When he was 14, Joshua Wong cofounded Scholarism, the Hong Kong student movement that successfully protested the introduction of a “patriotic” curriculum. Now he is one of the student leaders of the Hong Kong pro-democracy movement.

Despite facing continued violence from triad-affiliated gangsters, the occupation continues, always in a uniquely Hong Kong manner.

Today Joshua Wong turns 18, and he gains the right to vote. May he be able to use this right freely!

[Photo by Anthony Kwan, video by the New York Times]

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Discussion on how to regulate the 2017 elections has been going on for the last several months. A coalition of pro-democracy groups ran an informal referendum on the preferred system of election, gathering about 800,000 votes, or a fifth of the registered electorate. All the options in the referendum assumed no vetting process for the candidate, contrary to Beijing’s stance that any system for the 2017 election would only allow candidates pre-approved by the mainland government.

Afterwards (this happened during the summer), the pro-democracy groups organized an enormous rally, which had hundreds of thousands of participants, and announced plans to “occupy Central with love and peace” (Central contains the financial district) on October 1st if the Hong Kong legislature passed an election law in which candidates could run only with Beijing’s approval.

This was followed by an anti-democracy rally, partly attended by people bused in from across the border, which is a rather surreal notion; it’s like people are saying “we want our voices heard about the fact that we do not want our voices heard.”

A few days in advance of October 1st, a group of university students, some of them associated with group scholarism started a sit-in at a government building. Scholarism made news three years ago, when it (successfully) fought the proposal to introduce a “patriotic education” curriculum in grade school.

People have been facing the police with umbrellas and goggles to protect themselves from pepper spray.

The plaza in front of the government building, where the sit-in started, has been cleared, but for the whole weekend both Central and the neighboring district of Admiralty have been filled by thousands of protesters, day and night.

There is a petition at whitehouse.gov that has already exceeded the threshold required to receive a response, but that people might want to sign on.

Considering how the Chinese government feels about students rallying for democracy, there is reason to be worried.

[photos taken from Facebook, credits unknown]

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Those two days were busy, and he met Italian start-up founders in the area, went to a dinner at Stanford hosted by John Hennessy, presided the inauguration of a new Italian-language school, went to Twitter, Google, Facebook and Yahoo, and he met the local Italian research community.

For the last event, a few colleagues and I were asked to give a short presentation. Being not sure what to say to a prime minister, I asked a colleague who is the department chair at an Italian computer science department for some data on state funding of university research in computer science, and if there was a way to turn this data into a recommendation, and his response could be summarized as “we cannot be saved, there is no hope.” This might have made a good theme for a presentation, but instead I talked about the importance of fundamental research, and of working on ideas for which the technology is not ready yet, so that when the technology is ready the ideas are mature. Politicians are good at feigning attention when their mind is elsewhere, and he feigned it well.

Yesterday I was “interviewed” as part of the process to naturalize as an American citizen. Part of the interview is a test of the knowledge that an American is supposed to have. I liked to think that the officer would bring up a map of the world and ask me to point to France, then I would point to Brazil, and he would say, “great, now you are ready to be an American!” (Instead he asked how many US senators there are, when was the constitution written, and things like that.) The vibe was very different from any other interaction I have had with the American immigration system before; now it’s not any more “who are you, why are you stealing our jobs, and how do we know you are not a terrorist,” but it’s all “yay, you are going to be one us.”

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In a previous post we talked about the definition of the Ihara zeta function of a graph, and Ihara’s explicit formula for it, in terms of a determinant that is closely related to the characteristic polynomial of the adjacency matrix of the graph, so that the zeroes of the zeta function determine the eigenvalue of the graph. And if the zeta function of the graph satisfies a “Riemann hypothesis,” meaning that, after a change of variable, all zeroes and poles of the zeta function are in absolute value, then the graph is Ramanujan. Conversely, if the graph is Ramnujan then its Ihara zeta function must satisfy the Riemann hypothesis.

As I learned from notes by Pete Clark, the zeta functions of curves and varieties in finite fields also involve a determinant, and the “Riemann hypothesis” for such curves is a theorem (proved by Weil for curves and by Deligne for varieties), which says that (after an analogous change of variable) all zeroes and poles of the zeta function must be in absolute value, where is the size of the finite field. This means that one way to prove that a family of -regular graphs is Ramanujan is to construct for each graph in the family a variety over such that the determinant that comes up in the zeta function of is the “same” (up to terms that don’t affect the roots, and to the fact that if is a root for one characteristic polynomial, then has to be a root for the other) as the determinant that comes up in the zeta function of the variety . Clark shows that one can constructs such families of varieties (in fact, curves are enough) *for all the known algebraic constructions of Ramanujan graphs*, and one can use families of varieties for which the corresponding determinant is well understood to give new constructions. For example, for every , if is the number of distinct prime divisors of (which would be one if is a prime power) Clark gets a family of graphs with second eigenvalue . (The notes call the number of distinct prime divisors of , but it must be a typo.)

So spectral techniques underlie fundamental results in number theory and algebraic geometry, which then give us expander graphs. Sounds like something that we (theoretical computer scientists) should know more about.

The purpose of these notes is to explore a bit more the *statements* of these results, although we will not even begin to discuss their proofs.

One more reason why I find this material very interesting is that all the several applications of polynomials in computer science (to constructing error-correcting codes, secret sharing schemes, -wise independent generators, expanders of polylogarithmic degree, giving the codes used in PCP constructions, self-reducibility of the permanent, proving combinatorial bounds via the polynomial method, and so on), always eventually rely on three things:

- A multivariate polynomial restricted to a line can be seen as a univariate polynomial (and restricting to a -dimensional subspace gives a -variate polynomial); this means that results about multivariate polynomials can often be proved via a “randomized reduction” to the univariate case;
- A univariate polynomial of degree has at most roots, which follows from unique factorization
- Given desired roots we can always find an univariate polynomial of degree which has those roots, by defining it as .

This is enough to explain the remarkable pseudo-randomness of constructions that we get out of polynomials, and it is the set of principles that underlie much of what is below, except that we are going to restrict polynomials to the set of zeroes of another polynomial, instead of a line, and this is where things get much more complicated, and interesting.

Before getting started, however, I would like to work out a toy example (which is closely related to what goes on with the Ihara zeta function) showing how an expression that looks like the one defining zeta functions can be expressed as a characteristic polynomial of a linear operator, and how its roots (which are then the eigenvalues of the operator) help give bound to a counting problem, and how one can get such bounds directly from the trace of the operator.

If we have quantities that we want to bound, then the “zeta function” of the sequence is

For example, and this will be discussed in more detail below, if if a bivariate polynomial over , we may be interested in the number of solutions of the equation over , as well as the number of solutions over the extension . In this case, the zeta function of the curve defined by is precisely

and Weil proved that is a rational function, and that if are the zeroes and poles of , that is, the roots of the numerator and the denominator, then they are all at most in absolute value, and one has

How does one get an approximate formula for as in (2) from the zeroes of a function like (1), and what do determinants and traces have got to do with it?

Here is our toy example: suppose that we have a graph and that is the number of cycles of the graph of length . Here by “cycle of length ” we mean a sequence of vertices such that and, for , . So, for example, in a graph in which the vertices form a triangle, and are two distinct cycles of length 3, and is a cycle of length 2.

We could define the function

and hope that its zeroes have to do with the computation of , and that can be defined in terms of the characteristic polynomial.

Indeed, we have (remember, is a complex number, not a matrix):

and, if are the zeroes and poles of , then

How did we get these bounds? Well, we cheated because we already knew that , where are the eigenvalues of . This means that

Where we used . Now we see that the poles of are precisely the inverses of the eigenvalues of .

Now we are ready to look more closely at various zeta functions.

I would like to thank Ken Ribet and Jared Weinstein for the time they spent answering questions. Part of this post will follow, almost word for word, this post of Terry Tao. All the errors and misconceptions below, however, are my own original creation.

We will work up to the definition of the zeta function of curves in three steps: first we look at the zeta function for the integers, and review (all without proof) the formulas that relate it to the distribution of the primes. This will give us an idea of what to expect in other settings.

Then we define the zeta function for “the affine line over ,” in which integers are replaced by univariate polynomials in and primes are irreducible polynomials. We will get the “prime number theorem” that approximately a fraction of degree- polynomials are irreducible. The zeta function ends up being just (the inverse of) a linear function, and it has no roots whatsoever.

Finally, we get to curves: here the place of integers is taken by polynomial functions over the curve, and the primes are the prime ideals of the ring of those polynomial functions. After a few more miracles happen, understanding the zeta function will come down to understanding how many points are there on our curve when we think of it as being defined in instead of , for various choices of . This can also be seen as counting the number of fixed points of the -th power of the Frobenius function, and several more miracles allow us to count the number of such fixed points in terms of a polynomial which is in fact the characteristic polynomial of a linear operator, finally!, but there is a catch: this linear operator acts on the first homology group of the curve, and the stuff which is in that group is not vectors over the reals, so it’s not like our operator is a matrix of which we can compute the determinant just like that. Luckily, there is still a characteristic polynomials, and its roots are algebraic integers that happen to be roots of unity times .

**1. The zeta function for the integers **

Let’s start again with the zeta function defined over the integers. We have

which is well defined for all complex numbers such that . There is a unique function that agrees with the above definition for all such that , which is defined for all complex numbers, and which is meromorphic, meaning that is very nice (infinitely differentiable, equal to its Taylor series) except at a bounded number of place; we will call this extended function from now on. Whenever we define functions that don’t seem to make sense for some choice of inputs, such an extension is assumed.

We also define a normalized (and “smoothed”) indicator function of the primes, the von Mangoldt function such that if is a power of the prime , and if has at least two distinct prime factors. We have

and after some manipulations one gets the following relation between zeta function and von Mangoldt function:

which in turn leads to the explicit formula

where the summation ranges over the roots of the zeta function such that and, so if the Riemann hypothesis is true, which holds that such roots must have , we would have

we also have

and so, if the Riemann hypothesis is true, the number of primes between and would be plus or minus . (This is in fact an if-and-only-if.)

**2. The zeta function of the affine line **

Now we think about univariate polynomials over a finite field as taking the place of the integers. This is the case of the “affine line,” because we are talking about polynomials defined over all of which, as a 1-dimensional linear space, is a “line.” Our polynomials, however have arbitrary degree!

So is replaced by . Factorization, however, is defined over the natural numbers, not over . What should take the place of ? There is a simple answer (spoiler alert: monic polynomials), but let’s get there in a way that is easy to generalize.

When we are in a ring, like and , that is a set where we have operations of addition and multiplication, and where division is not always well defined (and hence factorization is interesting), the right objects to think about from the point of view of factorization are the *ideals* of the ring. An ideal is a subset that is a subgroup for addition and that is closed under multiplication. An ideal generated by an element , written , is the set of all multiples of , and it is called a *principal* ideal. It is easy to see, over the integers, that all ideals are principal.

(For a given ideal , compute the gcd of all the nonzero elements of the ideal: it is clear that is a subset of , because all elements of are multiples of , but can be obtained, using Euclid’s algorithm, as sums and products of elements of , and so it is in , which means that is a subset of .) The same is true in : all ideals are principal, and for the same reason: there is a version of Euclid’s algorithm for finding the gcd of polynomials. One can properly define factorization and primes only for ideals; in the case of integers, every ideal corresponds to the integer that generates it, and two different integers always define different ideals, except for . So we can use positive integers as representative for the ideals. In the case of polynomials, we have the same reasoning, except that iff is a scalar multiple of . So our representative for ideals will the *monic* polynomials, that is, polynomials whose leading coefficient is one. This answer our first question. Now comes another question: the formula for the zeta function over integers involves the number ; if is a monic polynomial, is this going to be replaced by ? But is a complex number and is a polynomial over a finite field, so how is that going to work out? Again, we look at this question in a general setting: we already said that we should think of the integer appearing in the formula for the zeta function as a representative of the ideal . Now, every ideal in a ring has a *norm*, which is the number of shifted copies of the ideal that we need to cover the whole ring, or, more precisely, the cardinality of the quotient . Clearly, the cardinality of the quotient is , because the quotient of by the multiples of is just the ring of integers mod , . What is the norm of a monic polynomial of degree over ? Think about this for a minute. The ring is made of polynomials with operations mod , so every element of is identified by the remainder of the division of by . This remainder is a polynomial of degree , defined by coefficients, and there are such possible remainders.

We conclude that the norm of is and we can finally write our zeta function

and it is common to make the change variable and work with

Now our “primes” are irreducible (in ) monic polynomials.

We still have a von Malgoldt function if and is irreducible, and otherwise. We have, for every monic polynomial ,

and the exponential identity

and manipulations as in the integer case lead to the explicit formula

where ranges over the roots of and the constant term depends on poles of . Nicely, we can just figure out what the zeta function is: there are exactly monic polynomials of degree , and plugging this fact into (3) we have

so there are no roots. One could view as where is the 1-dimensional identity matrix and is the 1-dimensional matrix whose only entry is . From (4), the explicit formula becomes

from which we get that degree- monic polynomials are irreducible.

This gives the analogy with the case of the integers, but there is a much more direct way of seeing that (5) is true.

Every monic polynomial over of degree is going to have roots in the algebraic closure of , counting multiplicities, and so it can be written as

note that while the roots may or may not belong to , all the coefficients of do belong to , once we expand the product. We can certainly take the roots to be a unique representation of , although we should be aware that not all -tuples of elements from the algebraic closure of represent a monic polynomial of degree with coefficients in . (There are infinitely many such tuples but only such polynomials.)

To understand which tuples correspond to monic polynomials in , in a way that will also gives a good understanding of the right-hand side of (5), we introduce the Frobenius function , which is well defined for every but also for every in any extension of . Three useful properties of this function are that:

- for every (if is prime this is Fermat’s little theorem), in fact, if belongs to an extension of ,
*if and only if*; - is bijective in any extension field ;
- is periodic with period in . This is just another way of saying that in , which is a special case of the first property.

(Many readers will recognize the above as a clumsy statement of the fact that, for every , the Galois group is cyclic of order , and that is a generator.)

From the first property we get that if is a root of and has coefficients in , then is also a root; but then also and so on. From the second property, we get that if we look at all the roots of , they all live in , and so the Frobenius function is going to be a permutation over (because it maps roots to roots, and it does so injectively).

Let us look at the cycles defined by over the multi-set . Each cycle needs to have a length that divides , because is -periodic by the third property. Let’s call two roots in the same cycle *equivalent* and let pick representatives , of classes of size , respectively. We see that each lives in and that, if we are given the we are implicitly being told all the roots (which we can rediscover by applying ) and now this is a much more economical representation of the polynomial. In fact this is a unique representation up to the choice of representatives, that is, a polynomial may be identified with the (multi)-set of equivalence classes described above. Each equivalence class is called a *closed point*.

We have reached a punchline: a monic polynomial of degree correspond to a single closed point of size , and the representation of a generic monic polynomial as a collection of closed points corresponds to the factorization of into a product of monic polynomials. Notice also that this argument shows that there can be *at most* monic polynomials of degree over . (Why?)

Now let’s forget about any specific polynomial , and let us look at all the closed points in . Suppose that we find closed points , each of size , where each must divide . Note that . Then each corresponds to an irreducible monic polynomial of degree such that is a polynomial of degree such that . In turn, these are all the monic polynomial of degree which are powers of irreducible polynomials, and so we have proved the exact formula

**3. The zeta function for an affine curve **

Now we have an affine irreducible curve in , that is, the set of points such that , where is a bivariate polynomial that is absolutely irreducible, that is, it has no non-trivial divisor even in the algebraic closure of . For example, the curve might be .

The role of the integers is now played by the *coordinate ring* of , which is . There are a couple of ways of thinking about it. One is the definition: an element is an equivalence class of polynomials, where two polynomials are equivalent if their difference is a multiple of . (In particular they need to agree on the zeroes of .) The other is to thing of an element of the coordinate ring as being a function defined on that is expressed as a polynomial. If were a line, then every element of the coordinate ring would just be a bivariate polynomial restricted to a line, and thus would be essentially a univariate polynomial. That is the case studied in the previous section.

When is not a line, the coordinate ring is more complex, and, in particular, it is not true that all ideals are principal. Although it is not possible any more to think of the ideals as polynomials, we can still think of them as being multi-sets of points. This time an ideal will be a *rational effective divisor*, that is a multi-set of points of that is Frobenius-invariant. The prime ideals will be closed points: divisors that are a single orbit of the Frobenius. Each divisor has a degree (the number of points counted with multiplicities) and its norm is .

We can then construct analogs of all the expressions we saw in the previous section. We have the definition of zeta function

that after the change of variable becomes

we have the von Malgoldt function which is equal to if is a power of and is prime, and if has at least two distinct prime divisors. We have the exponential formula of

which can be rewritten as

and finally we have

where the right-hand side is the number of points of –that is, solutions of ) in . If was a line, this number would just be . The end of the story of the Riemann hypothesis for curves is that it is more or less the same for every absolutely irreducible ; the number of points will be where the implied constant depends only on , as proved by Weil. His bound is where is the genus of the curve.

If this were all that we are interested in, Stepanov has given a fully elementary proof, and one can read about it in this monograph by Chen, Kayal and Wigderson in a treatment that does not talk about zeta functions, divisors, ideals and so on. The connection between zeroes of the zeta function and the size of is that if one can write the zeta function as a rational function and is the collection of roots, with multiplicities, of the polynomial in the numerator and the one in the denominator, one has

and Weil’s proof establishes that there at most roots, each of absolute value at most , thus the bound claimed above.

Although Stepanov’s technique gives an elementary way to bound , recall that our interest in this story is how to understand just the *statement* of Weil’s result as it concern a bound of to the absolute value of the eigenvalues of a certain operator.

Extend the Frobenius function to be defined over pairs of elements in the obvious way: ( maps to . Define to be the application of the Frobenius function times, thus . Recall that one of the properties of the Frobenius function is that if and only if . Consider , the set of points of in the algebraic closure of . Then is the number of *fixed points* of in the space .

In the complex case, if we have a curve , an algebraic function , and we want to count the number of fixed points of , we have the Lefschetz trace formula, which says that all we need to do is to estimate the eigenvalues of when viewed as a linear operator over the first homology group of — whatever that is. (See this blog post.)

Luckily, a Lefschetz trace formula exists in the finite fields case. Unfortunately, I need to end this post on a cliffhanger, because I don’t know how to describe the first homology group of (the formula involves the zero-th, the first, and the second homology group of , but only the first gives a non-trivial contribution).

So far, I am being told that it is not a vector space but a module, and that its ring of coefficients is made of -adic numbers for coprime with . The dimension is , and the Frobenius operator acts as a linear operator. I am also being told that “ what exactly are the elements of the group” is not the right question to ask, and that one should start from the properties that one wants this group to satisfy, verify that there is a construction that meets these properties, and then work with the properties and forget the construction.

In modules, one can still define a determinant and a characteristic polynomial for a linear operator, and here another miracle happens, and the coefficients of this polynomial end up being standard integers, its roots are algebraic integers, and so we talk about the roots being in absolute value.

I hope that there will be a future installment of this ongoing series, in which I explain what is the first homology group and I give some intuition about fixed point formulas in various settings, and, while I am there, maybe even explain what the Riemann-Roch theorem states.

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Here is a a first draft.

If you notice that work that you know of (for example, your work) is misrepresented or absent, please let me know and I will edit the document. (If possible, when you do so, do not compare me to Stalin and cc your message to half a dozen prominent people — true story.)

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It is shocking that the company decided that it was a good idea to destroy an investment they had made over several years; I am only familiar with the theory side of MSR SVC, which had great success in nurturing young talent and developing breakthrough ideas and results. The work on privacy led by Cynthia Dwork, for example, informed the thinking on privacy of higher management, and on how one could find new ways to balance European privacy law with the data collection necessary for advertising. This is one of the returns of having academic research in a company: not so that they can implement in C++ their algorithms, but so that they can generate and communicate new ways of thinking about problems. (Cynthia is one of the few people retained by Microsoft, but she has lost all her collaborators.) Microsoft’s loss will be other places’ win as the MSR SVC diaspora settles down elsewhere.

It is also shocking that, instead of planning an orderly shutdown, they simply threw people out overnight, which shows a fundamental indifference to the way the academic job market works (it can take a full year for an academic job offer to materialize).

I am not shocked by the class demonstrated by Omer Reingold; the moral stature of people is best seen in difficult moments. Omer has written a beautiful post about the lab, whose comment section has become a memorial to the lab, with people posting their personal remembrances.

Here at Berkeley and Stanford we will do our best to help, and we will make sure that everybody has some space to work. There will also be some kind of community-wide response, but it will take some time to figure out what we can do. Meanwhile, I urge everybody to reach out to their friends formerly at MSR SVC, make them feel the love of their community, and connect them to opportunities for both short term and long term positions, as they weigh their options.

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Known constructions of expanders produce Cayley graphs (or sometimes Schreier graphs, which is a related notion), because it is easier to analyze the spectra of such graphs. If is a group with operation and is the inverse of element , and is a symmetric set of generators, then the Cayley graph is the graph whose vertices are the elements of and whose edges are the pairs such that .

When the group is Abelian, there is good news and bad news. The good news is that the eigenvectors of the graphs are completely characterized (they are the characters of ) and the eigenvalues are given by a nice formula. (See here and here.) The bad news is that constant-degree Cayley graphs of Abelian groups cannot be expanders.

That’s very bad news, but it is still possible to get highly expanding graphs of polylogarithmic degree as Cayley graphs of Abelian groups.

Here we will look at the extreme case of a family of graphs of degree , where is the number of vertices. Even with such high degree, the weak version of the Alon-Boppana theorem still implies that we must have , and so we will be happy if we get a graph in which . Highly expanding graphs of degree are interesting because they have some of the properties of random graphs from the distribution. In turn, graphs from have all kind of interesting properties with high probabilities, including being essentially the best known *Ramsey graphs* and having the kind of *discrepancy* property that gives *seedless extractors for two independent sources.* Unfortunately, these properties cannot be certified by spectral methods. The graph that we will study today is believed to have such stronger properties, but there is no known promising approach to prove such conjectures, so we will content ourselves with proving strong spectral expansion.

The graph is the Paley graph. If is a prime, is the group of addition modulo , and is the set of elements of of the form , then the graph is just . That is, the graph has a vertex for each , and two vertices are adjacent if and only if there is an such that .

Consider , the finite field with elements of which is the additive group; of the nonzero elements of , half of them are quadratic residues, and thus elements of , each with two square roots, and half of them are not. Thus, our graph has degree .

The characterization of the eigenvalues of an Abelian Cayley graph tells us that we are going to have an eigenvalue for each , given by the formula

and so we just have to estimate the above sums. If we sum over all for in instead of just summering over we get

because every term in the formula for occurs twice, and we also get an extra 1 from the case . The sums are called Gauss (quadratic) sums, and Gauss himself proved that, for , we have . This means that, for , we have , and so we have the desired expansion.

Here is how we prove Gauss’s result.

For non-zero , define if is quadratic residue and otherwise. Then is a character of the multiplicative group of : indeed we have , and . A well-known name for is that it is the *Legendre symbol* .

To simplify notation, let us also call . Thus, we have

In the following, it will simplify notation to make be defined over all of , so we will set . Here is where comes in:

Lemma 1For ,

*Proof:* This is a one-observation proof: for every , we have

and so

where all the summations are over and, in the last line, we use the fact that for .

And now we do the last calculation.

Lemma 2For every non-trivial multiplicative character of and every non-trivial additive character of we have

where we extended to all of by defining .

*Proof:* We write down the sum and use the fact that is a multiplicative character and is an additive character.

Do a change of variable

and is always zero except, when , in which case it is .

(The above is not completely rigorous because we have “divided by zero.” A more precise accounting is to take the above sums for all but only the nonzero . In this case, after the change of variable, will range over and still only on . We can then bring back the case in the summation by noting that .)

So, putting it all together, we have for every , and so the second smallest eigenvalue of the adjacency matrix of the Paley graph is, in absolute value, at most , where is the degree.

Notice that all we used about our set of generators is that their indicator function is a multiplicative character.

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be the eigenvalues of the adjacency matrix of counted with multiplicities and sorted in descending order.

How good can the spectral expansion of be?

**1. Simple Bounds **

The simplest bound comes from a trace method. We have

by using one definition of the trace and

using the other definition and observing that counts the paths that go from to in two steps, of which there are at least : follow an edge to a neighbor of , then follow the same edge back. (There could be more if has multiple edges or self-loops.)

So we have

and so

The condition is necessary to get lower bounds of ; in the clique, for example, we have and .

A trace argument does not give us a lower bound on , and in fact it is possible to have and , for example in the bipartite complete graph.

If the diameter of is at least 4, it is easy to see that . Let be two vertices at distance 4. Define a vector as follows: , if is a neighbor of , and if is a neighbor of . Note that there cannot be any edge between a neighbor of and a neighbor of . Then we see that , that (because there are edges, each counted twice, that give a contribution of to ) and that is orthogonal to .

**2. Nilli’s Proof of the Alon-Boppana Theorem **

Nilli’s proof of the Alon-Boppana theorem gives

where is the diameter of . This means that if one has a family of (constant) degree- graphs, and every graph in the family satisfies , then one must have . This is why families of *Ramanujan* graphs, in which , are special, and so hard to construct, or even to prove existence of.

Friedman proves a stronger bound, in which the error term goes down with the square of the diameter. Friedman’s proof is the one presented in the Hoory-Linial-Wigderson survey. I like Nilli’s proof, even if it is a bit messier than Friedman’s, because it starts off with something that *clearly* is going to work, but the first two or three ways you try to establish the bound don’t work (believe me, I tried, because I didn’t see why some steps in the proof had to be that way), but eventually you find the right way to break up the estimate and it works.

So here is Nilli’s proof.

We are going to use essentially the same vector that we used to analyze the spectrum of the infinite tree, although the analysis will be a bit different.

Let be two vertices in at distance , and call . Let be a neighbor of . We say that the distance of a vertex from is the smallest of the shortest path distance from to and from to .

We construct a vector as follows:

- if
- if is at distance from

Note that this is more or less the same vector we constructed in the case of the tree. The reason for talking about the distance from two vertices instead of one, is that we want to say that every vertex is adjacent to at most vertices whose value of is strictly smaller; in the case of a tree, the root is exceptional because it has neighbors whose value of is strictly smaller. There will be a step in the proof in which this choice really makes a difference.

We claim

It turns out that, in order to prove (1) it is easier to reason about the Laplacian matrix than about the adjacency matrix. Define to be the (non-normalized) Laplacian of . We have the following nice expression for the quadratic form of

For every vertex let us call (for **s**maller) the set of neighbors of such that . We always have . Let us call the set of vertices at distance exactly from .

Now we do the calculations:

Finally,

because decreases (or at least does not increase) for increasing .

Putting everything together we have

and so

Now we are finally almost done: define a vector with the same construction we did for , but using the set as the reference for the distance, where is a neighbor of . We then have

It is clearly not possible for a vertex at distance from to also be at distance from , otherwise we would have a path of length from to , so the vectors and are non-zero on disjoint subsets of coordinate, and hence are orthogonal.

But we can say more: we also have , because there cannot be an edge such that both and , because otherwise we would have a path of length from to .

This means that if we take any linear combination of and we have

so we have found a two-dimensional set of vectors whose Rayleigh quotient is at most the above expression, and so

What did just happen? The basic intuition is that, as in the infinite tree, we set weights to go down by every time we get away from the “root,” and we would like to argue that, for every node , we have

by reasoning that one of the neighbors must be closer to the root, and hence have value , while the other neighbors are all at least . This bound fails at the “leaves” of the construction, which is fine because they account for a small portion of , but it also fails at the root, which is not adjacent to any larger vertex. In the case of the infinite tree this is still ok, because the root also accounts for only a small portion of ; in general graphs, however, the “root” vertex might account for a very large fraction of .

Indeed, the root contributes 1 to , and each set contributes . If the size of grows much more slowly than , then the contribution of the root to is too large and we have a problem. In this case, however, for many levels , there have to be many vertices in that have fewer than edges going forward to , and in that case, for many vertices we will have that will be much more than .

Although it seems hopeless to balance this argument and charge the weight of the edge in just the right way to and to , the calculation with the Laplacian manages to do that automatically.

**3. A More Sophisticated “Trace” Method **

The Hoory-Linial-Wigderson survey also gives a very clean and conceptual proof that

via a trace argument. I am not sure who was the first to come up with this idea.

Let us pick two vertices at distance , and call , and . Fix . Then and are orthogonal vectors, because the coordinates on which is nonzero correspond to vertices at distance from and the coordinates on which is nonzero correspond to vertices at distance from , and these conditions cannot be simultaneously satisfied. This means that .

Since is orthogonal to , we have

but we also have

Now, is the number of walks of length in that start at and get back to . In every -regular graph, and for every start vertex, the number of such walks is at least the corresponding number in the infinite -ary tree. This is clear if is a Cayley graph; it takes a few minutes (at least I had to think about it for a bit) to see that it holds in every graph.

Ok, then, what is the number of closed walks of length in the infinite -ary tree? This is a long story, but it is a very well-studied question and there is a bound (not the tightest know, but sufficient for our purposes) giving

so,

I put “trace” in quote in the title of this section, but one can turn the argument into an honest-to-God application of the trace method, although there is no gain in doing so. Pick an even number . We can say that

and

which gives

for , which is implied by the bound based on diameter that we proved above.

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