1. Lubotzky, Phillips, and Sarnak Construction of Ramanujan
Graphs
1 A Quick Review of Expanders
Recall that the edge isoperimetric or expanding constant h(X) for a general graph X is defined as
h(X) = inf
|∂F|
min{|F|, |V − F|}
: F ⊆ V, 0 ≤ |F| < ∞
which for a finite graph becomes
h(X) = min
|∂F|
|F|
: F ⊆ V, |F| ≤
|V |
2
And that a family of expanders is a sequence (Xn,k)k≥1 of k-regular graphs with |V (Xn,k)| → ∞
for n → ∞ and such that h(Xn,k) ≥ for all n and some > 0.
If k = µ0, µ1, . . . µn are the eigenvalues of the adjacency matrix of a k − regular graph X on n
vertices, then the following inequality due to Alon and Milman,
k − µ1
2
≤ h(X) ≤ 2k(k − µ1)
guarantees that a sequence of k-regular graphs (Xn,k) is a family of expanders if and only if k−µ1 >
for some > 0 and all n.
This raised the important question of how wide can we make the spectral gap between µ0 = k
and µ1. Recall that the Alon-Boppana bound says that for any sequence (Xn,k) of finite k-regular
graphs with |V (Xn,k)| → ∞ as n → ∞,
lim inf
n→∞
µ1 ≥ 2
√
k − 1
So that in the limit as the number of vertices goes to infinity, k − 2
√
k − 1 is the best we can hope
to do to bound the spectral gap.
This in turn motivates the definition of a Ramanujan Graph as a k − regular graph for which
µ1 < 2
√
k − 1. The name was coined by Lubotzky, Philips, and Sarnak and its origin will become
more apparent as we delve into their results.
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2. A natural question to ask is whether Ramanujan graphs of order n exist for every n. The answer
is yes, and is pretty easy to verify: just look at the spectra of Kn and Kn,n. Note however, that the
degree of these graphs is unbounded as n → ∞.
A question which has more meaning for applications (Kn isn’t very resource-efficient as a com-
munication network for instance) is “for a given fixed degree k, are there k-regular Ramanujan
graphs of arbitrarily large order?” This question is much more difficult to answer, and constructing
such a sequence of graphs is harder still.
The 1988 paper “Ramanujan Graphs” by Lubotzky, Philips, and Sarnak (LPS from here on),
supplies one of the first verified constructions of such a sequence. Margulis published a similar
construction independently in the same year, and the LPS construction has been generalized by
several authors subsequently, notably Chiu(1992) and Morgenstern(1994).
2 Outline of the LPS paper
There are essentially 5 important parts to the argument that the constructed graphs are Ramanujan,
in broadest outline:
1. Construct a multiplicative group Λ, and using some classical number-theoretic results, demon-
strate that it is a free group on p + 1 generators (where p is an odd prime, p ≡ 1 mod 4).
Observe that the Cayley graph of this group with respect to the generating set is thus iso-
morphic to the infinite p + 1-regular tree.
2. Construct a set of p + 1 matrices within PGL(2, Zq) (or possibly PSL(2, Zq)), depending on
whether p is a quadratic residue mod q). This set of generators thus determines a p+1-regular
Cayley graph on PGL(2, Zq), which we call Xp,q
.
3. Now, construct a homomorphism ϕ : Λ → PGL(2, Zq) (or possibly PSL), and observe that
ϕ is bijective on the respective generating sets. This ensures that the map induced on the
Cayley graph (the infinite tree) is indeed a graph homomorphism into the Cayley graph of
our generating set on PGL(2, Zq).
4. Demonstrate that ϕ is onto. This immediately tells us two important things:
– Our finite Cayley graph on PGL(2, Zq) (resp. PSL) is connected! This wasn’t obvious
before.
– We know what its order is! |PGL(2, Zq)| = q(q2
− 1) and |PSL(2, Zq)| = q(q2−1)
2
.
5. Define a specific linear operator on the space of functions on vertices of our Cayley graph.
Expand the action of this operator in terms of a spectral basis for the Laplacian operator. We
get an equation with, on one side, a sum over eigenfunctions (eigenvectors) of the Laplacian,
with coefficients depending on the Laplacian eigenvalues.
On the other side is a count rQ(pk
) of representations of pk
by a certain quaternary quadratic
form Q (which is the squared norm in a subgroup of our quaternion group Λ). Using known
(but very deep) number-theoretic results regarding this quadratic form, estimate the second
eigenvalue to show that Xp,q
is Ramanujan.
2

3. 3 Some Requisite Number theory
We will need a few classical (and one or two not-so-classical) results from number theory. These
will be the facts that tell us what is going on in our quaternion group Λ and will ultimately help
us with the left side of the equation in point 5 above.
• We say for any p, q ∈ N, “p is a quadratic residue mod q” if p ≡ a2
mod q for some a ∈ Zq.
• We define the Legendre symbol (for any p, q ∈ N) p
q
=



1 if p is a quadratic residue mod q
−1 if p is not a quadratic residue mod q
0 if p = 0 mod q
In particular we will need the fact that −1
p
= 1 ⇔ p ≡ 1 mod 4.
• We define H(Z) as the ring of integer quaternions, that is, quaternions with integer coefficients.
• We will frequently speak of the norm N(h) or |h| of h ∈ H(Z). In this context we will always
mean the algebraic norm, that is, the canonical multiplicative homomorphism from H(Z) into
Z, which is simply the square of the L2
norm.
This is where all the number-theoretic concern with quadratic forms comes in; the norm is a
sum of the four squares of the integer coordinates of an h ∈ H(Z).
• We will thusly need the classical Jacobi’s Theorem:
Theorem 3.1. The number of representations of a positive integer n as a sum of 4 squares
(which we denote r4(n)) is
r4(n) = 8
d|n,4 d
d
• And the much deeper Ramanujan’s Conjecture, now Deligne’s Theorem, which in particular
implies
Theorem 3.2. rQ(pk
) = C(pk
) + O (pk
) as k → ∞∀ ≥ 0, where
C(pk
) =



c1 d|pk d if p
q
= 1
c2 d|pk d if p
q
= −1 and k is odd
0 if p
q
= −1 and k is even
where Q(x1, x2, x3, x4) is the quadratic form x2
1 + 4q2
x2
2 + 4q2
x2
3 + 4q2
x2
4.
• Jacobi’s theorem tells us that there are exactly 8(p + 1) integer quaternions of norm p, for p
an odd prime. Note that if p ≡ 1 mod 4, we can surmise that
N(a0 + a1i + a2j + a3k) = a2
0 + a2
1 + a2
2 + a2
3 ⇒ exactly 1 of the ai is odd.
Because the norm is a multiplicative homomorphism into Z+
, we know that an element is a
unit iff its norm is 1, so there are 8 units, namely {±1, ±i, ±j, ±k}. These act as a group by
multiplication on the elements of norm p. If we choose any element α of norm p, it’s clear
that we can choose a unit so that a0 is positive and odd in α. Likewise, having chosen such
3

4. we see that its orbit contains 8 elements, so the 8(p + 1) elements of norm p have p + 1 orbit
representatives (which we can split into conjagate pairs). We call these
S = {α1, ¯α1, . . . , αk, ¯αk} where k = p+1
2
• Dickson(1922) showed that for integer quaternions of odd norm, we have a theory of GCD’s
and unique factorization. In addition we have that h ∈ H(Z) is prime ⇔ N(h) is prime.
Putting these facts together, it is not difficult to prove that
Theorem 3.3. Every α ∈ H(Z) of norm pk
with p an odd prime can be expressed uniquely
as
α = pr
Rm(S)
Where is a unit, Rm(S) is a reduced word on S (no conjugate pair appears consecutively),
and 2r + m = k.
4 PGL(2, Zq) and PSL(2, Zq)
We will need some elementary facts regarding these classical linear groups.
• For any field F, PGL(n, F) is defined as the quotient group GL(n, F)/Z(GL(n, F)), where
GL(n, F) is the set of invertible n × n matrices over F and Z(GL(n, F)) is the center, which
is simply the set of all scalar diagonal matrices having equal nonzero entries on the diagonal.
• PSL(n, F) is defined similarly except that we work only in the subgroup of matrices with
determinant 1, that is, SL(n, F). In other words, PSL(n, F) = SL(n, F)/Z(SL(n, F)).
• PGL(n, F) = PSL(n, F), (in the sense that any two coset representatives lie in the same
coset in PSL iff they lie in the same coset in PGL) ⇔ every element of F has an n-th root
in F (i.e. every element is the determinant of some scalar diagonal matrix).
• If PGL and PSL are finite, we have |PGL|
|PSL|
= [F×
: (F×
)n
], or equivalently the number of n-th
roots of unity in F.
• For a finite field Fq, we have an explicit count on |PGL(n, Fq)|:
|PGL(n, Fq)| =
(qn
− 1)(qn
− q) . . . (qn
− qn−1
)
(q − 1)
Which is easy to see by counting vectors and the subspaces they span in Fn
. In particular,
|PGL(2, Zq)| =
(q2
− 1)(q2
− q)
q − 1
= q(q2
− 1)
and combining this with the point above, we have
|PGL(2, Zq)| =
q(q2
− 1)
2
since the subgroup of quadratic residues has index 2 in Z×
q , or equivalently since ±1 are the
two square roots of unity in Zq. These will be the orders of our constructed Cayley graphs.
4

5. 5 Recap
We have now established the following:
• A set S = {α1, α1, α2, α2, . . . αp+1
2
, αp+1
2
} of p + 1 norm-p integer quaternions, the set of all
such with a0 odd and positive.
• Unique factorization of norm-pk
integer quaternions as εpr
Rm(S) where ε ∈ {±1, ±i, ±j, ±k}
is a unit, Rm(S) is a reduced word on S (no conjugate pair in succession), and 2r + m = k.
Existence follows from some old results on integer quaternions (α is prime ⇔ N(α) is), unique-
ness via a counting argument.
• A multiplicatively closed set Λ (2) of norm-pk
integer quaternions with a0 odd, a1, a2, a3 even.
Since this property is preserved under multiplication, we may immediately surmise from the
prior point that these factor uniquely in the above form with ε = ±1.
• A multiplicative group Λ(2) of equivalence classes of Λ (2) under the relation α ∼ β ⇔ pa
α =
±pb
β for some a, b ∈ N. Under ∼, we have αα = N(α) = pk
∼ 1, i.e. α−1
= α. Thus, the
set equivalence classes [S] of S is symmetric (closed under inversion), and Cay(Λ(2), [S]) is
undirected.
• Λ(2) is a free group on the generating set [S]; under the relation ∼, via unique factorization we
see that equivalence classes can be identified with the reduced words Rm(S), and multiplication
is by concatenation. So Cay(Λ(2), [S]) is the infinite p + 1-regular tree.
• A homomorphism ϕ : Λ(2) → PGL(2, Zq) or PSL(2, Zq), depending on whether p is a
quadratic residue modq. PGL = GL
Z(GL)
where Z(GL) is the center, the set of scalar matrices
(likewise, PSL = SL
Z(SL)
), and this ensures that ϕ is constant on equivalence classes in Λ(2)
(which contain a set of scalar multiples of a reduced word Rm(S)) and so is well-defined. We
can think of ϕ as factoring via σ ◦ π:
[α]
π
−→ α mod q
σ
−→
a0 + ia1 a2 + ia3
−a2 + ia3 a0 − ia1
where i is a square root of −1 in Zq (which exists iff we let q ≡ 1 mod 4). As sets,
Λ(2)
π
−→ H(Zq)/Z(H(Zq))
σ
−→ PGL(2, Zq) or PSL(2, Zq),
where we factor out the center (which is simply the scalars) in the image of π to ensure
ϕ is well-defined (constant on equivalence classes). ϕ also carries the norm function to the
determinant function, i.e. N(α) ≡ det(ϕ(α))(modq).
• ϕ maps [S] to a set of matrices with determinant pk
. Thus, if p
q
= 1 (p is a square
modq), every matrix in the image has square determinant, i.e. ϕ(α) = A(±I)U where U
has unit determinant and A is a scalar; i.e. im(ϕ) ⊆ (PSL(2, Zq). Otherwise, we know
im(ϕ) ⊆ (PGL(2, Zq).
• What remains to be determined is
5

6. 1. that ϕ is surjective. This will tell us the Cayley graph on PGL or PSL with generating set
ϕ([S]) is connected, as well as some structural information: Cay(ϕ([S]), PGL or PSL) is
bipartite when p
q
= −1 (since the generators take square-determinant cosets– in PSL,
of index 2– to non-square-determinant cosets, and vice-versa) Otherwise our Cayley
graph is not bipartite (since PSL(2, Zq) is simple, and any index-2 subgroup would be
normal).
2. that the second eigenvalue of the adjacency matrix of our Cayley graph does indeed
attain the Ramanujan bound, µ1 < 2
√
p.
6 The homomorphism ϕ
As noted above, ϕ factors as
Λ(2)
π
−→ H(Zq)/Z(H(Zq))
σ
−→ PGL(2, Zq) or PSL(2, Zq),
where π is the component-wise reduction homomorphism. σ is an isomorphism since given a coset
representative in PGL(2, Zq) we can map it thus:
a b
c d
→ a+d
2
+ i(a−d
2i
) + j(b−c
2
) + k(b+c
2i
)
working modq of course, with i2
= −1(modq) as before, and 2 has a multiplicative inverse since q
is odd.
In order to establish surjectivity it is sufficient to show that PSL(2, Zq) ⊆ im(ϕ), since PSL
has index 2 in PGL and certainly when p
q
= −1 we have im(ϕ) ∩ (PGL − PSL) = ∅. For any
unit-determinant matrix U over Zq then, we wish to find α ∈ H(Z) with ϕ(α) = U, a0 odd and
N(α) = pk
for some k. It will be fruitful to factor π further as
Λ(2)
π1
−→ H(Z2q)∗
/Z
π2
−→ H(Zq)∗
/Z
where π1 and π2 are the canonical reduction homomorphisms. Now, since σ is an isomorphism
taking norms to determinants, we can rephrase the question: given β = b0 + ib1 + jb2 + kb3 in
H(Zq) of norm 1(modq), we need α = a0 + ia1 + ja2 + ka3 ∈ H(Z) with α ≡ β mod q, a0 odd and
N(α) = pk
for some k.
At this point the authors cite a theorem of Malisev (1962) that says:
Theorem 6.1. Let f(x1, . . . xn) be a quadratic form in n ≥ 4 variables with integral coefficients and
discriminant d. Let (g, 2d) = 1. Then if m is sufficiently large (dependent only on the form f) with
(g, 2md) = 1, and (b1, . . . bn, g) = 1, f(b1, . . . bn) ≡ m(modg) then ∃(a1, . . . an) ≡ (b1, . . . bn)(modg)
with f(a1, . . . an) = m.
In our case, we take our quadratic form to be the norm of an integer quaternion with a1, a2, a3
even: f(x1, x2, x3, x4) = x2
1 + 4x2
2 + 4x2
3 + 4x2
4. Our discriminant (the determinant of the matrix
corresponding to the quadratic form) is 43
, so letting g = q and m = pk
, we certainly have (g, 2md) =
(q, 128pk
) = 1 since p = q.
6

7. By the theorem then, for k large enough and m = pk
≡ 1(modq), given (b0, b1, b2, b3) we have
a solution (a0, a1, a2, a3) ≡ (b0, b1, b2, b3)(modq) with f(a0, a1, a2, a3) = N(α) = m = pk
≡ 1 ≡
N(β)(modq). This establishes that ϕ is surjective.
Since the kernel of π1 is Λ(2q) = {α = a0 + ia1 + ja2 + ka3 : N(α) = pk
, a0 odd, 2q | a1, a2, a3},
we can now identify PGL(2, Zq), (resp. PSL) with Λ(2)/Λ(2q), considered as a group, and our
constructed Cayley graph Xp,q
as a homomorphic image of the infinite p+1-regular tree Λ(2) which
equates the vertices of Λ(2m) with the identity vertex. We also have the results about the structure
of Xp,q
in (1) above.
7