WEBVTT 00:00:16.533 --> 00:00:17.450 YUFEI ZHAO: All right. 00:00:17.450 --> 00:00:20.870 Last time we started talking about pseudorandom graphs, 00:00:20.870 --> 00:00:25.740 and we considered this theorem of Chung, Graham, and Wilson, 00:00:25.740 --> 00:00:30.470 which, for dense graphs, gave several equivalent notions 00:00:30.470 --> 00:00:34.800 of quasi-randomness that, at least the phase values, 00:00:34.800 --> 00:00:37.420 do not appear to be all that equivalent. 00:00:37.420 --> 00:00:41.870 But they are actually-- you can deduce one from the other. 00:00:41.870 --> 00:00:44.510 There was one condition at the very end which 00:00:44.510 --> 00:00:46.850 had to do with eigenvalues. 00:00:46.850 --> 00:00:51.125 And, basically, it said that if your second largest eigenvalue 00:00:51.125 --> 00:00:56.690 in absolute value is small, then the graph is pseudorandom. 00:00:56.690 --> 00:00:59.660 So that's something that I want to explore further today 00:00:59.660 --> 00:01:02.240 to better understand the relationship 00:01:02.240 --> 00:01:06.410 between eigenvalues of a graph and the pseudorandomness 00:01:06.410 --> 00:01:07.455 properties. 00:01:10.310 --> 00:01:12.860 For much of-- pretty much all of today, 00:01:12.860 --> 00:01:15.470 we're going to look at a special class of graphs 00:01:15.470 --> 00:01:17.420 known as n, d, lambda graphs. 00:01:22.080 --> 00:01:25.290 This just means we have n vertices, 00:01:25.290 --> 00:01:27.610 and we're only going to consider, mostly 00:01:27.610 --> 00:01:30.300 out of convenience, d regular graphs. 00:01:33.680 --> 00:01:36.110 So this will make our life somewhat simpler. 00:01:36.110 --> 00:01:38.210 And the lambda stands for that-- 00:01:38.210 --> 00:01:41.940 if you look at the adjacency matrix, 00:01:41.940 --> 00:01:45.860 and if you write down the eigenvalues 00:01:45.860 --> 00:01:56.380 of the adjacency matrix, then, well, 00:01:56.380 --> 00:01:57.630 what are these eigenvalues? 00:01:57.630 --> 00:02:03.030 The top one, because it's d regular, is equal to d. 00:02:03.030 --> 00:02:07.710 And lambda corresponds to the statement 00:02:07.710 --> 00:02:13.170 that all the other eigenvalues are, at most, 00:02:13.170 --> 00:02:16.760 lambda in absolute value. 00:02:16.760 --> 00:02:18.530 So the top one is equal to d. 00:02:18.530 --> 00:02:20.750 All the other ones in absolute value-- 00:02:20.750 --> 00:02:25.520 so it could be basically the maximum of these two-- 00:02:25.520 --> 00:02:28.890 is bounded above by lambda. 00:02:28.890 --> 00:02:32.970 And at the end of last time, we showed this expander 00:02:32.970 --> 00:02:48.205 mixing lemma, which, in this language, says that if G is n, 00:02:48.205 --> 00:02:58.540 d, lambda, then one has the following discrepancy type. 00:02:58.540 --> 00:03:01.770 So the randomness property, namely 00:03:01.770 --> 00:03:04.290 that if you look at two vertex sets 00:03:04.290 --> 00:03:07.170 and look at how many actual edges are between them compared 00:03:07.170 --> 00:03:09.330 to what you expect if this were a random graph 00:03:09.330 --> 00:03:14.050 of a similar density, then these two numbers are very similar, 00:03:14.050 --> 00:03:19.290 and the amount of error is controlled by your lambda. 00:03:19.290 --> 00:03:23.940 In particular, a smaller lambda gives you a more pseudorandom 00:03:23.940 --> 00:03:25.800 graph. 00:03:25.800 --> 00:03:27.360 So the second part of today's class, 00:03:27.360 --> 00:03:29.040 I want to explore the question of how 00:03:29.040 --> 00:03:31.590 small this lambda can be. 00:03:31.590 --> 00:03:36.170 So what's the optimal amount of pseudorandomness? 00:03:36.170 --> 00:03:39.150 But, first, I want to show you some examples. 00:03:39.150 --> 00:03:42.730 So, so far, we've been talking about pseudorandom graphs, 00:03:42.730 --> 00:03:45.580 and the only example, really, I've 00:03:45.580 --> 00:03:49.340 talked about is that a random graph is pseudorandom. 00:03:49.340 --> 00:03:50.100 Which is true. 00:03:50.100 --> 00:03:53.460 A random graph is pseudorandom with high probability, 00:03:53.460 --> 00:03:55.360 but some of the spirit of pseudorandomness 00:03:55.360 --> 00:03:58.390 is to come up with non-random examples, come up 00:03:58.390 --> 00:04:01.390 with deterministic constructions that give you 00:04:01.390 --> 00:04:03.920 pseudorandom properties. 00:04:03.920 --> 00:04:06.310 So I want to begin today with an example. 00:04:14.720 --> 00:04:17.690 A lot of examples, especially for pseudorandomness, 00:04:17.690 --> 00:04:20.779 come from this class of graphs called Cayley graphs, 00:04:20.779 --> 00:04:24.710 which are built from a group. 00:04:24.710 --> 00:04:27.440 So we're going to reserve the letter G for graphs, so I'm 00:04:27.440 --> 00:04:30.300 going to use gamma for a group. 00:04:30.300 --> 00:04:37.970 And I have a subset S of gamma, and S is symmetric, 00:04:37.970 --> 00:04:40.980 in that if you invert the elements of S, 00:04:40.980 --> 00:04:42.770 they remain in S. 00:04:42.770 --> 00:04:53.900 Then we define the Cayley graph given by this group 00:04:53.900 --> 00:05:01.400 and the set S to be the following graph, 00:05:01.400 --> 00:05:07.190 where V, the set of vertices, is just the set of group elements. 00:05:07.190 --> 00:05:16.240 And the edges are obtained by taking a group element 00:05:16.240 --> 00:05:20.340 and multiplying it by S to go to its neighbor. 00:05:26.480 --> 00:05:28.030 So this is a Cayley graph. 00:05:28.030 --> 00:05:30.670 And Cayley graphs are-- 00:05:30.670 --> 00:05:33.162 start with any group, start with any subset of the group, 00:05:33.162 --> 00:05:34.120 you get a Cayley graph. 00:05:34.120 --> 00:05:37.260 And this is a very important construction of graphs. 00:05:37.260 --> 00:05:39.630 They have lots of nice properties 00:05:39.630 --> 00:05:44.230 And, in particular, an example of a Cayley graph 00:05:44.230 --> 00:05:45.700 is a Paley graph. 00:05:49.830 --> 00:05:51.750 They're not related. 00:05:51.750 --> 00:05:54.650 So a Paley graph is a special case 00:05:54.650 --> 00:06:00.260 of a Cayley graph obtained by considering 00:06:00.260 --> 00:06:08.570 the group, the cyclic group mod p, where p is prime 1 mod 4. 00:06:12.110 --> 00:06:17.190 And I'm looking at S being the set of quadratic residues, mod 00:06:17.190 --> 00:06:17.690 p. 00:06:24.810 --> 00:06:28.580 It's actually nonzero quadratic residues. 00:06:28.580 --> 00:06:33.970 So elements of mod p that could be a square. 00:06:33.970 --> 00:06:38.130 So we will show in a second that this Paley graph has 00:06:38.130 --> 00:06:40.830 nice pseudorandom properties by showing 00:06:40.830 --> 00:06:45.030 that it is an n, d, lambda graph with lambda fairly small 00:06:45.030 --> 00:06:47.280 compared to the degree. 00:06:47.280 --> 00:06:49.800 Just a historical note-- 00:06:49.800 --> 00:06:55.240 so Raymond Paley-- so the Paley graph named after him-- 00:06:55.240 --> 00:06:59.380 he actually-- he was from the earlier part of 20th century. 00:06:59.380 --> 00:07:02.961 So from 1907 to 1932. 00:07:02.961 --> 00:07:05.210 So he died very young at the age of 26, 00:07:05.210 --> 00:07:07.220 and he actually died in an avalanche 00:07:07.220 --> 00:07:09.510 when he was skiing by himself in Banff. 00:07:09.510 --> 00:07:13.150 So Banff is a national park in Alberta in Canada. 00:07:13.150 --> 00:07:17.400 And when I was in Banff earlier this year for a math 00:07:17.400 --> 00:07:20.580 conference-- so there's also a math conference center there-- 00:07:20.580 --> 00:07:22.740 so I had a chance to go visit the-- 00:07:25.770 --> 00:07:27.000 Raymond Paley's tomb. 00:07:27.000 --> 00:07:33.596 So there's a graveyard there where you can find his tomb. 00:07:33.596 --> 00:07:37.920 And it's very sad that, in his short mathematical timespan, 00:07:37.920 --> 00:07:41.730 actually he managed to do a lot of amazing mathematical-- 00:07:41.730 --> 00:07:44.100 find a lot of amazing mathematical discoveries. 00:07:44.100 --> 00:07:46.890 And there are many important concepts named after him. 00:07:46.890 --> 00:07:50.760 So things like Paley-Wiener theorem, Paley-Zygmund, 00:07:50.760 --> 00:07:54.270 Littlewood-Paley, all this important ideas and analysis 00:07:54.270 --> 00:07:55.380 named after Paley. 00:07:55.380 --> 00:08:00.660 And Paley graph is also one of his contributions. 00:08:00.660 --> 00:08:06.390 So what we'll claim is that this Paley graph has the desired 00:08:06.390 --> 00:08:10.890 pseudorandom properties, in that if you look at its eigenvalues, 00:08:10.890 --> 00:08:21.120 then the top eigenvalues, except-- 00:08:21.120 --> 00:08:22.810 so except for the top eigenvalue, 00:08:22.810 --> 00:08:28.350 all the other eigenvalue are quite small. 00:08:28.350 --> 00:08:30.870 So keep in mind that the size of S 00:08:30.870 --> 00:08:33.020 is basically half of the group. 00:08:33.020 --> 00:08:34.770 So p minus 1 over 2. 00:08:34.770 --> 00:08:38.970 So for especially larger values of p, 00:08:38.970 --> 00:08:41.760 p's eigenvalues are quite small compared to the degree. 00:08:46.480 --> 00:08:48.780 So the main way to show that Cayley graphs 00:08:48.780 --> 00:08:54.090 like that have small eigenvalues is to just compute 00:08:54.090 --> 00:08:55.833 what the eigenvalues are. 00:08:55.833 --> 00:08:58.500 And this is actually not so hard to do for Cayley graphs, so let 00:08:58.500 --> 00:09:02.200 me do this explicitly. 00:09:02.200 --> 00:09:04.850 So I will tell you very explicitly a set 00:09:04.850 --> 00:09:08.220 of eigenvectors. 00:09:08.220 --> 00:09:10.690 And they are-- the first eigenvector 00:09:10.690 --> 00:09:13.540 is just the all 1's vector. 00:09:13.540 --> 00:09:15.850 The second eigenvector is the vector 00:09:15.850 --> 00:09:19.500 coming from 1, omega, omega squared, 00:09:19.500 --> 00:09:22.300 so omega to the p minus 1, where omega 00:09:22.300 --> 00:09:24.905 is a parameter of p-th root of unity. 00:09:29.760 --> 00:09:35.305 The next one is 1, omega square, omega fourth, 00:09:35.305 --> 00:09:38.090 all the way to omega p-- 00:09:38.090 --> 00:09:40.890 omega to the 2 times p minus 1. 00:09:40.890 --> 00:09:42.850 And so on. 00:09:42.850 --> 00:09:45.870 So I want to have-- 00:09:45.870 --> 00:09:46.440 yes, so OK. 00:09:46.440 --> 00:09:50.510 So I make this list, and I have p of them. 00:09:59.510 --> 00:10:01.230 So these are my eigenvectors. 00:10:01.230 --> 00:10:04.660 And let me check that they are actually eigenvectors. 00:10:04.660 --> 00:10:07.150 And then we can also compute their eigenvalues. 00:10:07.150 --> 00:10:12.580 So the top eigenvector corresponds to d. 00:10:12.580 --> 00:10:14.360 So the all 1's in a d regular graph 00:10:14.360 --> 00:10:17.180 is always an eigenvector with eigenvalue d. 00:10:17.180 --> 00:10:19.617 And the other ones, we'll just do this computation. 00:10:22.860 --> 00:10:25.570 So instead of getting confused with indices, 00:10:25.570 --> 00:10:28.000 let me just compute, as an example, 00:10:28.000 --> 00:10:37.712 the j-th coordinate of the adjacency matrix times V2. 00:10:37.712 --> 00:10:42.360 So the j-th coordinate, so what it comes to, 00:10:42.360 --> 00:10:45.630 is the following sum. 00:10:45.630 --> 00:10:52.110 If I run over S, then omega raised to j plus s. 00:10:52.110 --> 00:10:56.090 So S is symmetric, so I don't have to worry so much about 00:10:56.090 --> 00:10:57.080 plus or minus. 00:10:57.080 --> 00:10:58.878 So I say j plus s. 00:10:58.878 --> 00:11:00.920 So if you think about what this Cayley graph, how 00:11:00.920 --> 00:11:05.000 it is defined, if you hit this vector with that matrix, 00:11:05.000 --> 00:11:09.370 the j-th coordinate is that sum there. 00:11:09.370 --> 00:11:15.690 But I can rewrite the sum by taking out this common factor 00:11:15.690 --> 00:11:16.700 omega to j. 00:11:22.640 --> 00:11:28.890 And you see that this is the j-th coordinate of V2. 00:11:33.790 --> 00:11:34.970 And this is true for all j. 00:11:34.970 --> 00:11:38.030 So this number here is lambda 2. 00:11:44.640 --> 00:11:55.410 And, more generally, lambda k is the following sum, 00:11:55.410 --> 00:11:59.340 for k from 0-- 00:12:04.010 --> 00:12:08.768 so from k being 1 through p. 00:12:08.768 --> 00:12:11.580 So when you plug in k equals to 1, you just get d. 00:12:11.580 --> 00:12:17.450 And the others are sums of these exponential sums. 00:12:17.450 --> 00:12:19.610 Now, this is a pretty straightforward computation. 00:12:19.610 --> 00:12:21.193 And, in fact, we're not using anything 00:12:21.193 --> 00:12:22.440 about quadratic residues. 00:12:22.440 --> 00:12:29.076 This is a generic fact about Cayley graphs of z mod p. 00:12:29.076 --> 00:12:35.010 So this is true for all Cayley graphs S, not necessarily 00:12:35.010 --> 00:12:38.000 for quadratic residues. 00:12:38.000 --> 00:12:40.350 And the basic reason is that, here, you 00:12:40.350 --> 00:12:45.990 have this set of eigenvectors, and they do not depend on S. 00:12:45.990 --> 00:12:48.240 So you might know this concept from other places, 00:12:48.240 --> 00:12:50.880 such as circular matrices and whatnot, 00:12:50.880 --> 00:12:55.430 but this is true in this simple computation. 00:12:55.430 --> 00:13:02.110 So now we have the values of lambda explicitly. 00:13:02.110 --> 00:13:04.870 I can now compute their sizes. 00:13:04.870 --> 00:13:07.780 I want to know how big this lambda is. 00:13:07.780 --> 00:13:09.830 Well, the first one, when k equals to 1, 00:13:09.830 --> 00:13:15.720 it's exactly d, the degree, which is p minus 1 over 2. 00:13:15.720 --> 00:13:18.420 But what about the other ones? 00:13:18.420 --> 00:13:22.350 So, for the other ones, we can do a computation as follows. 00:13:22.350 --> 00:13:26.570 So note that I can rewrite lambda k 00:13:26.570 --> 00:13:30.320 by noting that if I take twice it 00:13:30.320 --> 00:13:34.550 and plus 1, then I obtain the following sum. 00:13:41.500 --> 00:13:45.550 Because here I am using the S as a set of quadratic residues. 00:13:45.550 --> 00:13:48.750 So if I consider this sum here, every quadratic residue 00:13:48.750 --> 00:13:52.050 gets counted twice, except for 0, which gets counted once. 00:13:55.110 --> 00:13:59.140 And now I would like to evaluate the size of this sum, 00:13:59.140 --> 00:14:00.850 this exponential sum. 00:14:00.850 --> 00:14:03.610 And this is something that's known as a Gauss sum. 00:14:07.940 --> 00:14:09.470 So, basically, a Gauss sum is what 00:14:09.470 --> 00:14:11.525 happens when you have something that's 00:14:11.525 --> 00:14:15.950 like a quadratic, an exponential sum with a quadratic dependence 00:14:15.950 --> 00:14:17.930 in the exponent. 00:14:17.930 --> 00:14:22.340 And the trick here is to consider the square of the sum. 00:14:26.540 --> 00:14:29.080 So the magnitude squared. 00:14:29.080 --> 00:14:32.480 Now if I expand the square-- 00:14:32.480 --> 00:14:36.930 so squaring is a common feature of many of the things 00:14:36.930 --> 00:14:38.370 we do in this course. 00:14:38.370 --> 00:14:40.900 It really simplifies your life. 00:14:40.900 --> 00:14:43.140 You do the square, you expand the sum. 00:14:43.140 --> 00:14:54.258 You can re-parameterize one of the summands like that. 00:14:54.258 --> 00:14:57.850 So do two steps at once. 00:14:57.850 --> 00:15:01.390 I'm re-parameterizing and I'm expanding. 00:15:01.390 --> 00:15:10.630 But now you see, if I expand the exponent, we find-- 00:15:16.520 --> 00:15:18.350 so that's just algebra. 00:15:18.350 --> 00:15:25.960 And now you notice that this sum here, the sum over a 00:15:25.960 --> 00:15:29.680 is equal to-- 00:15:29.680 --> 00:15:36.850 when b is nonzero, I claim that this sum is 0. 00:15:36.850 --> 00:15:41.030 And when b is nonzero, then I'm summing over some permutations 00:15:41.030 --> 00:15:43.910 of the roots of unity. 00:15:43.910 --> 00:15:49.150 So here I'm assuming that k is bigger than-- 00:15:49.150 --> 00:15:52.080 let's say here k is not 0. 00:15:52.080 --> 00:15:54.445 So I'm re-parameterizing k a little bit. 00:15:54.445 --> 00:15:56.020 So k is not 0. 00:15:56.020 --> 00:15:59.740 Then when b is not 0, the sum over a is 0. 00:15:59.740 --> 00:16:01.720 And otherwise it equals to p. 00:16:01.720 --> 00:16:05.880 So the sum over here equals to p. 00:16:09.090 --> 00:16:13.671 And, therefore, lambda k, lambda sub k-- 00:16:17.520 --> 00:16:20.270 how about if I-- 00:16:20.270 --> 00:16:22.980 so what should I change that to? 00:16:22.980 --> 00:16:30.480 So if I-- k is 0, then I want this to be lambda sub k plus 1. 00:16:30.480 --> 00:16:38.350 Then lambda sub k plus 1 is equal to plus/minus 00:16:38.350 --> 00:16:45.470 p plus 1 over 2, for all lambda not equal to 0. 00:16:51.800 --> 00:16:54.230 So, really, except for the top eigenvalue, 00:16:54.230 --> 00:16:56.570 which is just the degree, all the other ones 00:16:56.570 --> 00:17:01.140 are one of these two values, and they're all quite small. 00:17:01.140 --> 00:17:03.230 So this is an explicit computation showing you 00:17:03.230 --> 00:17:06.829 that this Paley graph is indeed a pseudorandom graph. 00:17:06.829 --> 00:17:08.599 It's an example of a quasi-random graph. 00:17:08.599 --> 00:17:10.198 Yes. 00:17:10.198 --> 00:17:12.875 AUDIENCE: Do we know what the sign is? 00:17:12.875 --> 00:17:15.250 YUFEI ZHAO: The question is, do we know what the sign is? 00:17:15.250 --> 00:17:17.990 So we actually-- so here I am not 00:17:17.990 --> 00:17:21.398 telling you what the sign is, but you can look up. 00:17:21.398 --> 00:17:23.190 Actually, people have computed exactly what 00:17:23.190 --> 00:17:24.750 the sign should be. 00:17:24.750 --> 00:17:28.000 And this is something that you can find in a number theory 00:17:28.000 --> 00:17:29.530 textbook, like Aaron and Rosen. 00:17:33.218 --> 00:17:35.810 Any more questions? 00:17:35.810 --> 00:17:38.270 There is a concept here I just want 00:17:38.270 --> 00:17:43.110 to bring out, that you might recognize sums like this. 00:17:43.110 --> 00:17:44.150 So this kind of sum. 00:17:46.830 --> 00:17:50.930 That's a Fourier coefficient. 00:17:50.930 --> 00:17:53.320 So if you have some Fourier transform, I mean, 00:17:53.320 --> 00:17:56.890 this is exactly what Fourier transforms look like. 00:17:56.890 --> 00:18:02.150 And it is indeed the case that, in general, 00:18:02.150 --> 00:18:13.560 if you have an Abelian group, then the eigenvalues 00:18:13.560 --> 00:18:22.550 and the spectral information of the corresponding Cayley graph 00:18:22.550 --> 00:18:25.537 corresponds to Fourier coefficients. 00:18:29.020 --> 00:18:32.080 And this is the connection that we'll see also 00:18:32.080 --> 00:18:35.710 later on in the course when we consider additive combinatorics 00:18:35.710 --> 00:18:38.470 and giving a Fourier analytic proof of Roth's theorem. 00:18:38.470 --> 00:18:42.820 And there Fourier analysis will play a central role. 00:18:42.820 --> 00:18:45.800 But this is actually-- this analogy, as I've written it, 00:18:45.800 --> 00:18:47.360 is only for Abelian groups. 00:18:47.360 --> 00:18:50.480 If you try to do the same for non-Abelian groups, 00:18:50.480 --> 00:18:55.530 you will get something somewhat different. 00:18:55.530 --> 00:18:58.790 So for non-Abelian groups, you do not 00:18:58.790 --> 00:19:01.790 have this nice notion of Fourier analysis, 00:19:01.790 --> 00:19:05.450 at least in the versions that generalizes what's 00:19:05.450 --> 00:19:07.450 above in a straightforward way. 00:19:07.450 --> 00:19:10.680 But, instead, you have something else, which many of you 00:19:10.680 --> 00:19:12.530 have seen before but under a different name. 00:19:12.530 --> 00:19:19.500 And that's representation theory, which, in some sense, 00:19:19.500 --> 00:19:21.540 is Fourier analysis, except, instead 00:19:21.540 --> 00:19:24.900 of one-dimensional objects and complex numbers, 00:19:24.900 --> 00:19:26.660 we're looking at higher-dimensional 00:19:26.660 --> 00:19:28.250 representations. 00:19:28.250 --> 00:19:31.120 So I just want to point out this connection, 00:19:31.120 --> 00:19:35.940 and we'll see more of it later on. 00:19:35.940 --> 00:19:36.980 Any questions? 00:19:41.780 --> 00:19:44.390 So let's talk more about Cayley graphs. 00:19:44.390 --> 00:19:46.970 So, last time, we mentioned these notions 00:19:46.970 --> 00:19:49.820 of quasi-randomness. 00:19:49.820 --> 00:19:51.980 And I said at the end of the class 00:19:51.980 --> 00:19:55.280 that many of these equivalences between quasi-random graphs, 00:19:55.280 --> 00:19:59.090 they fail for sparse graphs. 00:19:59.090 --> 00:20:02.220 If your density, if your x density is a constant, 00:20:02.220 --> 00:20:06.100 then the equivalences no longer hold. 00:20:06.100 --> 00:20:09.560 But what about for Cayley graphs? 00:20:09.560 --> 00:20:13.790 And, in particular, I would like to consider 00:20:13.790 --> 00:20:16.880 two specific notions that we discussed last time 00:20:16.880 --> 00:20:20.750 and try to understand how they relate to each other for Cayley 00:20:20.750 --> 00:20:21.250 graphs. 00:20:30.052 --> 00:20:33.848 So for dense Cayley graphs, it's a special case 00:20:33.848 --> 00:20:34.890 of what we did yesterday. 00:20:34.890 --> 00:20:37.430 So I'm really interested in sparser Cayley graphs, 00:20:37.430 --> 00:20:39.306 even down the degree. 00:20:39.306 --> 00:20:40.687 So even down the degree. 00:20:40.687 --> 00:20:42.270 So that's much sparser than the regime 00:20:42.270 --> 00:20:44.010 we were looking at last time. 00:20:44.010 --> 00:20:47.820 And the main result I want to tell you 00:20:47.820 --> 00:20:52.440 is that the DISC condition is, in a very strong sense, 00:20:52.440 --> 00:20:56.610 actually equivalent to the eigenvalue condition 00:20:56.610 --> 00:21:10.460 for all Cayley graphs, including non-Abelian Cayley graphs. 00:21:13.687 --> 00:21:15.520 So before telling you what the statement is, 00:21:15.520 --> 00:21:18.370 I first want to give an example showing you 00:21:18.370 --> 00:21:21.310 that this equivalence is definitely not true 00:21:21.310 --> 00:21:24.850 if you remove the assumption of Cayley graphs. 00:21:24.850 --> 00:21:27.830 For example, if you-- 00:21:27.830 --> 00:21:32.334 so example that this is false for non-Cayley. 00:21:37.080 --> 00:21:40.950 Because if you take, let's say, a large-- 00:21:40.950 --> 00:21:42.710 so let's say d regular graph. 00:21:42.710 --> 00:21:46.260 So let's say a large random d regular graph. 00:21:50.164 --> 00:21:53.390 d here can be a constant or growing with n, 00:21:53.390 --> 00:21:55.670 but this is a pretty robust example. 00:21:55.670 --> 00:22:00.842 And then I add to it an extra destroying copy of k sub d 00:22:00.842 --> 00:22:04.940 plus 1 that's much smaller in terms of number of vertices. 00:22:07.500 --> 00:22:09.360 The big, large random graph, well, 00:22:09.360 --> 00:22:11.370 by virtue of being a random graph, 00:22:11.370 --> 00:22:12.930 has the discrepancy property. 00:22:18.610 --> 00:22:22.320 And because we're only adding in a very small number 00:22:22.320 --> 00:22:25.710 of vertices, it does not destroy the discrepancy property. 00:22:25.710 --> 00:22:27.570 The discrepancy property, if you're just 00:22:27.570 --> 00:22:29.430 adding a small number of vertices, 00:22:29.430 --> 00:22:30.840 it doesn't change much. 00:22:30.840 --> 00:22:33.784 So this whole thing has discrepancy. 00:22:36.390 --> 00:22:38.190 However, what about the eigenvalues? 00:22:41.030 --> 00:22:43.040 Claim that the top two eigenvalues 00:22:43.040 --> 00:22:45.530 are in fact both equal to d. 00:22:45.530 --> 00:22:49.310 And that's because you have two eigenvectors, one which 00:22:49.310 --> 00:22:52.580 is the all 1's vector on this graph, another 00:22:52.580 --> 00:22:55.120 which is the all 1's vector on that graph. 00:22:55.120 --> 00:22:58.100 These two DISC components each give you a top eigenvector 00:22:58.100 --> 00:23:00.920 of d, so you get d twice. 00:23:00.920 --> 00:23:03.920 And, in particular, the second eigenvalue is not small. 00:23:07.200 --> 00:23:11.720 So the implication from DISC to eigenvalue really 00:23:11.720 --> 00:23:16.380 fails for non-Cayley graph for general graphs. 00:23:19.110 --> 00:23:21.890 The implication, the other direction is actually OK. 00:23:26.040 --> 00:23:30.240 In fact, the eigenvalue implies DISC is actually the content 00:23:30.240 --> 00:23:33.020 of the expander mixing lemma. 00:23:33.020 --> 00:23:40.920 So this follows by expander mixing lemma. 00:23:40.920 --> 00:23:43.470 And that's because, if you look at the expander mixing 00:23:43.470 --> 00:23:47.060 lemma for a Cayley graph-- 00:23:47.060 --> 00:23:50.700 or for a-- not for Cayley graph-- for-- 00:23:50.700 --> 00:23:52.890 if you have the eigenvalue condition, 00:23:52.890 --> 00:23:58.320 then, automatically, you would find that these two guys here 00:23:58.320 --> 00:23:59.380 are at most n. 00:24:02.520 --> 00:24:10.570 So if lambda is quite small compared to the degree, 00:24:10.570 --> 00:24:15.952 then you still have the desired type of quasi-randomness. 00:24:15.952 --> 00:24:18.160 So I'll make the statements more precise in a second. 00:24:21.420 --> 00:24:25.060 So the question is, how can we certify, how can we 00:24:25.060 --> 00:24:29.920 show that, in fact, DISC, which is seemingly weaker property, 00:24:29.920 --> 00:24:33.733 implies a stronger property of eigenvalue for Cayley graphs. 00:24:33.733 --> 00:24:35.650 And what is a special about Cayley graphs that 00:24:35.650 --> 00:24:38.350 would allow to do this, that the statement is generally 00:24:38.350 --> 00:24:40.758 false for non-Cayley graphs? 00:24:47.330 --> 00:24:49.430 So let me define-- 00:24:49.430 --> 00:24:53.290 so let me first tell you the result. 00:24:53.290 --> 00:24:56.015 So this is the result due to David Conlon 00:24:56.015 --> 00:24:57.920 and myself two years ago. 00:25:00.570 --> 00:25:05.200 So many of you may not have been to many seminar talks, where 00:25:05.200 --> 00:25:08.160 there's this convention in mathematics talks 00:25:08.160 --> 00:25:12.090 where you don't write out your full name, only by the initial. 00:25:12.090 --> 00:25:13.740 Although some kind of false modesty. 00:25:13.740 --> 00:25:17.280 But, of course, we all love talking about our own results, 00:25:17.280 --> 00:25:19.140 but somehow we don't like to write 00:25:19.140 --> 00:25:21.420 our own name for some reason. 00:25:21.420 --> 00:25:23.850 So here's the theorem. 00:25:23.850 --> 00:25:29.700 So I start with a finite group, gamma. 00:25:29.700 --> 00:25:34.168 And let me consider a subset S of gamma that is symmetric. 00:25:37.040 --> 00:25:39.320 And consider G the Cayley graph. 00:25:44.220 --> 00:25:49.060 Let me right n as the number of vertices, and d the size of S. 00:25:49.060 --> 00:25:52.370 So this is a d regular graph. 00:25:52.370 --> 00:25:54.662 Let me define the following properties. 00:26:01.550 --> 00:26:05.680 The first property, I'll call DISC with epsilon. 00:26:05.680 --> 00:26:07.930 So I give you an explicit parameter. 00:26:12.290 --> 00:26:14.960 The number of edges between x and y 00:26:14.960 --> 00:26:19.070 differs from the number of edges that you would expect. 00:26:19.070 --> 00:26:21.980 So as in the expander mixing lemma. 00:26:21.980 --> 00:26:25.490 So the DISC property is that this quantity is small relative 00:26:25.490 --> 00:26:26.980 to the total number of edges. 00:26:30.670 --> 00:26:35.710 The second property, which we'll call the eigenvalue property, 00:26:35.710 --> 00:26:50.110 EIG, is that G is an n, d, lambda graph, with lambda, 00:26:50.110 --> 00:26:52.796 at most, epsilon d. 00:26:52.796 --> 00:26:56.338 So lambda is quite small as a function of d. 00:26:59.326 --> 00:27:03.250 The conclusion of the theorem is that, up 00:27:03.250 --> 00:27:07.560 to a small change of parameters, these two properties 00:27:07.560 --> 00:27:08.790 are equivalent. 00:27:08.790 --> 00:27:13.800 In particular, eigenvalue implies a-- 00:27:13.800 --> 00:27:19.350 epsilon implies DISC of epsilon. 00:27:19.350 --> 00:27:23.430 And DISC of epsilon-- and this is 00:27:23.430 --> 00:27:26.160 the-- the second one is the more interesting direction-- 00:27:26.160 --> 00:27:29.550 it implies EIG. 00:27:29.550 --> 00:27:31.180 Well, you lose a little bit, but, 00:27:31.180 --> 00:27:33.200 at most, a constant factor. 00:27:33.200 --> 00:27:34.530 EIG of 8 epsilon. 00:27:40.680 --> 00:27:43.435 Any questions about the statement so far? 00:27:43.435 --> 00:27:46.070 And so, as I mentioned, this is completely false 00:27:46.070 --> 00:27:52.490 if you consider non-Cayley graphs. 00:27:52.490 --> 00:27:56.000 And we also, using expander mixing lemma, 00:27:56.000 --> 00:28:00.130 using that implication up there, this direction follows. 00:28:11.707 --> 00:28:13.290 One of the main reasons I want to show 00:28:13.290 --> 00:28:15.060 you a proof of this theorem is that it 00:28:15.060 --> 00:28:17.870 uses this tool which I think is worth knowing. 00:28:17.870 --> 00:28:21.030 And this is an important inequality known 00:28:21.030 --> 00:28:22.898 as Grothendieck's inequality. 00:28:35.460 --> 00:28:37.530 So many of you probably know Grothendieck 00:28:37.530 --> 00:28:41.100 as this famous French mathematician who 00:28:41.100 --> 00:28:44.040 reinvented modern algebraic geometry 00:28:44.040 --> 00:28:45.678 and spent the rest of his life writing 00:28:45.678 --> 00:28:47.220 tomes and tomes of text that have yet 00:28:47.220 --> 00:28:48.387 to be translated to English. 00:28:51.110 --> 00:28:53.600 But he also did some important foundational work 00:28:53.600 --> 00:28:56.480 in functional analysis before he became 00:28:56.480 --> 00:28:59.690 an algebraic geometry nerd. 00:28:59.690 --> 00:29:03.540 And this is one of the important results in that area that he-- 00:29:06.710 --> 00:29:08.580 so Grothendieck's inequality tells us 00:29:08.580 --> 00:29:13.140 that there exists some absolute constant k 00:29:13.140 --> 00:29:22.855 such that for every matrix A-- 00:29:22.855 --> 00:29:24.600 so a real-valued matrix-- 00:29:31.100 --> 00:29:33.020 we have that the-- 00:29:40.040 --> 00:29:41.330 so we have that, if you-- 00:29:41.330 --> 00:29:42.810 so here's the idea. 00:29:42.810 --> 00:29:46.070 Let's consider the supremum-- 00:29:46.070 --> 00:29:50.410 so let's consider the following quantity. 00:29:50.410 --> 00:29:51.530 This is a bilinear form. 00:29:55.310 --> 00:29:57.970 So this is a bilinear form. 00:29:57.970 --> 00:29:59.950 This is basically a-- 00:30:03.740 --> 00:30:07.770 so bilinear form, if you hit it by a vector x and y 00:30:07.770 --> 00:30:09.450 from the two sides. 00:30:09.450 --> 00:30:13.080 And I'm interested in what is the maximum value 00:30:13.080 --> 00:30:16.110 of this bilinear form if you are allowed 00:30:16.110 --> 00:30:23.480 to take x and y to be plus/minus 1-valued real numbers? 00:30:30.720 --> 00:30:32.920 So this is an important quantity, 00:30:32.920 --> 00:30:34.870 and it gives you a matrix. 00:30:34.870 --> 00:30:38.230 And it's basically asking you, you get a sign of plus or minus 00:30:38.230 --> 00:30:40.240 to each row and column, and I want 00:30:40.240 --> 00:30:42.323 to maximize this number here. 00:30:42.323 --> 00:30:43.990 This is an important quantity that we'll 00:30:43.990 --> 00:30:47.680 see actually much more in the next chapter on graph limits. 00:30:47.680 --> 00:30:49.420 But, for now, just take my word. 00:30:49.420 --> 00:30:51.580 This is a very important quantity. 00:30:51.580 --> 00:30:53.080 And this is actually a quantity that 00:30:53.080 --> 00:30:54.820 is very difficult to evaluate. 00:30:54.820 --> 00:30:56.800 If I give you a very large matrix 00:30:56.800 --> 00:30:59.710 and ask you to compute this number here, 00:30:59.710 --> 00:31:01.810 there is no good algorithm for it. 00:31:01.810 --> 00:31:05.710 And it's believed that there is no good algorithm for it. 00:31:05.710 --> 00:31:09.370 On the other hand, there is a relaxation 00:31:09.370 --> 00:31:13.130 of this problem, which is the following. 00:31:13.130 --> 00:31:19.430 It's still a sum, but now, instead of considering 00:31:19.430 --> 00:31:25.190 the bilinear form there, let's consider the xi's and yi's. 00:31:25.190 --> 00:31:32.600 Not-- take them not form real numbers, but take vectors. 00:31:32.600 --> 00:31:39.680 So let's consider the sum where I'm 00:31:39.680 --> 00:31:47.330 taking a similar-looking sum, except that xi's and yi's come 00:31:47.330 --> 00:31:51.770 from a unit ball in some vector space 00:31:51.770 --> 00:31:57.920 with an inner product, where B is the unit 00:31:57.920 --> 00:32:07.395 ball in some Rm, where here the dimension is actually not 00:32:07.395 --> 00:32:07.895 so relevant. 00:32:07.895 --> 00:32:10.470 The dimension is arbitrary. 00:32:10.470 --> 00:32:15.620 If you like, you can make m n or 2n because you only 00:32:15.620 --> 00:32:19.050 have that many vectors. 00:32:19.050 --> 00:32:22.160 So this quantity here, just by very definition, 00:32:22.160 --> 00:32:25.010 is a relaxation of the right-hand of this quantity 00:32:25.010 --> 00:32:25.510 here. 00:32:25.510 --> 00:32:27.620 So it's at least this large. 00:32:27.620 --> 00:32:30.650 So, in particular, if you have whatever plus/minus, 00:32:30.650 --> 00:32:34.160 you can always look at the same quantity with m equal to 1, 00:32:34.160 --> 00:32:35.660 and you obtain this quantity here. 00:32:35.660 --> 00:32:38.310 But this quantity may be substantially larger. 00:32:38.310 --> 00:32:43.880 So the x and y's have more room to put themselves in 00:32:43.880 --> 00:32:45.220 to maximize the sum. 00:32:48.330 --> 00:32:52.650 And Grothendieck's inequality tells us that the left-hand 00:32:52.650 --> 00:32:57.210 side actually cannot be too much larger than the right-hand 00:32:57.210 --> 00:32:58.110 side. 00:32:58.110 --> 00:33:02.650 It exceeds it by, at most, a constant factor. 00:33:02.650 --> 00:33:04.870 So, in other words, the left-hand side, 00:33:04.870 --> 00:33:15.060 which is known as a semi-definite relaxation, 00:33:15.060 --> 00:33:17.910 you are not losing by more than a constant factor 00:33:17.910 --> 00:33:20.730 compared to the original problem. 00:33:20.730 --> 00:33:22.500 And this is important in computer science 00:33:22.500 --> 00:33:24.210 because the left-hand side turns out 00:33:24.210 --> 00:33:27.960 to be a Semidefinite Program, an SDP, which 00:33:27.960 --> 00:33:32.740 does have efficient algorithms to compute. 00:33:32.740 --> 00:33:35.610 So you can give a constant factor approximation 00:33:35.610 --> 00:33:38.430 to this difficult compute but important quantity 00:33:38.430 --> 00:33:41.030 by using semidefinite relaxation. 00:33:41.030 --> 00:33:43.020 And Grothendieck's inequality promises us 00:33:43.020 --> 00:33:44.540 that it is a good relaxation. 00:33:50.560 --> 00:33:53.050 You might ask, what is the value of k? 00:33:53.050 --> 00:33:55.790 So I said there exists some constant k. 00:33:55.790 --> 00:33:58.720 So this is actually a mystery. 00:33:58.720 --> 00:34:04.000 So the current proofs have been improved over time. 00:34:04.000 --> 00:34:06.340 And Grothendieck himself proved this theorem, 00:34:06.340 --> 00:34:08.889 but it constantly has been improved over time. 00:34:08.889 --> 00:34:11.469 And, currently, the best-known result 00:34:11.469 --> 00:34:20.604 is something along the lines of k roughly 1.78 works. 00:34:20.604 --> 00:34:25.719 But the optimal value, which is known as Grothendieck's 00:34:25.719 --> 00:34:33.310 constant, is unknown. 00:34:40.600 --> 00:34:42.317 So this is Grothendieck's constant. 00:34:42.317 --> 00:34:43.900 Actually, this, what I've written down 00:34:43.900 --> 00:34:47.949 is what's called the real Grothendieck's constant. 00:34:47.949 --> 00:34:49.780 Because you can also write a version 00:34:49.780 --> 00:34:52.810 for complex numbers and complex vectors, 00:34:52.810 --> 00:34:55.130 and that's the complex Grothendieck's constant. 00:34:55.130 --> 00:34:55.630 Yes. 00:34:55.630 --> 00:34:57.518 AUDIENCE: Is there a lower bound that's 00:34:57.518 --> 00:34:59.305 known [INAUDIBLE] greater than 1? 00:34:59.305 --> 00:35:00.790 YUFEI ZHAO: Is there a lower bound that is known? 00:35:00.790 --> 00:35:01.290 Yes. 00:35:01.290 --> 00:35:03.430 It's known that it's strictly bigger than 1. 00:35:03.430 --> 00:35:04.922 AUDIENCE: Do we know [INAUDIBLE]?? 00:35:04.922 --> 00:35:06.880 YUFEI ZHAO: So there are some specific numbers, 00:35:06.880 --> 00:35:07.840 but I forget what they are. 00:35:07.840 --> 00:35:08.632 You can look it up. 00:35:12.350 --> 00:35:13.500 Any more questions? 00:35:16.080 --> 00:35:18.790 So we'll leave Grothendieck's inequality. 00:35:18.790 --> 00:35:20.650 We'll use it as a black box. 00:35:20.650 --> 00:35:22.440 So if you wish to learn the proof, 00:35:22.440 --> 00:35:23.810 I encourage you to do so. 00:35:23.810 --> 00:35:26.700 There are some quite nice proofs out there. 00:35:26.700 --> 00:35:29.520 And we'll use it to prove this theorem here 00:35:29.520 --> 00:35:31.360 about quasi-random Cayley graphs. 00:35:44.890 --> 00:35:49.040 So let's suppose DISC holds. 00:35:55.880 --> 00:35:58.160 So what would we like to-- what do we like to show? 00:35:58.160 --> 00:36:04.910 We want to show that this eigenvalue condition holds. 00:36:04.910 --> 00:36:07.570 And we'll use the-- 00:36:07.570 --> 00:36:11.240 some min-max characterization of eigenvalues. 00:36:11.240 --> 00:36:13.400 But, first, some preliminaries. 00:36:13.400 --> 00:36:19.700 Suppose you have vectors x and y which have plus/minus 1 00:36:19.700 --> 00:36:21.065 coordinate values. 00:36:24.270 --> 00:36:34.810 Then, by letting-- so let's consider the following vectors, 00:36:34.810 --> 00:36:37.820 where I split up x and y according 00:36:37.820 --> 00:36:40.503 to where they're positive and where they're negative. 00:36:46.070 --> 00:36:50.740 So, here, these are such that x plus is equal to-- 00:36:50.740 --> 00:36:55.396 so if I evaluate it on a coordinate g, then it's 1. 00:36:55.396 --> 00:37:04.861 So if x sub g is plus 1, and 0 otherwise. 00:37:04.861 --> 00:37:12.720 xg sub minus is 1 if x sub g is minus 1. 00:37:12.720 --> 00:37:14.150 0 otherwise. 00:37:14.150 --> 00:37:19.360 So x splits into x plus minus x minus, and y splits 00:37:19.360 --> 00:37:22.450 into y plus minus y minus. 00:37:25.710 --> 00:37:39.310 Let's consider a matrix A where the g comma h entry of A 00:37:39.310 --> 00:37:41.970 is the following quantity. 00:37:41.970 --> 00:37:49.090 I have the set S, and I look at whether g inverse h lies in S. 00:37:49.090 --> 00:37:50.960 And I can consider an indicator of that. 00:37:50.960 --> 00:37:52.880 So it's 1 or 0. 00:37:52.880 --> 00:38:01.010 And then subtract d over n so that this value has mean 0. 00:38:01.010 --> 00:38:03.950 So this is a matrix. 00:38:03.950 --> 00:38:10.720 And now if I consider the bilinear form, 00:38:10.720 --> 00:38:14.680 hit A from left and right with x and y, 00:38:14.680 --> 00:38:22.330 then the bilinear form splits according to the plus 00:38:22.330 --> 00:38:24.835 and minuses of the x's. 00:38:39.230 --> 00:38:41.330 And I claim that each one of these terms 00:38:41.330 --> 00:38:42.950 is controlled because of DISC. 00:38:45.674 --> 00:38:54.010 So, for example, the first term is, 00:38:54.010 --> 00:38:55.820 if you expand out what this guy is-- 00:38:55.820 --> 00:38:57.070 so here's an indicator vector. 00:38:57.070 --> 00:38:58.480 That's an indicator vector. 00:38:58.480 --> 00:39:00.400 And if you look at the definition, 00:39:00.400 --> 00:39:05.830 then this is precisely the number of edges between x plus 00:39:05.830 --> 00:39:10.960 and y plus minus d over n times the size 00:39:10.960 --> 00:39:14.410 of x plus times the size of y plus, 00:39:14.410 --> 00:39:21.140 where x plus is the set of group elements 00:39:21.140 --> 00:39:26.236 such that x sub g is 1, and so on. 00:39:34.668 --> 00:39:35.660 All right. 00:39:43.110 --> 00:39:47.830 So the punchline up there is that this quantity-- 00:39:47.830 --> 00:40:01.520 so this quantity is, at most, by discrepancy, epsilon dn. 00:40:05.020 --> 00:40:08.410 So this sum here, by triangle inequality, 00:40:08.410 --> 00:40:12.920 is, at most, 4 epsilon dn. 00:40:53.580 --> 00:40:56.548 All right. 00:40:56.548 --> 00:40:59.920 So, so far, we've reinterpreted the discrepancy property. 00:40:59.920 --> 00:41:02.460 And what we really want to show is that this graph 00:41:02.460 --> 00:41:05.255 satisfies eigenvalue condition. 00:41:05.255 --> 00:41:07.005 So what does that actually mean to satisfy 00:41:07.005 --> 00:41:08.510 the eigenvalue condition? 00:41:08.510 --> 00:41:11.440 So by the min-max characterization 00:41:11.440 --> 00:41:21.670 of eigenvalues, it follows that the maximum of these two 00:41:21.670 --> 00:41:23.440 eigenvalues, which is the quantity that we 00:41:23.440 --> 00:41:33.010 would like to control, is equal to the following. 00:41:33.010 --> 00:41:41.400 It is equal to the supremum of this bilinear form 00:41:41.400 --> 00:41:46.770 when x and y are unit-length vectors. 00:41:53.240 --> 00:41:56.900 And this is simply because A is the matrix-- 00:41:56.900 --> 00:41:59.390 it's not the adjacency matrix. 00:41:59.390 --> 00:42:01.570 A is not the adjacency matrix. 00:42:01.570 --> 00:42:04.790 A is the matrix obtained by essentially taking 00:42:04.790 --> 00:42:07.250 the adjacency matrix and subtracting 00:42:07.250 --> 00:42:09.500 that constant there. 00:42:09.500 --> 00:42:12.900 And subtracting that constant gets rid of the top eigenvalue. 00:42:12.900 --> 00:42:15.552 And what you remained is whatever that's left. 00:42:15.552 --> 00:42:17.510 And you want to show that whatever you remained 00:42:17.510 --> 00:42:20.550 has small spectral radius. 00:42:20.550 --> 00:42:22.910 So we would like to show that this quantity here 00:42:22.910 --> 00:42:23.600 is quite small. 00:42:27.880 --> 00:42:29.370 Well, let's do it. 00:42:29.370 --> 00:42:33.360 So give me a pair of vectors, x and y. 00:42:37.340 --> 00:42:44.290 And let's set the following quantities, 00:42:44.290 --> 00:42:49.090 where I take a twist on this x vector 00:42:49.090 --> 00:42:57.480 by rotating the coordinates, setting 00:42:57.480 --> 00:43:04.720 x super s sub g, the coordinate g, to be x sub sg. 00:43:04.720 --> 00:43:08.820 So x is a vector indexed by the group elements, 00:43:08.820 --> 00:43:14.520 and then rotating this indexing of the group elements by s. 00:43:14.520 --> 00:43:18.180 So that's what I mean by superscript s. 00:43:18.180 --> 00:43:24.904 And, likewise, y superscript s is defined similarly. 00:43:32.140 --> 00:43:36.760 So I claim that these twists, these rotations, 00:43:36.760 --> 00:43:40.820 do not change the norm of these vectors. 00:43:40.820 --> 00:43:43.080 And that should be pretty clear, because I'm simply 00:43:43.080 --> 00:43:47.540 relabeling the coordinates in a uniform way. 00:43:47.540 --> 00:43:50.130 And, likewise, same for y. 00:44:00.660 --> 00:44:09.970 So I would like to show this quantity up here is small. 00:44:09.970 --> 00:44:13.120 So let's consider two unit vectors. 00:44:25.450 --> 00:44:28.410 And consider this bilinear form. 00:44:28.410 --> 00:44:42.490 If I expand out this bilinear form, it looks like that. 00:44:42.490 --> 00:44:45.080 I'm just writing it out. 00:44:45.080 --> 00:44:52.390 But now let me just throw in an extra variable of summation. 00:44:52.390 --> 00:45:04.270 What we'll do is essentially look at the same sum, 00:45:04.270 --> 00:45:18.620 but now I add in an extra s, and put this s over here. 00:45:18.620 --> 00:45:21.450 So convince yourself that this is the same sum. 00:45:24.220 --> 00:45:27.890 So it's simply re-parameterizing the sum. 00:45:27.890 --> 00:45:28.890 So this is the same sum. 00:45:33.590 --> 00:45:36.080 But now, if you look at the definition of A, 00:45:36.080 --> 00:45:37.986 there's this cancellation. 00:45:41.720 --> 00:45:44.800 So the two s's cancel out. 00:45:44.800 --> 00:45:46.360 So let's rewrite the sum. 00:45:49.630 --> 00:45:54.541 1 over n, then g, h, s, all group elements. 00:45:54.541 --> 00:46:04.040 Then-- now, if I bring this summation of s, 00:46:04.040 --> 00:46:08.880 now I bring it inside, and then you see that what's inside is 00:46:08.880 --> 00:46:16.695 simply the inner product between the two vectors, x sub g-- 00:46:16.695 --> 00:46:17.695 between the two vectors. 00:46:24.490 --> 00:46:26.730 So this is-- so what's inside is simply 00:46:26.730 --> 00:46:36.600 the product, inner product, between these two. 00:46:36.600 --> 00:46:42.100 So I may need to redefine. 00:46:45.840 --> 00:46:46.340 Yes. 00:46:46.340 --> 00:46:47.510 So when you're looking at-- when you're 00:46:47.510 --> 00:46:48.930 talking about non-Abelian groups, 00:46:48.930 --> 00:46:50.630 it's always a question of which side 00:46:50.630 --> 00:46:53.160 should you multiply things by. 00:46:53.160 --> 00:46:57.220 And you guys are OK? 00:46:57.220 --> 00:46:59.770 Or I need to change this s to over here. 00:46:59.770 --> 00:47:03.650 But anyway, it should work. 00:47:03.650 --> 00:47:04.250 Yes, question. 00:47:04.250 --> 00:47:05.995 AUDIENCE: yh [INAUDIBLE]. 00:47:05.995 --> 00:47:06.620 YUFEI ZHAO: yh. 00:47:06.620 --> 00:47:07.606 Thank you. 00:47:17.480 --> 00:47:19.830 Yes, I think-- 00:47:19.830 --> 00:47:20.820 OK. 00:47:20.820 --> 00:47:21.320 Question. 00:47:21.320 --> 00:47:26.912 AUDIENCE: [INAUDIBLE] 00:47:26.912 --> 00:47:29.900 YUFEI ZHAO: Great. 00:47:29.900 --> 00:47:32.883 So maybe I need to switch the definition here, 00:47:32.883 --> 00:47:35.050 but, in any case, some version of this should be OK. 00:47:35.050 --> 00:47:35.550 Yes. 00:47:35.550 --> 00:47:37.220 So figure it out later in the notes. 00:47:39.800 --> 00:47:45.010 But now-- OK. 00:47:45.010 --> 00:47:48.190 So you have this-- 00:47:48.190 --> 00:47:49.770 we have this here. 00:47:49.770 --> 00:47:55.720 And if you look at this quantity here, 00:47:55.720 --> 00:47:57.850 it is the kind of quantity that comes up 00:47:57.850 --> 00:48:00.521 in Grothendieck's inequality. 00:48:03.565 --> 00:48:05.190 So this is basically the left-hand side 00:48:05.190 --> 00:48:08.550 of Grothendieck's inequality. 00:48:08.550 --> 00:48:12.230 What about the right-hand side of Grothendieck's inequality? 00:48:12.230 --> 00:48:14.790 Well, we already controlled that. 00:48:14.790 --> 00:48:16.340 We already controlled that because we 00:48:16.340 --> 00:48:20.340 said, whenever you have up there little x and little y-- 00:48:26.000 --> 00:48:31.900 so the conclusion of this board was that-- 00:48:31.900 --> 00:48:34.210 let me erase over here. 00:48:34.210 --> 00:48:41.160 So the conclusion of this board was that this bilinear form 00:48:41.160 --> 00:48:45.690 is bounded by, at most, 4 epsilon d, 00:48:45.690 --> 00:48:52.450 for all x and y being plus/minus 1 coordinate valued. 00:48:52.450 --> 00:48:57.700 So combining them by Grothendieck, 00:48:57.700 --> 00:49:03.830 we have an upper bound, which is the Grothendieck constant times 00:49:03.830 --> 00:49:05.510 4 epsilon-- 00:49:09.030 --> 00:49:11.870 so 4 epsilon dn. 00:49:15.570 --> 00:49:16.820 There's a-- sorry. 00:49:16.820 --> 00:49:19.250 There's an n missing here. 00:49:23.480 --> 00:49:26.720 And, therefore, because the Grothendieck constant 00:49:26.720 --> 00:49:30.846 is less than 2, we have a bound of 8 epsilon d. 00:49:34.580 --> 00:49:38.370 And this shows that this variational problem, which 00:49:38.370 --> 00:49:42.450 characterizes the largest eigenvalue in absolute value, 00:49:42.450 --> 00:49:47.090 is, at most, 8 epsilon d, thereby 00:49:47.090 --> 00:49:50.450 implying the eigenvalue property. 00:49:58.430 --> 00:50:00.680 So the main takeaway from this proof, two things. 00:50:00.680 --> 00:50:03.810 One is Grothendieck's inequality is a nice thing to know. 00:50:03.810 --> 00:50:06.170 So it's a semidefinite relaxation 00:50:06.170 --> 00:50:09.050 that changes the problem, which is initially somewhat 00:50:09.050 --> 00:50:11.750 intractable, to a semidefinite problem which 00:50:11.750 --> 00:50:13.730 is both, from a computer science point of view, 00:50:13.730 --> 00:50:15.590 algorithmically tractable, but also has 00:50:15.590 --> 00:50:17.680 nice mathematical properties. 00:50:17.680 --> 00:50:19.580 And for this application here, there's 00:50:19.580 --> 00:50:21.380 this nice trick in this proof where 00:50:21.380 --> 00:50:26.660 I'm symmetrizing the coordinates using the group symmetries. 00:50:26.660 --> 00:50:32.170 And that allows me to obtain this characterization showing 00:50:32.170 --> 00:50:35.680 that eigenvalue condition and this discrepancy condition 00:50:35.680 --> 00:50:38.900 are equivalent for Cayley graphs. 00:50:42.228 --> 00:50:43.270 Let's take a quick break. 00:50:46.465 --> 00:50:47.340 Any questions so far? 00:50:51.320 --> 00:50:54.220 So we've been talking about n, d, lambda graphs. 00:50:54.220 --> 00:50:55.250 So d regular graphs. 00:50:55.250 --> 00:50:57.170 And the next question I would like to address 00:50:57.170 --> 00:51:02.536 is, In an n, d, lambda graph, how small can lambda be? 00:51:02.536 --> 00:51:06.840 So smaller lambda corresponds to a more pseudorandom graph. 00:51:06.840 --> 00:51:09.830 So how small can this be? 00:51:09.830 --> 00:51:12.650 And the right kind of setting that I want you to think about 00:51:12.650 --> 00:51:15.770 is think of d as a constant. 00:51:15.770 --> 00:51:19.370 So think of d as a constant, and n getting large. 00:51:24.540 --> 00:51:29.830 So how small can lambda be. 00:51:32.860 --> 00:51:36.340 And it turns out there is a limit to how small it can be. 00:51:36.340 --> 00:51:46.520 And it is known as the Alon-Boppana bound, 00:51:46.520 --> 00:51:50.480 which tells you that if you have a fixed d-- 00:51:50.480 --> 00:52:00.440 and so G is an n-vertex graph with adjacency matrix 00:52:00.440 --> 00:52:08.620 eigenvalues lambda 1 through lambda n, 00:52:08.620 --> 00:52:11.940 sorted in non-increasing order. 00:52:11.940 --> 00:52:17.550 Then the second largest eigenvalue has to be at least, 00:52:17.550 --> 00:52:23.190 basically, 2 root d minus 1 minus a small error term, 00:52:23.190 --> 00:52:24.640 little on-- 00:52:24.640 --> 00:52:30.590 little o1, where the little o1 goes to 0 as n 00:52:30.590 --> 00:52:31.996 goes to infinity. 00:52:38.190 --> 00:52:46.700 So the Alon-Boppana bound tells you that the lambda cannot be 00:52:46.700 --> 00:52:49.120 below this quantity here. 00:52:49.120 --> 00:52:51.050 And I want to explain what is the significance 00:52:51.050 --> 00:52:54.080 of this quantity, and you will see it in the proof. 00:52:54.080 --> 00:52:56.080 And this quantity is the best possible. 00:52:56.080 --> 00:52:59.930 And it also says what do we know about the existence of graphs 00:52:59.930 --> 00:53:04.980 which have lambda 2 close to this number. 00:53:04.980 --> 00:53:07.580 So this is the optimal number you can put here. 00:53:07.580 --> 00:53:08.080 Question. 00:53:08.080 --> 00:53:09.747 AUDIENCE: Does it say anything about how 00:53:09.747 --> 00:53:11.813 negative lambda n can be? 00:53:11.813 --> 00:53:13.230 YUFEI ZHAO: Question-- does it say 00:53:13.230 --> 00:53:15.630 how negative lambda n can be? 00:53:15.630 --> 00:53:17.313 So I'll address that in a second, 00:53:17.313 --> 00:53:19.730 but, essentially, if you have a bipartite graph and lambda 00:53:19.730 --> 00:53:22.478 n equals to minus lambda 1. 00:53:22.478 --> 00:53:30.706 AUDIENCE: [INAUDIBLE] 00:53:30.706 --> 00:53:32.900 YUFEI ZHAO: More questions? 00:53:32.900 --> 00:53:36.780 So I want to show you a proof and, time permitting, 00:53:36.780 --> 00:53:39.320 a couple of proofs of Alon-Boppana bound. 00:53:39.320 --> 00:53:43.970 And they're all quite simple to execute, but the-- 00:53:43.970 --> 00:53:46.100 I think it's a good way to understand how 00:53:46.100 --> 00:53:47.560 these special techniques work. 00:53:56.490 --> 00:53:59.990 So, first, as with all of the proofs that we did concerning-- 00:53:59.990 --> 00:54:01.130 or most of them-- 00:54:01.130 --> 00:54:03.560 concerning eigenvalues, we're looking 00:54:03.560 --> 00:54:10.500 at the Courant-Fischer characterization 00:54:10.500 --> 00:54:14.340 of eigenvalues. 00:54:14.340 --> 00:54:25.980 It suffices to show, to exhibit some vector z-- 00:54:29.850 --> 00:54:33.000 so a nonzero vector-- 00:54:33.000 --> 00:54:39.010 such that z is orthogonal to the all 1's vector 00:54:39.010 --> 00:54:49.350 and this quotient is at least the claimed bound. 00:54:53.170 --> 00:54:54.920 So by the Courant-Fischer characterization 00:54:54.920 --> 00:54:57.650 of the second eigenvalue, if you vary 00:54:57.650 --> 00:55:00.020 over all such d that are orthogonal to the unit 00:55:00.020 --> 00:55:04.250 vector, then the maximum value this quantity attains 00:55:04.250 --> 00:55:06.450 is equal to lambda 2. 00:55:06.450 --> 00:55:09.080 So to show the lambda 2 is large, 00:55:09.080 --> 00:55:14.270 it suffices to exhibit such a z. 00:55:14.270 --> 00:55:18.550 So let me construct such a z for you. 00:55:18.550 --> 00:55:22.380 So let r be a positive integer. 00:55:22.380 --> 00:55:29.490 And let's pick an arbitrary vertex v. So v 00:55:29.490 --> 00:55:32.500 is a vertex in the graph. 00:55:32.500 --> 00:55:48.074 And let V sub i denote vertices at distance exactly i from V. 00:55:48.074 --> 00:55:55.670 From-- yes, from V. So, in particular, V0 is equal to V-- 00:55:55.670 --> 00:55:57.610 and I can just draw you a picture. 00:55:57.610 --> 00:56:05.110 So you have V0, and then the neighbors of V0, 00:56:05.110 --> 00:56:13.907 and each of them have more neighbors. 00:56:16.829 --> 00:56:18.780 Like that. 00:56:18.780 --> 00:56:26.270 So I'm calling V0 this stuff, big V0. 00:56:26.270 --> 00:56:31.470 And then big V1, V sub 2, and so on. 00:56:34.900 --> 00:56:39.130 So I'm going to define a vector, which I'll eventually 00:56:39.130 --> 00:56:41.560 make into z, by telling you what is 00:56:41.560 --> 00:56:48.440 the value of this vector on each of these vertices. 00:56:48.440 --> 00:56:53.920 I will do this by setting very explicitly-- 00:56:53.920 --> 00:57:03.660 so set x to be a vector with value 00:57:03.660 --> 00:57:14.820 x sub u to be wi, where wi is d minus 1 raised to power 00:57:14.820 --> 00:57:23.740 minus i over 2 whenever u lies in set big V sub i. 00:57:23.740 --> 00:57:27.580 So u is distance exactly i from V. I set it to this number. 00:57:27.580 --> 00:57:33.630 So notice that they decrease as you get further away from V. 00:57:33.630 --> 00:57:41.436 And I do this for all distances less than r. 00:57:44.760 --> 00:57:47.670 So this is my x vector. 00:57:47.670 --> 00:57:53.860 And I set all the other vect-- all the other corners 00:57:53.860 --> 00:58:02.050 to be 0 if the distance between u and V is at least r. 00:58:04.990 --> 00:58:06.690 So that gives you this vector. 00:58:06.690 --> 00:58:10.590 And I would like to compute that quotient over there 00:58:10.590 --> 00:58:12.450 for this vector. 00:58:12.450 --> 00:58:22.760 And I claim that this quotient here is at least 00:58:22.760 --> 00:58:24.060 the following quantity. 00:58:33.970 --> 00:58:36.577 But this is a computation, so let's just do it. 00:58:36.577 --> 00:58:37.410 So why is this true? 00:58:40.570 --> 00:58:45.210 Well, if you compute the norm of x-- 00:58:45.210 --> 00:58:47.460 so I'm just taking the sum of the squares 00:58:47.460 --> 00:58:49.530 of these coordinates. 00:58:49.530 --> 00:58:57.130 Well, that comes from adding up these values. 00:58:57.130 --> 00:59:01.370 So for each element in the i-th neighborhood. 00:59:01.370 --> 00:59:05.010 So I have wi squared. 00:59:05.010 --> 00:59:14.620 And if I look at that quantity up there, so what is this? 00:59:14.620 --> 00:59:17.120 A is the adjacency matrix. 00:59:17.120 --> 00:59:21.740 So over here, A is the adjacency matrix. 00:59:25.460 --> 00:59:35.170 So this quantity, I can write it as a sum over all vertices u. 00:59:35.170 --> 00:59:39.010 And I look at x sub u, and now I sum again 00:59:39.010 --> 00:59:50.600 over all neighbors of u, and consider x sub u prime. 00:59:50.600 --> 00:59:52.190 It's that sum there. 00:59:52.190 --> 00:59:57.570 But this sum, I have some control over, because it is-- 00:59:57.570 --> 01:00:00.000 so what's happening here? 01:00:00.000 --> 01:00:04.250 I claim it has at least the following quantity. 01:00:04.250 --> 01:00:05.660 Consider where u is. 01:00:05.660 --> 01:00:07.660 So u could be-- 01:00:07.660 --> 01:00:11.000 I mean, it's only nonzero if u lies 01:00:11.000 --> 01:00:13.031 in the r minus 1th neighborhood. 01:00:16.800 --> 01:00:21.480 So in that neighborhood, I have V sub i possible choices 01:00:21.480 --> 01:00:24.960 for the vertex u. 01:00:24.960 --> 01:00:31.990 For that choice, this x sub u is w sub i. 01:00:31.990 --> 01:00:35.230 But what about all its neighbors? 01:00:35.230 --> 01:00:38.280 So it could have neighbors, well, 01:00:38.280 --> 01:00:40.400 in the same set going left. 01:00:40.400 --> 01:00:44.500 But there's-- so there's one neighbor going left, 01:00:44.500 --> 01:00:47.380 and all the other neighbors are-- 01:00:47.380 --> 01:00:50.080 maybe it's in the same set, maybe it's in the next set. 01:00:50.080 --> 01:00:53.110 But, in any case, I have the following inequality. 01:00:53.110 --> 01:00:57.730 There's one neighbor in the same-- 01:00:57.730 --> 01:01:01.660 in the left, if you look at that picture just now. 01:01:01.660 --> 01:01:04.900 And then all the remaining neighbors 01:01:04.900 --> 01:01:12.640 have x sub u primes at least w sub i plus 1, 01:01:12.640 --> 01:01:14.860 because these weights are decreasing. 01:01:14.860 --> 01:01:17.840 So I can-- the worst case, so to speak, 01:01:17.840 --> 01:01:23.390 is if you-- all the neighbors point to the next set. 01:01:23.390 --> 01:01:24.760 So I had that inequality there. 01:01:28.600 --> 01:01:30.990 There's an issue. 01:01:30.990 --> 01:01:42.900 Because if you go to the very last set, 01:01:42.900 --> 01:01:45.000 if you go to the very last set and think 01:01:45.000 --> 01:01:49.410 about what happens, when i in in that very last set, 01:01:49.410 --> 01:01:55.260 I'm overcounting neighbors that no longer has weights. 01:01:55.260 --> 01:01:58.260 So I need to take them out. 01:01:58.260 --> 01:02:02.815 So I should subtract d minus 1 times-- 01:02:05.680 --> 01:02:09.590 and so this is the maximum possible weight 01:02:09.590 --> 01:02:12.010 sum I could have-- 01:02:12.010 --> 01:02:15.360 maximum possible overcount. 01:02:15.360 --> 01:02:18.430 So each product here has t minus 1 neighbors at most. 01:02:18.430 --> 01:02:19.330 All right. 01:02:19.330 --> 01:02:21.740 So this is-- should be pretty straightforward if you 01:02:21.740 --> 01:02:23.000 do the counting correctly. 01:02:23.000 --> 01:02:26.960 But now let's plug in what these weights are. 01:02:26.960 --> 01:02:30.620 And you'll find that this sum here, this quantity, 01:02:30.620 --> 01:02:32.780 is equal to-- so the key point here 01:02:32.780 --> 01:02:36.890 is that this thing simplifies very nicely if you consider 01:02:36.890 --> 01:02:39.500 what this is. 01:02:39.500 --> 01:02:42.680 So what ends up happening is that you 01:02:42.680 --> 01:02:45.250 get this extra factor of 2 root d minus 1. 01:02:45.250 --> 01:03:01.160 And then the sum minus 1/2 of V sub [INAUDIBLE].. 01:03:01.160 --> 01:03:05.320 It's pretty straightforward computation 01:03:05.320 --> 01:03:09.490 using the specific weights that we have. 01:03:09.490 --> 01:03:17.490 And one more thing is that notice that this-- 01:03:17.490 --> 01:03:22.250 so notice that the sizes of each neighborhood cannot expand 01:03:22.250 --> 01:03:30.050 by more than a factor of d minus 1, because, well, 01:03:30.050 --> 01:03:34.280 you only have d minus 1 outward edges going forward at each 01:03:34.280 --> 01:03:36.390 step. 01:03:36.390 --> 01:03:43.260 And, as a result, I can bound this guy. 01:03:43.260 --> 01:03:48.510 And so what you find is that this whole thing here 01:03:48.510 --> 01:03:53.826 is that least 2 times root d minus 1. 01:03:53.826 --> 01:03:56.520 The main term is the sum. 01:04:03.500 --> 01:04:08.750 And this here is less than each individual summand. 01:04:08.750 --> 01:04:12.440 So I can do 1 minus 1 over 2r. 01:04:19.150 --> 01:04:21.930 Putting these two together, you find the claim. 01:04:28.835 --> 01:04:30.400 All right. 01:04:30.400 --> 01:04:33.730 So I've exhibited this vector x, which 01:04:33.730 --> 01:04:36.160 has that quotient property. 01:04:36.160 --> 01:04:39.840 But that's not quite enough, because we need a vector-- 01:04:39.840 --> 01:04:42.100 so it's called z up here-- 01:04:42.100 --> 01:04:46.530 that is orthogonal to the all 1's vector. 01:04:46.530 --> 01:04:51.940 And that you can do, because if the number of vertices 01:04:51.940 --> 01:04:56.320 is quite a bit larger than-- 01:04:59.230 --> 01:05:11.080 compared to the degree, then I claim that there exists u and v 01:05:11.080 --> 01:05:12.640 vectors-- 01:05:12.640 --> 01:05:18.520 vertices that are at distance at least 2r. 01:05:22.480 --> 01:05:25.340 So if I let-- 01:05:30.150 --> 01:05:32.220 this is the size of this tree. 01:05:32.220 --> 01:05:35.520 So if you have-- 01:05:35.520 --> 01:05:37.950 everything is within distance r-- 01:05:37.950 --> 01:05:44.738 distance 2r from a vertex, then they all lie on this tree edge. 01:05:44.738 --> 01:05:46.780 If you count the number of vertices in that tree, 01:05:46.780 --> 01:05:49.840 it's what I have-- the sum I've written here. 01:05:49.840 --> 01:05:55.820 So if I consider these two vectors-- 01:05:55.820 --> 01:06:05.250 so be-- so x be the vector obtained above, which is, 01:06:05.250 --> 01:06:06.440 in some sense-- 01:06:06.440 --> 01:06:13.780 and I'm being somewhat informal here-- centered at v. 01:06:13.780 --> 01:06:19.280 And if I let y be the vector but I center it now at the vector-- 01:06:19.280 --> 01:06:27.020 at u, then I claim that, essentially, x and y 01:06:27.020 --> 01:06:32.830 are supported on disjoint vertex sets that have 01:06:32.830 --> 01:06:35.950 no edges even between them. 01:06:35.950 --> 01:06:40.020 So, in particular, this inner product-- 01:06:40.020 --> 01:06:43.870 this bilinear form-- not inner product but this bilinear 01:06:43.870 --> 01:06:45.100 form-- 01:06:45.100 --> 01:06:59.120 is equal to 0, since no edge between the supports of x 01:06:59.120 --> 01:07:00.056 and y. 01:07:04.550 --> 01:07:09.700 So now I have two vectors that do not interact, 01:07:09.700 --> 01:07:12.760 but both have this nice property above. 01:07:12.760 --> 01:07:14.998 And now I can take a linear combination. 01:07:18.590 --> 01:07:22.030 Let me choose a constant c-- 01:07:22.030 --> 01:07:24.860 so it's a real constant-- 01:07:24.860 --> 01:07:31.315 such that this z equal to x minus cy has-- 01:07:37.035 --> 01:07:39.110 and I can choose this constant. 01:07:39.110 --> 01:07:41.080 So x and y are both non-negative entries. 01:07:41.080 --> 01:07:43.840 They're both nonzero, and I can choose this constant c 01:07:43.840 --> 01:07:45.900 so that it is-- 01:07:45.900 --> 01:07:48.840 this z is orthogonal to the all 1's vector. 01:07:48.840 --> 01:07:53.570 And I now I have this extra property I want. 01:07:53.570 --> 01:07:55.830 But what about the inner products? 01:07:55.830 --> 01:08:01.140 Well, these two vectors, x and y, they do not interact at all. 01:08:01.140 --> 01:08:05.270 So their inner products split just fine, 01:08:05.270 --> 01:08:08.450 and the bilinear form splits just fine. 01:08:19.020 --> 01:08:30.810 So you have this inequality here, as desired. 01:08:30.810 --> 01:08:37.479 And r, notice that I can take r going 01:08:37.479 --> 01:08:43.740 to infinity as n going to infinity, because d is fixed. 01:08:43.740 --> 01:08:47.399 So if n goes to infinity, then r can go to infinity, roughly 01:08:47.399 --> 01:08:48.661 a logarithmic n. 01:08:52.990 --> 01:08:55.980 And that proves the Alon-Boppana bound. 01:08:55.980 --> 01:08:58.540 And just to recap, to prove this bound, 01:08:58.540 --> 01:09:01.990 we needed to exhibit by the Courant-Fischer some vector 01:09:01.990 --> 01:09:03.355 with a nice-- 01:09:03.355 --> 01:09:06.760 this quotient such that this quotient is large. 01:09:06.760 --> 01:09:11.350 And we exhibit this quotient by constructing the vector 01:09:11.350 --> 01:09:15.399 explicitly around the vertex and finding two such vertices that 01:09:15.399 --> 01:09:17.689 are far away from constructing these two vectors, 01:09:17.689 --> 01:09:19.660 taking the appropriate linear combination 01:09:19.660 --> 01:09:22.510 so that the final vector is orthogonal to the unit vector, 01:09:22.510 --> 01:09:26.270 to the all 1's vector, and then showing 01:09:26.270 --> 01:09:31.220 that the corresponding bilinear form has-- is large enough. 01:09:34.520 --> 01:09:35.396 Any questions? 01:09:38.140 --> 01:09:42.370 I want to show you a different proof which gives you 01:09:42.370 --> 01:09:47.229 a slightly worse result, but the proof is conceptually nice. 01:09:47.229 --> 01:09:51.569 So let me give you a second proof which 01:09:51.569 --> 01:09:54.864 is slightly weakening. 01:10:00.298 --> 01:10:03.290 And just that we'll show-- 01:10:03.290 --> 01:10:06.076 so we'll show that-- 01:10:06.076 --> 01:10:09.930 so the earlier proof showed that lambda 2 is quite large. 01:10:09.930 --> 01:10:17.100 But, next, we'll show that the max of lambda 2 and the lambda 01:10:17.100 --> 01:10:19.800 n is large. 01:10:24.500 --> 01:10:28.290 So not that the second largest eigenvalue is large, 01:10:28.290 --> 01:10:30.550 but the second largest eigenvalue in absolute value 01:10:30.550 --> 01:10:31.050 is large. 01:10:31.050 --> 01:10:34.290 So it's slightly weaker, but, for all intents and purposes, 01:10:34.290 --> 01:10:36.710 it's the same spirit. 01:10:36.710 --> 01:10:39.850 So I'll show this one here. 01:10:39.850 --> 01:10:42.040 And this is a nice illustration of what's 01:10:42.040 --> 01:10:48.940 called a trace method, sometimes also a moment method. 01:10:56.570 --> 01:10:59.090 Here's the idea. 01:10:59.090 --> 01:11:03.610 As we saw in the proof relating the quasi-randomness of C4 01:11:03.610 --> 01:11:06.380 and eigenvalues, well, C4's are-- 01:11:06.380 --> 01:11:09.320 eigenvalues are related to counting closed walks 01:11:09.320 --> 01:11:10.430 in a graph. 01:11:10.430 --> 01:11:14.540 And so we'll use that counting closed walks in a graph. 01:11:14.540 --> 01:11:21.110 And, specifically, the 2k-th moment of the spectrum 01:11:21.110 --> 01:11:28.610 is equal to the trace of the 2k-th power, which 01:11:28.610 --> 01:11:34.790 counts the number of closed walks of length exactly 2k. 01:11:49.680 --> 01:11:53.700 So to lower bound the left-hand side, 01:11:53.700 --> 01:11:56.880 we want to lower-bound the right-hand side. 01:11:56.880 --> 01:12:02.830 So let's consider closed walks starting at a fixed vertex. 01:12:02.830 --> 01:12:10.950 So the number of closed walks of length 01:12:10.950 --> 01:12:25.390 exactly 2k starting at a fixed vertex v. Here 01:12:25.390 --> 01:12:26.800 we're in a d regular graph. 01:12:26.800 --> 01:12:32.730 So here we are in a d regular graph. 01:12:32.730 --> 01:12:36.530 I claim, whatever this number is-- it maybe different 01:12:36.530 --> 01:12:42.860 for each d-- it is at least the same quantity if I 01:12:42.860 --> 01:12:46.070 do this walk in an infinite d regular tree. 01:12:53.280 --> 01:12:55.710 So infinite d regular tree is what? 01:12:58.530 --> 01:13:00.390 This is an infinite d regulator tree. 01:13:05.659 --> 01:13:09.300 We just start with the vertex, and go out d regular. 01:13:09.300 --> 01:13:12.880 So why is this true? 01:13:12.880 --> 01:13:14.665 So think about how you walk. 01:13:14.665 --> 01:13:15.820 So let me just explain. 01:13:15.820 --> 01:13:17.868 This is, I think, pretty easy once you 01:13:17.868 --> 01:13:18.910 see things the right way. 01:13:18.910 --> 01:13:22.360 So start with a vertex v. Think about how you walk. 01:13:22.360 --> 01:13:26.140 And whatever way you can walk, well, you 01:13:26.140 --> 01:13:31.740 can walk the same way on the infinite d regular tree. 01:13:31.740 --> 01:13:34.020 Well, I mean, sorry. 01:13:34.020 --> 01:13:37.560 Whatever walk you can do an infinite d regular tree, 01:13:37.560 --> 01:13:43.110 if you label the first vertex, the first edge, second edge, 01:13:43.110 --> 01:13:44.730 if you do a corresponding labeling 01:13:44.730 --> 01:13:46.350 on your original graph, you can do 01:13:46.350 --> 01:13:49.440 that walk on your original graph. 01:13:49.440 --> 01:13:52.020 Although the original graph may have some additional walks, 01:13:52.020 --> 01:13:54.110 namely things that involve cycles, 01:13:54.110 --> 01:13:56.575 that are not available on your tree. 01:13:56.575 --> 01:13:58.200 But, certainly, every walk, you can do. 01:13:58.200 --> 01:14:01.400 Every closed walk you can do on a tree, 01:14:01.400 --> 01:14:06.150 you can do the same walk on your graph. 01:14:06.150 --> 01:14:07.530 So you can make this more formal. 01:14:07.530 --> 01:14:09.447 So you can write down a bijection or injection 01:14:09.447 --> 01:14:12.192 to make this more formal, but it should be fairly convincing 01:14:12.192 --> 01:14:13.400 that this inequality is true. 01:14:16.940 --> 01:14:18.580 But this is just a number. 01:14:18.580 --> 01:14:21.740 So this is a number of 2k walks in a d regular tree starting 01:14:21.740 --> 01:14:22.410 on the vertex. 01:14:22.410 --> 01:14:23.993 And this number has been well studied, 01:14:23.993 --> 01:14:27.600 and we don't need to know the precise number. 01:14:27.600 --> 01:14:29.430 We just need to know some good lower bound. 01:14:29.430 --> 01:14:31.980 And here is one lower bound, which is that there's at least 01:14:31.980 --> 01:14:34.080 a Catalan number, the k-th Catalan number, 01:14:34.080 --> 01:14:43.230 times d minus 1 to the k, where C sub k is the k-th Catalan 01:14:43.230 --> 01:14:45.867 number, which is equal to-- 01:14:50.176 --> 01:14:53.860 so k 2k choose k divided by k plus 1. 01:14:53.860 --> 01:14:55.750 So let me remind you what this is. 01:14:55.750 --> 01:14:58.980 A wonder, has many combinatorial interpretations, 01:14:58.980 --> 01:15:02.140 and it's a fun exercise to do bijections between them. 01:15:02.140 --> 01:15:06.870 But, in particular, C3 is the equal to 5, 01:15:06.870 --> 01:15:13.070 which counts the number of ups and down walks of length 01:15:13.070 --> 01:15:17.423 6 that never dip below the horizontal line 01:15:17.423 --> 01:15:18.090 where you start. 01:15:23.780 --> 01:15:27.310 So, then, this corresponds to going away 01:15:27.310 --> 01:15:32.200 from the root versus coming back to the root. 01:15:32.200 --> 01:15:34.380 Soon you have at least that many ways. 01:15:34.380 --> 01:15:37.210 And when you are moving away from the root, 01:15:37.210 --> 01:15:42.196 you have d minus 1 choices on which branch to go to. 01:15:42.196 --> 01:15:43.172 OK, good. 01:15:49.030 --> 01:15:56.460 Given that, the right-hand side is at least, then, 01:15:56.460 --> 01:16:02.700 n, the number of vertices, times the quantity above related 01:16:02.700 --> 01:16:03.780 to Catalan numbers. 01:16:11.310 --> 01:16:14.630 On the other hand, the left-hand side is at most-- 01:16:14.630 --> 01:16:16.730 here we're using that 2k is an even number-- 01:16:16.730 --> 01:16:23.220 is at most d to the 2k plus all the other eigenvalues that are 01:16:23.220 --> 01:16:25.740 most lambda in absolute value. 01:16:25.740 --> 01:16:28.920 So let me call this quantity lambda. 01:16:32.570 --> 01:16:39.310 Rearranging this inequality, we find that lambda to the 2k 01:16:39.310 --> 01:16:46.430 is at least this number here. 01:16:56.390 --> 01:17:00.516 Just here, I'm changing n minus 1 to n. 01:17:00.516 --> 01:17:01.930 So we have that. 01:17:01.930 --> 01:17:04.180 And now what can we do? 01:17:04.180 --> 01:17:10.900 We let n go to infinity and k go to infinity slowly enough. 01:17:10.900 --> 01:17:16.690 So if k goes to infinity and n goes to-- 01:17:16.690 --> 01:17:21.160 so k goes to infinity with n, but not too quickly. 01:17:21.160 --> 01:17:25.260 But k is little of log n. 01:17:25.260 --> 01:17:28.450 And we find that this quantity here 01:17:28.450 --> 01:17:30.550 is essentially 2 to the k-- 01:17:33.784 --> 01:17:34.730 2 to the 2k. 01:17:37.250 --> 01:17:40.410 And this guy here is little o1. 01:17:40.410 --> 01:17:45.850 So lambda is at least 2 root d minus 1 minus little o1. 01:17:51.050 --> 01:17:53.570 That proves, essentially, the Alon-Boppana bound, 01:17:53.570 --> 01:17:56.990 although a small weakening because we are-- 01:17:56.990 --> 01:17:59.780 this big eigenvalue, you might find 01:17:59.780 --> 01:18:03.230 might actually be very negative instead of very positive. 01:18:03.230 --> 01:18:03.830 But that's OK. 01:18:03.830 --> 01:18:09.200 For applications, this is not such a big deal. 01:18:09.200 --> 01:18:10.790 These are two different proofs. 01:18:10.790 --> 01:18:13.400 And now, we think about, are they really the same proof? 01:18:13.400 --> 01:18:14.930 Are they different proofs? 01:18:14.930 --> 01:18:16.800 Are they related to each other? 01:18:16.800 --> 01:18:18.450 So it's worth thinking about. 01:18:18.450 --> 01:18:20.750 They look very different, but how 01:18:20.750 --> 01:18:22.500 are they related to each other? 01:18:22.500 --> 01:18:24.510 And one final remark. 01:18:24.510 --> 01:18:26.730 You already saw two different proofs as to-- 01:18:26.730 --> 01:18:28.340 I mean, that shows you this number, 01:18:28.340 --> 01:18:30.630 and you see where this number comes from. 01:18:30.630 --> 01:18:32.750 And let me just offer one final remark on where 01:18:32.750 --> 01:18:35.330 that number really comes from. 01:18:35.330 --> 01:18:39.580 And it really comes from this infinite d regular tree. 01:18:42.630 --> 01:18:45.300 So it turns out that 2 root d minus 1 01:18:45.300 --> 01:18:55.320 exactly is the spectral radius of the infinite d regular tree. 01:18:58.120 --> 01:19:00.400 And that is the reason, in some sense, 01:19:00.400 --> 01:19:03.790 that this is the correct number occurring Alon-Boppana bound. 01:19:07.380 --> 01:19:11.310 This is-- if you've seen things like algebraic topology 01:19:11.310 --> 01:19:16.040 or topology, this is a universal cover for d regular graphs. 01:19:16.040 --> 01:19:20.330 So I won't talk more about it, but just some general remarks, 01:19:20.330 --> 01:19:22.310 and you already saw two different proofs. 01:19:22.310 --> 01:19:24.540 So beginning of next time, I want to wrap this up 01:19:24.540 --> 01:19:25.740 and to show you-- 01:19:25.740 --> 01:19:28.590 to explain some-- what we know about 01:19:28.590 --> 01:19:32.540 are there graphs for which this bound is tight? 01:19:32.540 --> 01:19:36.240 And the answer is yes, and there are lots of major open problems 01:19:36.240 --> 01:19:39.403 as well related to what happens there. 01:19:39.403 --> 01:19:40.820 And then, after that, I would like 01:19:40.820 --> 01:19:42.830 to start talking about graph limits. 01:19:42.830 --> 01:19:45.760 So that's the next chapter of this course. 01:19:45.760 --> 01:19:47.610 OK, good.