Paley graphs and the Ramanujan graphs of Lubotzky, Phillips, and Sarnak [15] and Margulis [16], are Cayley graphs.  It is not hard to lift this result to all vertex-transitive graphs, where a graph is said to be vertex transitive if the automorphism group of the graph acts transitively on the vertices. Note that Cayley graphs are always vertex transitive, while the example of the Petersen graph shows that the converse is not true.
for all S, T ⊆ V (Γ). Corollary 1.5 then says that Γ is an (n, d, λ )-graph with λ ≤ 8εd. In turn, the expander mixing lemma shows that for all S, T ⊆ V (Γ). Thus, Corollary 1.5 may be seen as saying that in vertex-transitive graphs the discrepancy condition (1) bootstraps itself to the stronger condition (2). For comparison, we note also a result of Bilu and Linial [2] which says that if Γ is an n-vertex d-regular graph satisfying the stronger discrepancy condition (2), then it is an (n, d, λ )-graph with λ = O(εd log(2/ε)). Our result shows that in Cayley graphs, one can derive the stronger conclusion λ = O(εd) from the weaker condition (1).
We begin the process of proving Theorem 1.4 and Corollary 1.5 in the next section by introducing two matrix norms, the cut norm and the spectral norm, which capture the properties of having small discrepancy and having small second eigenvalue. The rough idea then will be to show that if A is the adjacency matrix of a vertex-transitive graph Γ, the spectral norm of A agrees with a semidefinite relaxation of the cut norm.
We note that some similar ideas have been used in this context before. In particular, a semidefinite relaxation was used by Alon et al. [1] to prove the related result that if a graph G has small discrepancy, then one can remove a small fraction of the vertices so that in the remaining graph all eigenvalues, except the largest, are small.

Norms and relaxations
For any vector x = (x 1 , . . . , x n ) ∈ R n , we define its p-norm, 1 ≤ p ≤ ∞, by In particular, we write |x| := |x| 2 for the Euclidean norm. The inner product on R n is x, y := x * y = x 1 y 1 + · · · + x n y n .
We define a number of matrix norms. The most familiar one is the spectral norm: where the second equality above follows from an application of the Cauchy-Schwarz inequality. Equivalently, A equals the largest singular value of A. If A is symmetric, then A is also the maximum of the absolute values of the eigenvalues of A.
The notion of ε-uniformity is captured by the cut norm: It is not hard to see that the cut norm can be expressed as a linear relaxation: a st x s y t = sup To see the validity of this reformulation, note that the expression inside the absolute value is linear individually in each of x 1 , . . . , x m , y 1 , . . . , y n , and hence its extremum is attained when all these variables are {0, 1}-valued, which is equivalent to the earlier formulation with subsets S and T .
Returning to the discussion of matrix norms, we further relax the cut norm by allowing each x s and y t to be numbers in [−1, 1]: a st x s y t .
This norm is equivalent to the cut norm: 1] m and y + , y − ∈ [0, 1] n , and then apply the triangle inequality and (4). Finally, we consider a semidefinite relaxation, which we shall refer to as the Grothendieck norm: a st x s , y t , where (H, ·, · ) is any Hilbert space 1 and B(H) is the unit ball in H, containing all points in H of norm at most 1. In fact, since we are working with m + n vectors, one can assume that H = R m+n in the definition above.
A key fact about the Grothendieck norm is that it is equivalent to the ∞ → 1 norm. Though this result was proved by Grothendieck [8], it was first stated in this form by Lindenstrauss and Pełczyński [14]. We refer the interested reader to the surveys [11,17] for further information.
Theorem 2.1 (Grothendieck's inequality). There exists a constant K G such that for all real-valued matrices A, We use the symbol K G (the real Grothendieck constant) to denote the optimal constant in the above inequality.
The best current upper bound on the real Grothendieck constant is K G < π 2 log(1+ √ 2) = 1.78 . . . , due to Braverman, Makarychev, Makarychev, and Naor [4] (see Haagerup [9] for the best constant in the complex case). Combining (5) with Grothendieck's inequality (and using 4K G ≤ 8), we have that is, the cut norm and the Grothendieck norm are equivalent.
The spectral norm provides an upper bound on the Grothendieck norm: for any A ∈ R m×n , The proof is by a straightforward application of the Cauchy-Schwarz inequality. 2 In general, the spectral norm and the Grothendieck norm are inequivalent, i.e., √ mn A / A G can be arbitrarily large, as shown by Example 1.2. Our main theorem below shows, somewhat surprisingly, that the two norms agree for matrices associated to vertex-transitive graphs.
From now on we consider only square matrices. Let A = (a s,t ) 1≤s,t≤n . We say that a permutation σ of [n] is an automorphism if A σ (s),σ (t) = A s,t for all s,t ∈ [n]. We call a subgroup G of the permutations of [n] transitive if for every s,t ∈ [n] there is some σ ∈ G such that σ (s) = t. We say that A is vertex-transitive if its group of automorphisms is transitive on [n].
Theorem 2.2. A G = n A for all n × n vertex-transitive real-valued matrices A.
Using (6), we obtain the following corollary relating the cut norm and the spectral norm, from which Corollary 1.5 clearly follows.

Corollary 2.3. A cut ≤ n A ≤ 8 A cut for all n × n vertex-transitive real-valued matrices A.
As discussed in the introduction, the proof will proceed through an analysis of Cayley graphs, which form the subject of the next section.

Cayley graphs
We consider a weighted version of Cayley graphs. Let f : G → R. Define a weighted directed graph Γ = Cay(G, f ) with vertex set G and edge weights Γ : G × G → R given by Γ(g, h) := f (gh −1 ) for all g, h ∈ G. Let A( f ) denote the associated matrix, whose rows and columns are indexed by G, with a g,h = Γ(g, h) = f (gh −1 ) for all g, h ∈ G. The natural analogue of the assumption that S is symmetric (giving undirected Cayley graphs) would be f (g) = f (g −1 ), implying that A( f ) is symmetric, though we will not need to make such an assumption here.
We define various norms for functions f : G → R corresponding to the matrix norms defined in Section 2. We use the averaging measure on G. Define the p-norms for functions on G by The spectral norm is defined as follows, agreeing with the matrix version (up to normalization): 2 We may assume that H = R k for some k ≤ m + n and we have x 1 , . . . , x m , y 1 , . . . , y n ∈ B(H) such Our main result now is that the spectral norm agrees with the Grothendieck norm for functions on a group.
Theorem 3.1. f = f G for every function f : G → R on a finite group G.
We thank Assaf Naor and Prasad Raghavendra for independently suggesting the following short proof of Theorem 3.1 after Y. Z. presented an initial version of this work at the Simons Symposium on Analysis of Boolean Functions in April 2016. We include their proof here with permission. Our original proof using representation theory can be found in the appendix.
Proof. Let f : G → R. Let x, y : G → R with x 2 ≤ 1 and y 2 ≤ 1 be such that Define x g (h) = x(gh) and y g (h) = y(gh) for all g, h ∈ G. We view x g and y h as vectors in the unit ball in L 2 (G) equipped with inner product x, y = E g∈G x(g)y(g) for all x, y ∈ L 2 (G). We have Combining with f G ≤ f from (7), we obtain f G = f .

Transitive graphs
Now we show that Theorem 3.1 on functions f : G → R implies Theorem 2.2 on vertex-transitive matrices. The idea is that we can lift the edges of a transitive graph to a Cayley graph over its automorphism group and this operation preserves all the norms that we are interested in.
Recall that A = (a s,t ) s,t∈[n] is an n × n vertex-transitive matrix if the group of automorphisms of A is transitive on [n]. Let G be any transitive subgroup of automorphisms (it could be the whole automorphism group or any transitive subgroup). Here every element g ∈ G ≤ S n is some permutation of [n], i.e., g : [n] → [n], and the product of two elements g, h ∈ S n is defined by (gh)(s) = h(g(s)) for all s ∈ [n]. We have a g(s),g(t) = a s,t for all s,t ∈ [n] and g ∈ G. Proof. The proofs of these two identities are essentially the same. Let H denote either an arbitrary Hilbert space (for the Grothendieck norm) or simply R (for the spectral norm). For any x 1 , . . . , x n , y 1 , . . . , y n ∈ H, define x, y : G → H by x(g) = x g (1) and y(g) = y g(1) for all g ∈ G. Then a s,t x s , y t . (9) Furthermore, we have n −1 ∑ s∈[n] |x s | 2 = E g∈G |x(g)| 2 , and similarly with y. This shows n f ≥ A and Conversely, given x, y : G → H, we can set x 1 , . . . , x n , y 1 , . . . , y n ∈ H to be x s = E g∈G:g(1)=s x(g) and y s = E g∈G:g(1)=s y(g) for all s ∈ [n]. Then (9) still holds. We also have n −1 ∑ s∈[n] |x s | 2 ≤ E g∈G |x(g)| 2 by convexity, and similarly with y. This shows that n f ≤ A and n 2 f G ≤ A G .

Bipartite analogues
There are also variants of our results in the bipartite setting. Given a group G of order n and a function f : G → R, the bipartite Cayley graph Γ = Bip(G, f ) is the bipartite graph between two copies of G with edge weights f : G × G → R given by Γ(g, h) = f (gh −1 ). We write B( f ) for the associated matrix, whose rows and columns are both indexed by G, with b g,h = Γ(g, h) = f (gh −1 ). Since we did not require that f be symmetric in Theorem 3.1, we can apply it directly to prove that B( f ) G = n B( f ) for any f : G → R. An analogue of Theorem 1.4 then follows provided we replace the second largest eigenvalue with the largest singular value of B(Γ) − d n J, where J is the all-1's matrix. We also have an analogue of Theorem 1.5. We say that a bipartite graph Γ between sets U and V of orders m and n, respectively, is bitransitive if the group of automorphisms of Γ is transitive on each part. For example, the graph between any two consecutive layers of the Boolean cube satisfies this condition. A slight amendment to the proof of Lemma 4.1 then allows us to show that if B is an m × n bitransitive real-valued matrix, then B G = √ mn B . The claimed analogue of Theorem 1.5 then follows easily, with the second largest eigenvalue now replaced by the largest singular value of B(Γ) − pJ, where p is the density of Γ.

A Representation-theoretic proof
We include in this appendix our original proof of Theorem 3.1, that f = f G for f : G → R, using representation theory. The proof follows the spirit of Gowers' work on quasirandom groups [7] and is a natural generalization of the proof for the abelian case [12].

A.1 Abelian groups
As a warm up, we give a short proof in the abelian setting, which is significantly easier. This proof is instructive, though not strictly required for the general setting.
Here and throughout the appendix, we will work with functions f : G → C and with complex analogues of our various norms. For the most part, the definitions of these norms are the same as in the real case or the same after some minor modification. For example, letting D := {z ∈ C : |z| ≤ 1} denote the unit disk in the complex plane, the complex ∞ → 1 norm for functions f : G → C is defined by The essence of the proof below can be found in Kohayakawa-Rödl-Schacht [12] (where the proof is attributed to Gowers). Here we give a more streamlined presentation.
for every function f : G → C on a finite abelian group G. Proof. It is easy to see that f C ∞→1 ≤ f from the definitions, as x 2 ≤ 1 for all x : G → D. It is well known that f = |E g∈G f (g)χ(g)| for some multiplicative character χ : G → C. On the other hand, since Similarly to (5), it can be shown that the cut norm is equivalent to the complex ∞ → 1 norm for complex-valued matrices. Together with Theorem A.1, this implies that the cut norm and the spectral norm are equivalent for abelian Cayley graphs.
Remark. For non-abelian groups G, it is possible to have f > f C ∞→1 . If an undirected weighted Cayley graph on G admits a coordinate-wise uniformly bounded eigenbasis, then one has f = O( f C ∞→1 ) by the above argument. 3 However, as will be addressed in future work, not all Cayley graphs admit such eigenbases. Consequently, the above proof does not extend immediately to the non-abelian case. The proof given below instead generalizes the above proof by replacing the multiplicative characters with group representations.
In the next three subsections, we recall some standard mathematical tools used in our proof. The reader familiar with these tools should feel free to skip ahead to Section A.5.

A.2 Group representation theory
We recall some basic facts from group representation theory. They can be found in any standard textbook on the subject, e.g., [18].
Let G be a finite group, not necessarily abelian. We write G for the set of distinct irreducible representations of G. An irreducible representation ρ ∈ G of dimension d ρ is a homomorphism ρ : G → U(d ρ ), where U(n) is the group of unitary n × n matrices (so that ρ(g −1 ) = ρ(g) * ). Note that by a standard averaging trick, we need to only consider unitary representations.
We will need the following orthogonality result known as Schur's lemma.
Theorem A.2 (Schur's lemma). Let G be a finite group and let ρ, σ ∈ G. Let M be any d ρ × d σ matrix with complex entries. Then

A.3 Non-abelian Fourier analysis
Let us first recall the Fourier transform for abelian groups. Let G be a finite abelian group and G the group of multiplicative characters χ : G → C. For any function f : G → C, we can define its Fourier transform f : G → C by f (χ) = E g∈G f (g)χ(g). We then have the inversion formula f (g) = ∑ χ∈ G f (χ)χ(g).
From now on, let G be any finite group, not necessarily abelian, and G the set of distinct irreducible unitary representations ρ : G → U(d ρ ) of G. For any f : G → C and any irreducible representation ρ ∈ G, define the Fourier transform of f at ρ by f (ρ) = E g∈G f (g)ρ(g).
Note that f (ρ) is a d ρ × d ρ matrix, so its dimension varies with ρ.  3 We thank Assaf Naor for this remark.
If A and B are d × d matrices and b 1 , . . . , b d are the columns of B, then AB 2