 Okay, thanks everybody for making it out today today our speaker is Mike Tate from Villanova University who will be talking about spectral terrain problems. Go ahead and take us away Mike. Okay, thanks. Yeah, so I have a I'm wearing this shirt. I have a soft spot for South Carolina ever since my, my friend played soccer for South Carolina so she gave me the shirt 15 years ago and ever since then. Yeah, I feel, I feel good about South Carolina. And so, wait, can you guys see my screen. Okay, here we go. And so, yeah, since then I call, I definitely am team, the real USC here. So, yeah, so I'm going to tell you about. Let me move this so I can see you and see my screen at the same time. Okay, so I'm going to talk about spectral terrain problems. Feel free to stop me or interrupt or whatever. So I'm going to talk about spectral terrain problems. And I want to start by talking about turn on problems. This is sort of the main function and extremal graph theory so it's the throne number for a graph so the ideas you you have forbidden graph F. And you're playing this optimization game so you want to maximize the number of edges in an n vertex graph. And you're subject to the constraint that you don't see F as a sub graph so it's it's an optimization subject to a constraint. And so, this is the densest triangle free graph that you can have on nine vertices. It has 20 edges, I guess and and there's no triangle on there and you can sort of check that you can't do better than this. And you can see there's no triangle because it's a bipartisan graph right so like I, you can see it in two ways right like there's by the pigeonhole principle, if you take any three vertices then one of them has or two of them have the same color and so there's no edge between those two or you can see it this is a bipartisan graph so it's too colorful. And so any sub graph of it is also too colorful which means that there's no triangle in there. And this is generalizable, you can kind of see that this construction a complete bipartisan graph is has no triangle so it gives you a lower bound on this optimization problem it's like construction so it gives you a lower bound on the maximum. And mantel serum is essentially that you can't do better than that so that's the content of and tells him is that if you have one more edge than this. Then you have a triangle and then tell something slightly stronger that this is the only external graph so so if you really have a graph that has the maximum number of edges, then it's a complete bipartisan graph and the parts are balanced. So to run generalize this you can see the, the construction is sort of obvious how to generalize I'll take an R part type graph. So each part has size and over R, and this you can see that it has no KR plus one again from the pigeonhole principle or from this graph is our colorful so it has no. Some graphs are also are colorful so so this gives me a lower bound for my strong number for bidding KR plus one, and the content of terms there and is that it's also an upper bound, and trying to them also gives this stability result that for n large enough if. If you really have a graph that's KR plus one free that has this many edges, then it's a complete multi part type graph that's balance. And this is generalized to graphs. If I'm forbidding an arbitrary graph of chromatic number R plus one again the construction is the same and you see, you see from not the pigeonhole principle anymore but from the chromatic number argument that this graph has no sub graphs of chromatic number So this graph has roughly one minus one over R times n squared over two edges, and it's it's G free and so it gives me a lower bound for this number, and this kind of sell this error stone theorem is a really strong theorem it shows that that's also an upper bound for. An arbitrary for being a graph of chromatic number R plus one. So this is really nice it gives us an asymptotic formula for this problem, most of the time, the only time that it doesn't asymptotically solve the problem is when this term is zero so an artist one. And then, we don't have an asymptotic formula anymore and this just says that the charon number is little lower and squared in that case and so if my forbidden graph is bipartite and we don't really know what's happening at least by this theorem, and, and in the case that the if it's not bipartite then we have this nice asymptotic formula. And when, when the, when the forbidden graph is bipartite the problem is like very fundamentally different. There's the, the, the error stone theorem says that the, the charon number of a bipartite graph is little low and squared but you can actually do much better like this, the charon number of any bipartite graph is really sub sub quadratic. So I'm going to try to show a theorem theorem that says the charon number of a complete bipartite graph is bounded above by something that looks like end of the two minus one over S. And so since any graph F, like if I have a bipartite graph F here, F is contained in suitably large complete bipartite graph and so it's trying number is bounded above by the time number of that graph. So we sort of have this like dichotomy between these problems where, if the chromatic number of my forbidden graph is at least three then I have these nice constructions that asymptotically solve the problem. Their product like what I mean by that is there's sort of a small number of sets of vertices and how a vertex behaves only depends on what set it's in. So if I'm forbidding something that's not a complete graph the, the extremal graphs are not quite complete multi part type graphs but up to little low and squared edges, they are so if you kind of like take a step back, it looks like a product like construction. And the forbidden graph is bipartite, not only is the toron function very different, but the extremal graphs are also very different so in the cases that we sort of know what's happening when I prevent a bipartite graph. You have these, you have these constructions that come from geometry and algebra and productive planes and generalized polygons and things like that. A lot different than a complete multi part type graph so generally these, these constructions of extremal graphs for forbidden bipartite graphs they look. They're describing algebraic way they, they look sort of pseudo random in a certain sense you can quantify what that means. It's not at all like, like a toron. So they say they really look very different. Okay, so, so that's kind of the extremal graph three part I want to do a spectral version of this extremal graph three problem. So in spectral graph theory the goal is to associate a matrix with your graph and then properties of the matrix, you want to deduce properties of the graph and so basically I'm going to give you some matrix that's corresponding to your graph. I'm going to compute its eigenvalues and eigenvectors and from those eigenvalues and eigenvectors I want to say oh well my graph must look like whatever like some properties of my guy. So in this talk, I'll consider the adjacency matrix there are lots and lots of matrices that you can associate with a graph the adjacency matrix is maybe the most well studied one and and that's what I'll be focused on. So this matrix is an end by end matrix if I have a graph with and vertices. It is the zero one matrix you, you put a one in the IJ entry if vertex I is adjacent to vertex J and otherwise you put so it's this nice zero one matrix, it is symmetric. So if you notice this definition is is symmetric and in terms of I and J. So by is equal to a sub J I, and when you have a real symmetric matrix that has these very nice properties in particular it has a full set of real eigenvalues and real eigenvectors. But the eigenvectors you can take them to be an orthonormal an orthonormal basis of our end. And these properties are very nice when you're trying to use linear algebra to say something about about your graph. So this matrix is very nice for the reason that you can use these big linear algebra theorems to prove graph theory. And so you have these eigenvalues, you always have this full set of eigenvalues and so you want to know something about your graph from this set of values. Okay, so this is kind of the spectral version of the toron problem so so the question is, I want to give you a graph, the same thing I'm going to forbid F so I'm looking at every rest. And now the toron problem was maximize the number of edges. Now I'm going to say I want to maximize the largest eigenvalue of the adjacency. So I'm maximizing the spectral radius of the adjacency matrix of the graph. Okay, so this question was kind of a more general version of this question was first posed by reality and so high they these are called reality so high problems. There are questions as maximize lambda one over some family of graphs. So any question that's of that form kind of a very general question you take a family of graphs and I want to know which one in that family has the largest spectral radius. So you can think of like Stanley's mound is maximizing lambda one over the family of graphs that has images, for example so that's a that's kind of a classical example of this type of theorem. And the family that we're maximizing over in this case is the family of n vertex F free graphs. And I'll define this, this function the spectral extremal number of my graph F and so it's the maximum spectral radius overall and vertex at three drops. So, usually people use capital EX to know to denote the family of extremal graphs and that's wrong problem so the family of graphs that have the maximum number of edges and are at three, and I'll use the same same kind of notation I use capital sbx to denote the family of spectral So, I may, during this talk I may say edge extremal or Toronto extremal to talk about this family, and I may say spectral extremal to talk about this family. Okay, so why ask this question. And, and I guess really what I want to say in this slide is, is, I want to explain why I'm calling this a spectral Toronto problem so you know that the questions sort of look the same but are they really related. The answer is yes. The spectral radius of a graph is related to the number of edges in the graph so somehow these two questions are capturing capturing similar information about the graph. So this is sort of how the spectral radius is related to the number of edges as a lower down as an end as an upper down so so one of the really nice things about adjacency matrices is is that that's the, the, particularly, particularly the largest value of adjacency matrix is that you can think of it as an optimization problem. So this is called the Rayleigh portion this x transpose a x over x transfer is x. And it turns out that the spectral radius of of a graph is given by the maximum over this Rayleigh portion you can see this because I mean it follows because there is a full set of eigenvectors. So you can kind of expand x as a linear combination of eigenvectors and then and then down every, every eigenvalue by the largest one essentially that's how it works. And so, this is sort of gives you a, it gives you another way to think about the spectral radius of a graph it's some information about the graph that's in terms of a different optimization problem. The idea of this is that the spectral radius of a graph is an upper down for the average degree of a graph because if I, you know, I'm taking the maximum over all vectors here, if I take x is the all ones back there. And then when I do this extra transpose a x. What happens when x is the all ones back there is every edge is just counted twice. And so you got twice the number of edges and that the length of this one is one is one is and so this is exactly the average degree of the graph and so you get this very nice down that that if I tell you the spectral radius of a graph it this really immediately tells me something about the death it tells me something about the density of the graph. It's an upper bound in terms of the degrees and to see that you sort of look at the eigenvector eigenvalue equation so you look at it as a, well it's a matrix equation that you look at it as a system of equation so you have this matrix equation. And you can think about it as a system of equations for, for every kind of row here there's an equation. And so what this says is when I, when I expand that row. Oh, so I expand this this row dot axis is this lambda times this entry. I get, I get this equation. And so you can kind of think about it as like I mean what's happening here is you have, you have this ex you, and the entry here has to sort of be an average scaled by lambda of the entries of its neighbors. So it's kind of this like dynamical system. And, and I mean this tells you something about about this lambda one it's constrained by this process where where every vertex sort of has to have its neighbors. Its neighbors being equal to a scaled version of its own entry. And if you look at, if you look at this where I look, if you look at kind of this picture where I take this to be the maximum eigenvector entry. And you, okay you get some over the neighbors of this vertex of the eigenvector entries, each one is bounded by the maximum. And this sum is bounded by the degree of you times the times the maximum eigenvector entry and so you get the spectral radius time to maximum degree is at most the degree of that vertex times the maximum eigenvector entry and so you get an upper bound of the maximum degree for the spectral radius here. Right so you, you now have these, these two bounds that for example when the graph is regular this, this tells you that immediately right so so the spectral radius is sort of bounded between the average and the maximum degree. If the neighbors are close together then you have a lot of information, if they're far apart then it's not so clear what's happening. But since the spectral radius is related to the degrees in this way. It sort of makes sense to ask this about to call this a spectral parameter, I might say. I'm just going to say that you can play the same game with, like we, we, we found out something about the spectral radius by, by looking at the eigenvector eigenvalue equation for a, and we looked at kind of this, the entry of the vertex, as compared to the entries of its neighbors, but you are starting with larger powers of the matrix. So you get this kind of eigenvector eigenvalue equation, where this is now landed for the k times x. And what's happening is you're sort of, you have this vertex and its entry needs to be a scaled version of when you kind of take walks of length k. What's happening at the end, at the end points of walks of walks of length k because a to the k sort of count these walks of length k right. And so you can, you can sort of play the same game and ask, well can I get structural information about the graph from looking at higher moments of the spectral radius basically. And the answer is yes and you can actually prove some like several theorems using this kind of set up so you can prove mental Sam you can prove Stanley's down you can prove some stronger versions of Stanley's down and a couple other things so it's it's it's sort of surprising that kind of this this sequence of eigenvalue should tell you so much information about the structure of your graph but it turns out that it does. Okay, so I want to say what kind of spectral Tehran theorems have been proved before so one of the big ones was by Nikki four of in the early 2000s and he said that okay if my forbidden graph is the complete graph. Then the spectral radius of this graph is bounded above by the spectral radius of the Toronto so this is that complete our part site, complete balanced our part. So, this Tehran graph was the, the edge extremal graph if I want to forbid AR plus one and maximize the number of edges, then I should choose this graph and make before us the term says that also it's the spectral extremal graph so if I want to forbid AR plus one and maximize the spectral radius of my graph and also I should choose this complete multi parts like that. So it's kind of this like product like construction. He also. Oh, and so, since, since the spectral radius is an upper bound on the average degree, any upper bound on the spectral extremal number of a graph automatically gives you an upper bound on the at on the extreme So this one of the motivations for doing this kind of study is that it can actually. It can imply Tehran type results and it can actually even strengthen Tehran type results so this implies that the number of edges in an extremal graph is actually less than the number of edges in a Tehran graph you have to be a little bit careful with the parity of that and stuff but Tehran actually implies Tehran's it's stronger than Tehran's. He also proved a spectral version of the curve are you sure from here and and his spectral version of the curve are you sure from here and actually implies the best known edge version of the curve are you sure from here so it is. This theorem implies your right he's improvement of the privilege to show them. So these are kind of two sort of motivating theorems that show you why you might want to prove something like this because you can you can prove stronger versions of classical Okay, so I want to give kind of an overview of what's happening in the edge extremal case and in the spectral extremal case. So if we kind of the first thing that you should study is for bidding for bidding pay our plus one so it's Iran's theorem says, take a complete multi part type graph. And Nikki for us theorem says also take a complete multi part type graph of balance sizes. So these are the same, the same. What about an odd cycle so triangle or even a larger odd cycle. So it's known that the edge extremal graph, as long as an is large enough the edge extremal graph is a complete bipartisan graph. And Nikki for all of us are actually something I mean I think this was known before Nick for but it's also known that the spectral extremal version of this problem if I forbid an odd cycle then as long as an is large enough. An odd cycle graph is also a complete a complete bipartite graph on an even on a balance number of parts in each chart. And so again the answers are the same. In this talk I'm always going to think about and being large enough like this, you know some some strange things might happen if I'm forgetting a long cycle and and it's small then this is not extremal but I want to I want to know what happens when I'm getting really large. Okay, so when I forbid a complete graph, the answers are the same. When I forbid an odd cycle the answer to the same so maybe it's the same problem, but um, it turns out that it's not and the answer, well it's not clear from this example but this is actually sort of what motivated my work in this. In this topic I was sort of trying to maximize my radius of planar graphs and other planar graphs and you know planar graphs have these nice characterizations that they don't have certain minors and so I'm trying to maximize my radius overall KR minor free graphs. And this question is that there's a there's a natural KR minor free graph so if I take a complete graph on our minus two vertices and I join it to an independent set. So this is an okay or minor. And it is this kind of the natural version. And it turns out that this is the, this is the spectral extremal graph, if, and is large enough. And, and that's a theorem that I heard with Josh Coven and a few years ago, five years ago or so. And for the, for the edge extremal question, it's not clear. So this is a conjecture of Seymour, that the same thing should be true. And it's true when R is at most six or at most seven or something like that. And it's conjectured that by Seymour that for fixed R if n is large enough, then this is the extremal graph but it seems difficult to prove people don't know how to prove this. You might say well okay did the spectral theorem imply this one right so that would kind of be that would be really good. But unfortunately, this does give you an upper bound the spectral theorem gives you an upper bound on the number of edges and a KR minor free graph but it's not good in this case so basically this graph is very far from being regular and so the very large degree is make it so that the spectral radius is far away from the average degree so so unfortunately the spectral theorem, it doesn't really tell you anything about the edge extremal version, but yeah that's sort of one of the motivations of studying this. Okay so this is an example where we know the spectral extremal graph but we don't know the edge extremal graph. So this is an example where in some sense we know more about the spectral version and this is an example where the extremal graphs are very different actually. So, you can check so so the edge extremal graphs for C4 come from projected planes they're these graphs with that are very close to regular they're highly pseudo random the degrees are roughly and into the one half. And you can check that the spectral radius of a C4 free graph is also bounded above by something like and to the one half. So it's very close like it's one time to the one half plus something of lower order. And so you can ask well are the extremal graphs the same. But it turns out they're not even though the projected plane graphs and the actual spectral extremal graphs, they have very very similar largest eigenvalue but they look completely different so that this is a conjecture of making for all that was proved not too long ago I think like maybe 10 years ago or something that the graphs. Maximizing spectral radius over C4 free graphs are the friendship. So you take this triangle intersecting on a vertex. And this is true for all and so it's it's harder to show when I guess the number of vertices even because you have this extra edge sticking out. But it's true for all and the extremal graphs for the edge case are not known all the time so they're known in certain situations when we know things about projected planes and stuff like that but they're not known all the time. So this is an example where we know more about the spectral case than than the edge case but but kind of be sort of understand them both pretty well. So what about C2K free. So this is a really difficult question in the edge case, and it turns out it's also a really difficult question in the, in the spectral extremal case. So the conjecture, the conjecture for the right hand side here is that I should take. So they want to prove it C2K I think I take a complete graph on a minus one vertices, mostly joined with an independent set, and I put in one extra edge here maybe. And, but we don't know this is true. So this is true for C6 and it's unknown for the other cases so this is a conjecture and make people out that the proof for C6 is pretty recent like it's just maybe a year ago. And it seems quite difficult to prove it in general. But it seems maybe more trackable than the than the edge case. We really don't know what's going on here so we don't even know kind of the order of magnitude of what's happening, except in very in certain cases so when I'm forgetting C4 C6 and 10 and we have some understanding of what's happening, and the other cases we don't and the extremal graphs, they look very different than this case so here we're also kind of looking at these. When we know what they, or let's say the best constructions that we know they look like these generalized poly on basically. But yeah, both sides now, we don't know what's happening basically. And then here, this is another case where we sort of have no idea what's happening. We don't even have a conjecture as to what's happening here, the on the right hand side on the left hand side the edge extremal grass. We know it depends on the value of essence so if T is much larger than us then you can use these projective norm graphs, which again they're kind of the grass coming from geometry so it's the intersection of some surfaces or something. And then we've got the graphs book, very pseudo random basically. Not at all like this product type instruction. There's not even a guess on the right hand side what they should be like I don't even have intuition for what the extremal graph should be like. And that's kind of like I want to talk about intuition so how can you guess what the extremal graph should be like how how is this function behaving we saw that kind of sometimes it's the same graph that the same extremal graphs of the as the spectrum radius, sometimes it's not sometimes very different here here the graphs are very different but the spectral radius and the average degree are very close together and to into examples, even though the graphs are very different here. That's not the case so the spectral radius here is much bigger than the average degree on the on the left. And here we don't know. Okay, so so that's kind of like your arching question here is, when are these questions the same. How can we decide what we think the extremal graph should be. And how can we go about proving those things. And so that's kind of big picture what's happening and I kind of want to say, you know, by looking at a few specific graphs where we were able to answer this question. Okay, so one thing to notice is that the, the functions that you're trying to optimize they just incentivize slightly different things so the edge. The edge problem asks you to maximize the number of edges right so it's just every edge that you have is in one more, and every edge counts the same amount so you just want to maximize this here, the spectral radius you're trying to maximize this sort of weird quadratic form. And it turns out if you if you think about this a little bit like, the edges are the same. It helps me to add edges to vertices that already have lots of edges so we kind of want to. It helps to concentrate the edges on one vertex or on a small set of vertices more than than more than in the edge case that the distribution of the edges in that in that case it doesn't matter at all. It doesn't matter you have to max you know it's it's something different that's happening. Okay, so I want to look at this when I forbid a few different things so the so the first one is I want to forbid the friendship graph now so F pay is this friendship graph on a fixed number of triangles. The other way to think about it is this is a vertex that has pay edges in its neighborhood. Okay, so that's that's kind of one way I want to think about this. The version of this problem was solved in the 90s. The extremal function looks like this so kind of, even though the air dish stone theorem tells us the assing products of it you can still ask more specific questions it's so interesting to try to figure out what exactly is the exact run function number and and what are the extremal gaps and so if you're this group of people determine determine these and the internal graph sort of look like this it depends of pay is odd is even but let me do pay is odd. Basically you put a large complete bipartite graph. Okay, so that's this time. And then I just, I want to put two complete graphs of size to get this joint complete graph. And so right now I want to think of n is going to infinity. So what this means is essentially the graph is a complete bipartite graph and I'm putting just a small number of extra edges. Okay, so that's how I want to think about these extremal graphs. And so pay is a constant and that's going to infinity. And it turns out that in this case. Maybe it's not so surprising when the edge when the edge extremal graphs were complete bipartisan graphs and also the spectral extremal graphs were complete bipartisan graphs. And the first thing I want to say is that's true in this case as well. So if you have a spectral extremal graph. It has this many others. And I think there's kind of a general thing that's true here is that anytime the edge extremal graph is a terrain graph plus a constant number of edges. Then this should be true the spectrum extremal graph should be also have the maximum number of edges overall every box. Yeah I'll explain why I think this this is true. I think this is true in general, this is actually sort of interesting like you may have a, like I think it's even possible to prove this if you don't know what the edge extremal graphs are so you could, you could, you know, possibly know that the edge extremal graph satisfies this property but not know exactly what the extremal graphs are. And I think still this is true and potentially you could even prove it even if you don't know what the extremal graphs are. So one, one version of this theorem when we replace the 01 by zero is that. So, if I have a color critical graph, then it's known that for n large enough, this is an old theorem of some other that's if, if, if you have a color critical graph then for n large enough the extremal graph is just a terrain graph. And Nikki four I've showed that this is true in the spectral problem as well. So when I, this conductor is true in those cases. Okay, and the other graph that I want to look at is a wheel on T vertices so it's a, it's a, it's a vertex that has a cycle of blank T minus one in the middle of the wheel on T vertices. Notice that if T minus one is even, then this wheel is for dramatic and it's color critical and so this theorem applies. So the edge extremal graph and the spectral extremal graph. In this case, is just a, a three part type graph, complete three part type graph. On the other hand, if the total number of vertices is odd, then the, the cycle inside the neighborhood is even and so it's a three chromatic graph and it's not critical and so that these terms on apply and so we want to ask this question for for in this case. This problem was solved quite recently so for pay equals to this was solved in 2018 and the extremal graphs look like this, you take a, make this like a bigger, you take a complete bipartite graph. And you put a matching in both sides, what a matching in both sides. So if I have a complete bipartite graph and I put a vertex of degree two on top internal degree two, then you'll have, you'll have a wheel on five vertices. And that's kind of how I want to think of these constructions is sort of, we know, like asymptotically what the extremal graphs look like. So I really want to think of it as, if I start with a complete bipartite graph then what, what else am I allowed to add. It's not trivial to show that that's actually the case that's where all the work is but that's kind of the intuition that you should have. When, when pay is bigger than to the structure changes of the extremal graphs. So when pay is bigger than to what you want to do is take a complete bipartite graph on not quite balanced but close. And what I want to put up here is like up here on basically I'm not allowed to have a vertex of internal degree pay and I'm not allowed to have a path on to pay minus one vertices. So I can just put, let's say this joint copies of KK in here. And then I want to put a single edge on this side. If N is not, if the part size here is not divisible by K, then you have to do something quite a different but let's just assume for simplicity. So you kind of the structure changes in the W5 case like there's significantly many edges on both sides and then in the larger wheels case there's the edge, the internal edges only come on one side. So the first theorem is this is a joint work with SEBI and our PhD students to hear the side. And our theorem is that in the W5 case, the extremal graphs are the same. So the extremal graph is this in the spectral case. But when pay is bigger than six they're actually disjoint. So we basically we show that the extremal graphs have very similar structure to here like it's still is true that the extremal graph is roughly a complete bipartite graph. And it's still is true that you should sort of put this one edge here and put these something here that doesn't create this wall. Something with a linear number of vertices here that doesn't create this wall. So this is still true, but the size of the parts are not the same actually. So somehow like the structure is very similar but the families themselves are actually just gone. So here you have some, you can do calculus or it's an access and the part sizes are slightly off from each other. And here, it actually is a little bit confusing like it depends on parody of and stuff like that but the sizes are closer to that let's say they're off by at most For chaos. Three, so for w seven, it's also the case that the extremal graphs are the same. And we don't act we don't know for his four and five there's sort of one, they should be the same but there's one technical detail that there's a lemma that doesn't go through and we don't know how to fix it. Okay, so let me sort of briefly say how do you prove something like this. Right. So, let's say we have a graph that we think might be extremely You kind of Yeah, you sort of go in stages, showing that your graph is closer and closer to this extremal graph so you automatically have a lower bound on the spectral radius right like if you have a graph you think of the screen wall the spectral radius of that graph gives you a lower bound on this parameter. And that constraints what your extremal graphs look like right if I know my spectral radius is at least an over two then I know a decent amount about my graph I know it's fairly dense I know it doesn't have this happen so you can sort of say something about it. Once you have kind of rough structure of our graph, then you can say something about the eigenvectors of the graph. Like the structure tells you something about the eigenvector entries and the eigenvector entries also tell you something about the structure so you can kind of sort of bounce back and forth between structural information and eigenvector information. And you sort of can get more and more precise information until you are close enough to your graph. And then potentially you can alter to show that you have some sort of stability result basically like you can alter to show that the graph is actually what you think it is. Okay, so let me just remind you when when I'm forbidding a friendship graph I'm aiming for a complete bipartite graph plus a constant number that just two small cleats in there. So when I'm forbidding a wheel on five vertices I'm aiming for a complete bipartite graph plus a matching image part. And when I'm aiming for a wheel on a larger number of vertices I'm aiming for a complete bipartite graph with kind of a spanning, let's say spanning k regular graph on one side and a single edge on the other side. So that's that's kind of what I'm aiming for. You notice that if I take a step back from all of these constructions they all look like complete bipartisan graphs. The details are sort of to show that yeah this complete bipartite graph is really there. And once you kind of know the complete bipartite graph is there then you can try to argue that what's inside satisfies these properties that you think it is. So let's kind of think about how does this work in this specific case. This is kind of a broad outline of what's happening, I want to say, I want to show you how to do something like this so so this is kind of this piece, this piece of the argument. So you have this lower down and that tells you kind of roughly what the graph looks like. And so in these cases, you have a lower down to then ever to this tells me that let's just do a complete bipartite graph so so the goal here is to show I am this complete bipartite graph plus some other things. The first part of this is kind of like to get an approximation of that I want to say I have a very large maximum cut to kind of take a step back from the graph it looks like a complete bipartisan graph. After that you sort of show that okay I'm going to do this more precise argument. And I want to give some details of this because I think it's, it's nice to see how you can argue that the extremographs in the spectral case and the face are similar. Okay, because that's not always the case right like some of some of these some of these problems, the two families are completely different. But when they're the same, like, if you think they're the same how do you show us right so this is this is kind of the details that I want to give you right now. So the first is to get a lower bound on the on the spectrum radius. Okay, this one's easy. If I have a complete bipartite graph. Then the cycle radius of this graph is an over two, let's say Anna's even. So it's an over two the graph is regular and and there's no app in this graph of course so so this is a lower down so whatever my extremograph is, it has lambda one is at least an over two. And we need a structural lemma. Okay, so this is a statement that's true in any graph. The number of edges in any graph is bounded below by this function of the spectral radius and the number of time. Okay, so what that means is that if the spectral radius is large and a number of triangles is small, then this is giving me a good lower bound on my number of items. Okay, this argument is using what I was saying earlier you kind of do this eigenvector eigenvalue equation, where I'm counting walks of length two. So for a squared, and they're kind of two types of walks of length to I can kind of walk from my vertex and go outside of my neighborhood or I can kind of walk from my vertex and go inside of my neighborhood. And I do the eigenvector when I when I'm counting the passive length to basically this edge gets walked on twice I can go to this way, or I can go this way. Whereas this edge gets walked on only once when I do the passive one to there's only one way to do this to have this edge be in a walk of length to from my vertex. And so that's sort of where this number of triangles coming from that's why it comes into this this theorem. This theorem implies. It implies and I'll say I'm stronger than 1000 and it implies maybe maybe sandwich down as well, but but it implies. And it's tight for certain cases of breath so this this theorem is best possible. Now, in our specific cases. I'm going to use this to show that I have a very good lower bound so so this Lambda one squared is at least I'm squared over four that's our lower bound on on the cycle radius. And now, because our graphs, like what are they forbidding if I have, if I'm a friendship graph that means I'm forbidding a matching in the neighborhood of any vertex, a matching of size k. So that means there can't be very many edges in the neighborhood of vertex that means they're entering. If I'm forbidding a w to pay plus one and there's no cycle of length to pay in the neighborhood by the even circuit down even circuit theorem, there are not too many edges in the neighborhood and this means there's not very many triangles. So in particular there are sub cubically many triangles so you can use. You can use the triangle removal Emma. Yeah, this is this is what's if if I think about this as adding edges to a complete bipartite graph, this is what I'm not allowed to add in the part so if I really knew that my graph was complete bipartisan, then I'm not allowed to add these these things in the the details the hard part is showing that this intuition is really correct so like here. Here I'm just. I just know that I have a large maximum part it's not necessarily complete bipartisan graph and so to show that really the graph mostly looks like this that that's where the details are. You can you can use. There's a stability here with your right so you're kind of using the triangle removal Emma and the stability here with your right that says, if you have close to and this where there were four edges and you have no triangle and then there's this very large maximum time. And so I'm using the triangle removal Emma to say that I can remove a small number of edges and have a graph with no triangles but close to and square to report edges and then I'm using the stability Once I know this that sort of my graph doesn't have it and my graph has this many edges and my graph doesn't have very many triangles, then essentially like in order for that to happen in order for you to have this very large maximum time, but well it's not necessarily complete bipartisan yet, but if you take a step back it sort of looks like most of the vertices have to have close to and over to neighbors. If most of the vertices have close to and over to neighbors then you can't really have a vertex of large internal degree. So if I have a vertex that has epsilon times and neighbors, then it's going to be adjacent to lots of these vertices, like the whole neighborhood will be adjacent to lots of these vertices, and then you can kind of embed you graph. And so sort of most vertices are behaving how you want them to, and you sort of want to show that all the vertices behave how you want them to so it's kind of a progressive thing, some in some cases that it's only one or two steps and in the, in the large case like it's, it's pretty hard to show in this you kind of have to progressively get closer and closer to your graph over several different steps. So but but essentially you sort of slowly make your way to the to the part where, like, all the vertices are essentially the hate of how you want to. If all of the vertices are behaving how you want to so every vertex kind of does not have very many neighbors in its own part and has most of the neighbors in the other part. And then you, if you know that much about the graph and you know a lot about your eigenvector as well. So you can, like the graph right now, essentially looks close to regular. And what that means is that the eigenvector should be close to the concept of a regular graph a graph is regular if and only if its first eigenvector is the all one sector. And so this is saying, well, my graph is close to regular so the first eigenvector should be close to the all one sector. So, so in particular, you can sort of show that they're all yeah they're all very close to each other. And now, yeah completing the proof. Usually you can do it, or you can, you can usually. So basically, in the cases that you are showing that the spectral extremo graph and the edge extremo graph are the same. So basically the idea is that okay well, all the vertices are looking the same which means that the eigenvector is constant. And that means when I look at this so if the eigenvector was actually constant. Then when I look at this quadratic form it's equivalent to counting address right if it's content each some and it's the same. If it's close to constant then this quadratic form well it's almost the same as counting edges. And if, you know, if, if, if almost is really very close then, then, then I can't have fewer edges and the spectral extreme. So if, if the if these two, if these two functions are really close enough so this quadratic function and the number of edges if they're really that close then they, they have to be the same. Okay, so this is kind of big picture how these proofs go. I mean that even the one where the, the, the spectral extremo and the edge extremo graphs are disjoint, it's still actually satisfies this thing where where the eigenvector is close to constant but you know for whatever reason it's it's not quite the same as counting at this so so it's not, it's not always straightforward when you get the guy it's not always the same problem. Okay, so what do we do next. There's other forbidden graphs of course they're like this, this, I mean this is a very general type of question. I, there are, there are lots of open problems like the, you know, the making forest and vector about C2K free grass and that's a very nice question. And any, any question where we know that the family of sort of edge extremo graphs, it would be nice to ask well is it the same as the spectral extremo face or or not. It's very interesting and then the same or one. One thing is our, our technical lemma about getting the large maximum pipe that really only works when pay is to so so this pounding. When we did this counting this theorem is tight in certain cases but if you try to do the same thing and graph that essentially looks three part type or larger. You can essentially look by part site and the town and site. If you have basically if you have a graph that has a lot of triangles so if I have cubically many triangles, then something that is going wrong. So we sort of need a version of this theorem that is tight when your graph is essentially three part type or larger. Okay, so. We don't know how to do this I suspect though the rest of the method works like all of kind of the big picture why it works it should still work. We don't know what's happening when you add linearly many edges so that was the case with both with all the wheels you're adding linearly many edges to a complete bipartite graph. In some cases it's the same as the drawing graph in some cases the families are distraught so I'm. I'm not really sure what's going on. I don't really understand. I think it would be very nice to investigate say what what do K s s s free graphs look like. That would be a nice question. And then there's this last question. This is sort of just an annoying thing. I don't know what's happening with W 9 and W 11. We still kind of know that they roughly look extremal but. Let's say in the W nine case I can't really rule out the case that our graphs kind of looks like this triangles on one side and matching on the other. I mean, if you compare the sector radius of this graph to the spectral radius of the graph we actually think is extremal which is a four is on this side and one adjunct. If you compare these this one wins this one wins but it's it's very close. Okay, and so because it's so close. There's not a lot of room for error when we're kind of doing this. This cleaning up of our graph and it's kind of getting closer and closer to the to what we think is the extremal graph there's there's not a lot of room to make mistakes there and so we kind of can't quite show that it actually should be this. I know it probably is. Okay, so that's why I want to stop. Thank you. Thanks Mike. If we can all thank our speaker in some way and then we'll go ahead and open it up for some questions. We have any questions for our speaker. I got a question for you. So, you said, couple of slides back right that the nearly constant eigenvector for the extremal graph. It wasn't clear to me under what circumstances that happens or that you even expect that to happen. Yeah, I don't have a good general answer for you. In all of these cases that we've been able to do this. At this point we know a lot about the structure so so in every case where we've been able to say something like this it's like the graph is little o n squared edges away from a Toronto death. And so at that point you can say, you can say that the eigenvector is close to constant. Yeah, it's a it's a question I don't know how to answer in general like I don't, I can't give you a kind of structural here. If you graph of the structure then the, and the eigenvector is close to constant I think that's a that's a really nice question but it's. It's hard to figure out how to say something like that. It's close to the question of when it's an edge version right so that when when is the extremal graph or the maximal number of edges when is that nearly regular or something. I guess, is that not so well understood. I. Yeah I don't know I mean yeah it's like sort of if you if you fix any forbidden graph and kind of think about what's happening you might. You can usually deduce whether the graph is going to be close to regular very far from regular but how to do it in general I don't, I don't know. And they're definitely related questions but it's also not clear to me that they're the same question right like you can have a. Yeah, yeah it's not it's not clear to me what exactly is controlling the eigenvector entries. Yeah. Any other questions for Mike. Okay, if not, thanks again Mike, and thanks everybody for making it out and have a good weekend everybody. Okay, thanks for listening.