1874:(in that article, the eigenvalues are again from the Laplacian, so lower bounds on the second-smallest eigenvalue of the Laplacian are lower bounds on the spectral gap). Thus they're Cheeger inequality analogues for the two vertex-expansion notions. I'm sure this is not well-explained in the article and should be expanded, especially history and context. The discussions of the different notions could also be separated, with a first part about edge expansion and spectral expansion, and a second part about other notions of expansions.
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6000:(1) the WP article does not assume the graph to be connected, nor equipped with a probability measure. As a matter of fact, Cheeger's formulas work with unconnected graphs, and even the formulas established by my contradictor work as well! I never studied probability measures of graphs, so I don't know whether this requirement has implications or not. Anyway: the disputed paragraph should add those assumptions.
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I think verifiability should mean 'obvious to experts', in this case experts in spectral graph theory / expander graphs. This means it can be verified by people with a rigorous math background, like yourself, with some effort. It can be verified by average wikipedians if they learn linear algebra and math first. As an example, I think it'd take serious effort for me to verify any technical claim made in
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3578:. I don't want to restrict the article to one type of expansion. The reason why this wikipedia article is useful (for me) is precisely because it puts several different ways of looking at expander graphs into context. However, I think it'd be a good idea to have more informal sections preceding the 'definitions' section, like the 'history' section on
2576:, and here is why: Let's take the complete graph, regular, degree d, d + 1 vertices. Let's take W to be all vertices but one. The edge boundary of W is made of d edges, while the outer vertex boundary is made of only one vertex (the missing one). I think this example proves that in this case you don't get a better value than
6234:
I am explaining the problems of this article, because it is being explained to me that there is no need to do efforts to be understood or verified by the average reader, which is in complete opposition with my view of the role of
Knowledge, not to speak about the "no original work" or "verifiability"
6212:
First, please use the "Show preview" button -- your habit of making dozens of unimportant edits is extremely irritating, at least to me. Second, if you'd like to improve this article, by all means do so; but the problems with this article have nothing to do with your original complaint, which I hope
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I don't say I did something perfect, the French version is lacking the expander mixing lemma, the random walks and the geometrical interpretation of expander graphs. Also, the
Margulis construction would need some explanation wrt why it works well. But there's a chance that even those advanced topics
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I'd be surprised if the average wikipedia reader can even understand the statement. This is a specialized article for people with a strong math background. I thank you very much for diligently checking that the way I stated the inequality is now correct, and for pointing out a mistake I made earlier.
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With all due respect, we are not in a research paper. We are on wikipedia, and the average wikipedia's reader is not able to read research papers. My own definition of routine is reasonings and calculations that the average wikipedia reader would be able to do by matching the sources to the wikipedia
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as the sort of routine change-of-notation that should be allowed here. It is not reasonable to require us to use the exact notation present in our sources (for one thing, because the sources likely do not all use the exact same notation themselves) and so this sort of conversion is both necessary and
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I completly disagree that the sources should be verifiable by experts. They should be verifiable by any motivated reader. And a normal reaction if what is written in the source does not match what is written in the article is: "oh, this is not a valid source for that article", you can't expect people
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I am sorry, but I agree with David
Eppstein above and disagree with Maths Poetry. Errors do happen (I note and fix my own errors here from time to time), but this is not a reason to declare changing notation OR. The notation in Bobkov et al. is different from that in our article, therefore it is not
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Yes. There are so many points that differ from one author to another. For example, vertex boundary version edge boundary, normalized eigenvalues or not, regular graphs or connected graphs or all graphs in the expander graph definition, limits on the maximum degree or not in that definition, constant
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If M is the adjacency matrix, and A is the admittance/Laplacian matrix, then every eigenvalue λ of M corresponds to an eigenvalue (d-λ) of A, where d is the degree, so they are essentially equivalent concepts. There are also some bounds related to expanders that are expressed not in terms of the 2nd
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isn't for "people with a strong math background". It completly relies on examples, plain text explanations, graphics, and can be understood by anyone who basically understood what an unoriented graph is. Which proves that it was possible to do something else than this catalogue of formulas that the
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Further agreed: translating between different notations is tough but is necessary to make a readable encyclopedia article; the fact that sometimes factors of 2 get lots in this process is evidence of human frailty, not original research. Putting the word "wrong" in boldface repeatedly doesn't make
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From your description, it still seems routine to me. The criterion for me is: would I be embarrassed to call it a "lemma" and supply an explicit proof if I were writing this in a research paper? In this case, yes. Everything that you break down into steps 1-8 is just a trivial change of variables,
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I have shown you it is very easy to miss some point, and we both know that the tiniest detail in maths can make the better-looking constructions collapse. As a matter of fact, your original formulas were wrong, which proves, if needed, that a) this was not such a "routine" as you said, and b) that
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You've been browsing through Bobkov's article and assumed this variable A in Bobkov had the meaning of that variable B in Hoory, without even knowing whether the hypothesis were exactly the same... Also, you can't assume that people verifying those sources have a deep understanding of the subject
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The definition is much too implicit. More or less all graphs are expander graphs; the question is how "expandery" are they. The numbers defined in the article all measure how much of an expander graph a particular graph is. Some results require a particular lower bound on these numbers for the
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Yes, the
English article about expander graphs is very technical and is made to be understood only by specialists. Now you should ask to yourself: had it to be so? Isn't the purpose of wikipedia's math articles to do good vulgarization? Obviously, this one has failed that mission. Knowledge math
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It is made by artificially matching notions and quantities in one paper (Bobkov) with notions in another paper (Hoory). Ylloh has keept this part under silence when presenting the facts above. He is also keping under silence the fact that he has to use more formulas to reach his result (like the
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The
Cheeger inequalities for vertex expansion seem to be rather non-standard in the literature, so maybe we should indeed deemphasize them and move the definitions of vertex expansion and its inequalities to its own section about "vertex expansion". Also we should make clear the history of these
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I have shown that it was very dangerous to do such personal research, since wrong formulas have given wrong informations during one year and a half, which would not have happened if their author had restrained himself to doing compilation of secondary sources, as everyone does in
Knowledge but
4158:
While the explanation in the footnote may or may not be sufficient, or may even be confusing for some readers, I don't think this constitutes original research. All that happens is a simple translation from an inequality for the smallest positive eigenvalue of the
Laplacian (in the notation of
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The derivations involved consist of streamlining the notation and reordering terms. I expanded the footnote to explain the situation. Verifiability, yes, but verifiable by who? I think verifiability by experts should be enough (which includes you, for which reason the clarification footnote is
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with the second smallest eigenvalue of the
Laplacian did not make the proof collapse: I tried to build a counter example and the formula still holds. Nevertheless, you have to admit this is another detail you had overseen, which supports my claim that all of this derivation is non-obvious and
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If the graph is finite and connected, then 0 occurs with multiplicity 1 as only constant functions are harmonic, so in this case there does not seem to be any problem with (5). If the graph is not connected, for the same reason there is an obvious problem with (5). Am I missing something?
3003:
This policy does not talk about math articles. The thing we're talking about, even though it is a bit confusing, can be derived using very basic algebraic manipulations. Once we agree on the correctness, I'd prefer not to remove the
Cheeger inequalities for vertex expansion. The algebraic
4515:
Is the average reader ready to do that work of decomponing one's reasoning into little pieces and confronting it to sparse sentenses in a high-level mathematics document? Of course not. That's why there is no room for inferring new theorems on wikipedia. That's why sources must be
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did not come up. Weaker versions of these inequalities are maybe more well-known and appeared in Alon's "Eigenvalues and
Expanders" (1986), where vertex expansion was called 'magnification' and the notation was less streamlined. Even better bounds are shown in Theorem 4.6 of
3626:
In other sections of this discussion page (see above) I point out other things that I think are missing, or other points where I think there is room for improvement. Maybe what is missing the most is a definition of "expander graph" : after all, this paper is about expander
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This policy allows routine mathematical calculations, such as adding numbers, converting units, or calculating a person's age, provided there is consensus among editors that the calculation is an obvious, correct, and meaningful reflection of the sources."
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verifiable, i.e. the sentence or formula must appear in the source. Maybe with different notation, but the notation change must be obvious, here it's a discutable variable substitution and also the use of other inequities that appear somewhere else in the
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To be honest, I am getting irritated too, and I will leave you alone in your tiny club of experts. Have a lot of mathematical fun together in the high spheres. From my part, I have finished trying to help on the English speaking wikipedia. Goodbye.
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Cheeger proved an inequality for manifolds. Later on, "Cheeger-like" inequalities were proven for graphs by other people. Many people refer to these inequalities now just as "Cheeger inequalities", even though Cheeger did not prove
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I can see why vertex expansion is more interesting than edge expansion from an intuitive point of view: in a computers network, you are interested in how many computers you can reach, not how many wires you go
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This was not a simple restatement. It involved (and still involves) a lot of interpretation, understanding of advanced graph theory, reasoning, inferring from several formulas, computations, and restatement.
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Then you twisted the formulas in algebraic operations that are all but obvious to the average reader, so even if they wanted to verify your formulas they *could* not, even if those operations were correct.
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Feel free to bring more contributors in, the more the merrier. I think they will give me right on this : no original work. If it's not in the source, don't imagine it, and don't write it in the wikipedia
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These formulas, that Ylloh pretended to be proved by "routine" calculations, stayed in the article from November 9, 2010 to March, 27, 2012, until I tediously showed him a mistake in his original work.
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My first problem with that is that deriving the text on the English Knowledge from Bobkov's paper is not straightforward at all. In fact, what you did exactly matches the definition of "original work".
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I tried to find a paper that would tell me which inequality is Cheeger's and which one is Buser's and did not find any. Anyway, we were definitely using opposite names. Thanks for pointing out that.
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834:"expander" properties to be useful, but some are more in the context of a family of graphs, and then all that is required is that there exists some lower bound on these numbers ("expander family").
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manipulations we're dealing with are trivial, especially compared with the proof of these inequalities. Yet I see your point, and I'd like to have an opinion on this from other wikipedia editors.
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Graph theory is unique among mathematics in the sense that someone with very little technical knowledge can have an intuitive understanding of a lot of things, why don't you take profit of that?
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What kind of a definition is that? What are these "contexts"? Do you mean that the authors don't agree on the definition of the spectral gap? And if so, why? The cheeger inequality clearly uses
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In this section, I will detail Ylloh's proof of the disputed formulas, so everyone can make his/her opinion whether this is Original Research or just innocent rephrasing of existing material.
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What if Ylloh was not available for answering my questions, and be proven wrong, and "fixing" his original work? What if he had left wikipedia ? These wrong formulas would have stayed forever.
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I agree that it'd be nice to also state lower bounds for vertex expansion in terms of the spectral gap (like Buser's inequality). The edge expansion lower bound trivially carries over to
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Remove the "Original Work" banner if you feel like, I really don't care. I don't feel it has be proven at all that this is not original work, so, no, I am not willing to drop this claim.
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with the small exception that Tkuvho calls out, that we should justify the multiplicity of 0. The rest is just combining two nearby inequalities from the same source in an obvious way. —
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point of view (basically, the one of Shlomo Hoory's paper), and keep saying "one could define things differently", "similar relations exist for...", "definition might vary", etc. See
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So instead of embracing all point of views, I just describe one, and let the reader dig for the other ones. Not perfect a solution, of course, probably just a matter of tastes...
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The fact that personal work should not be on wikipedia is a general principle. I suppose that you grasp the motivation for this rule way better now that I exhibited this mistake.
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It's not radically different, it's a factor of 2. I introduced the factor of 2 already when I corrected the mistake in the article. The most recent version should be correct.
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6006:(5) This is obviously dubious, but I did not succeed in building a counter-example out of that constatation... This does not mean it is correct to do such assumptions though.
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The most we can do with respect to maths is to copy the existing theorems from the sources, and sched some light on them, but we cannot infer new theorems by ourselves.
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and never seem to discuss the issue. Different authors do use different definitions based on what is more convenient for them. And there are many different choices:
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eigenvalue of the laplacian matrix, while there's no such equivalent thing with Hoory (think about the case where two first eigenvalues of the adjacency matrix are
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A disconnected graph satisfies the inequalities in question (since the expansion of the smallest connected component is 0, and the right-hand sides are also 0).
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I also added examples. I don't give as many details as the English page (by far), but at least people might understand what it is all about. At least I hope so.
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that it is a simple transformation, or a simple change in the notation, that's more to it. Also, I am still not fully convinced that current result is correct.
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It defines a lot of things, and gives a lot of properties, but does not define the object described in its title. Or I did not find this definition. --
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The formulas that are in the article are in NO external source. You can search everywhere, you won't find it anywhere in the scientific litterature.
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difference, if I am not wrong. I think there's a possibility the article did some overinterpretation of Hoory here. Please correct me if I'm wrong.
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Sorry if I offended you. Please replace "your web site" with "a mathematical review with reading committee" and it will sound much more graceful.
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In the spectral expansion section, you mention that lamba = max_{1 <= i <= n}(lambda_1, ..., lambda_{n-1}) but we have lambda_{i+1} : -->
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1901:. Even if this reasoning is true, it's original research, and therefore you have to put it onto your web site, but not on Knowledge. Sorry.
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No. Cheeger's inequality is an upper bound on h, while Buser's inequality is a lower bound on h, both in terms of the spectral gap. See
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6137:. Yet I am happy that people contribute technical content there, and I am confident that some experts have verified claims made there.
4170:) to an inequality for the second-largest eigenvalue of the adjacency matrix. I would greatly appreciate the opinion of other editors.
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3724:. See also the discussion above, consisting of 80 edits this user made here on the talk page. The two inequalities in question are:
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formulas in a maths article since November, 9, 2010? If you had restrained from doing original work, that would not have happened.
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A directed graph can also be interpreted as a balanced bipartite graph (with all edges going from one copy of V to another copy).
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Show me a scientific publication where these formulas are shown, and you could put them here. Meanwhile: no sources, no sugar. --
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Also, the degree is an integer and the logarithm is likely not an integer. So I doubt that's even an equal sign that is meant.
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I clarified this a little bit. I agree that the article should have a single consistent definition. Cheeger's inequality uses
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For the record, I don't agree on correctness yet, there are still so much thing that could make that result of yours wrong.
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I have edited the disputed footnote a bit, to make the ref-s more specific. Note that combining the BHT bound in terms of
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To avoid the impression of a "mess" in the article (sorry for the bad word), I took the following approach: I just take
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are not in the paper of Bobkov et. al. -- or I did not find them. This paper is used as a source for those formulas.
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2486:(theirs), it means that the formulas in the article that you derived (with what was supposed to be routine !!!) are
2009:, whatever its name. It enables to prove that a given graph is expander in a given ratio. We have nothing alike for
750:= lambda_i. So is lambda = lambda_1 ? In that case, how is the spectral gap d - lambda_1 different from d - lambda?
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related articles on Knowledge. If you would like to participate, please visit the project page, where you can join
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5034:(5) Assuming that the smallest non-zero eigenvalue and the second smallest eigenvalue are the same, we can write
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Here is the definition I added to French page: (translating it into English, sorry if that's not correct English)
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That's also the conclusion I reached. Readers will judge whether this point is just "routine" considerations. --
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Also, there are three different notions referred to in this proof that are usually represented with the symbol
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I suppose that d = log | V | or d = log | E | is meant. G itself is (V, E) and does not even have a cardinal.
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current "expander graph" article has turned into. Also, every formula that lies in the French article can be
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is also very far from OR by synthesis, since it is stated explicitly in the 3rd displayed formula on p.154.
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Here is an example of one of the things that might still go wrong and that Ylloh eluded: Bobkov assumes his
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is the smallest positive eigenvalue of the Laplacian. If you want, this fact can be added to the footnote.
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degree in the expander families or not, etc. The list of all variations on this theme is really impressive.
2171:, and I find your proposal to put it on my webpage to be offensive. Nevertheless I appreciate your opinion.
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Finally, to be able to have something that is really comparable to Cheeger's inequality, we would need a
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Almost 12 years later, there is still no definition here on the English page! Please, someone fix this.
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As remarked in the footnote, it's implied by Theorem 1 of Bobkov et al.. They use a different notation.
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Following the critique of MathsPoetry, I looked at the beginning of the article. The definition says:
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isn't it the second eigenvalue of the admittance matrix rather than adjacency matrix that matters ? --
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Do we agree that this level of demonstration + calculations is far, far away from this definition? --
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in the mentioned sources, without the need for a committee of experts to emit a doctrinal agreement.
2284:. He uses vastly different notation, though, and I think the more modern vertex expansion notation (
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6003:(2) Notice that Bobkov et. al.'s result is more general, since it does not assume a regular graph.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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bound, which makes me think that this whole work of mixing Cheeger and Bobkov is a bit pointless.
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4940:(1) Let G = (V, E) be a finite, undirected, connected graph equipped with a probability measure
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This very illustrative example should prevent anyone from the temptation of doing original work.
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from the inequations here, we are losing accuracy on the bound. I don't know "how much" though.
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2514:???? That's in the manifold (continuous case), for the "other" Cheeger constant, isn't it? --
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1521:, as you pretend. To prove this second assertion, you give a complicated proof in a footnote.
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Ping. I still don't see where they say the spectral gap is d - λ. Original research too ? --
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Correct, but your proof of it collapses. And I think that you'll have to introduce a factor
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routine calculations at all, there is room for interpretation and error. ... as I proved it.
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5652:{\displaystyle 2\cdot (d-\lambda _{2})\geq {\frac {({\sqrt {1+h_{\text{out}}}}-1)^{2}}{2}}}
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boring typos or arithmetic errors into something important or into violations of policy. --
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Don't think I'm lazy, if English were my mother language I would add this stuff myself. --
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say that the spectral gap is d - λ. Or did I miss it? As a matter of fact, a small lambda
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This article about expander graphs does not even give a definition of an expander graph.
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Yes, but section 2.4 still does only say "a small λ (or a large spectral gap)", it does
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With respect to the need of informal stuff, motivation, and history, I do agree. Also,
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5483:{\displaystyle \lambda _{\infty }\geq {\frac {({\sqrt {1+h_{\text{out}}}}-1)^{2}}{2}}}
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3817:{\displaystyle h_{\text{out}}(G)\leq \left({\sqrt {4(d-\lambda _{2})}}+1\right)^{2}-1}
1413:{\displaystyle \lambda _{\infty }\geq {\frac {({\sqrt {1+h_{\text{out}}}}-1)^{2}}{2}}}
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Requested articles/Applied arts and sciences/Computer science, computing, and Internet
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Ylloh asked me to comment here. To me, what he describes seems to fall clearly under
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I don't think any of this is original research because the changes can be considered
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See two discussion sections above for another problem with d = log | G | lines... --
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That does not contradict with what is claimed in the current version of the article.
5162:(it's a general property of regular graphs, that can be demonstrated rather easily)
4823:. For the sake of our mental health, I will distinguish them clearly and call them
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no definition of an "expander graph", while it is the subject of the article (!!!)
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So yes, the precise definition of 'spectral gap' depends on authors and contexts.
772:, I must be missing something, seems like you're making a wholly different graph.
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An expander is a finite, undirected multigraph that has good expansion properties
5981:{\displaystyle h_{\text{out}}(G)\leq ({\sqrt {4\cdot (d-\lambda _{2})}}+1)^{2}-1}
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I'm sorry that this is confusing. The area does not have a streamlined notation.
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Bobkov et al. use the Laplacian instead of the adjacency matrix. All eigenvalues
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5723:{\displaystyle {\sqrt {2\cdot (d-\lambda _{2})}}\geq {\frac {h_{\text{in}}}{2}}}
4336:, i.e. the unconnected regular graph, in this case the definitions don't match).
3582:, or a 'motivation' section that talks about applications and intuitions first.
2280:
expansion properties. My understanding is that they originate in the article of
1922:, not for the spectral gap :-). Cheeger's inequality is about a lower bound for
444:
4213:
Knowledge:Scientific citation guidelines#Examples, derivations and restatements
538:
for the complement of a vertex set standard? It makes my eyes sore to look at
6138:
6080:
4684:
4197:
3669:
3583:
3450:
3005:
2939:
2907:
2530:
2501:
2392:
1935:
1879:
1164:
597:
421:
4134:
to denote twice the smallest positive eigenvalue of the Laplacian matrix of
2924:
if you want to prove anything, which will radically change your formulas. --
2272:
I agree that the Bobkov et al. source is not the best. Maybe Theorem 4.6 of
215:
4545:
only allowed but actually desired to change it before citing their result.
4435:{\displaystyle h_{\text{out}}(G)\leq ({\sqrt {2(d-\lambda _{2})}}+1)^{2}-1}
2115:
eigenvalue of the Laplacian? All that I can see is a definition of (their)
1265:{\displaystyle h_{\text{out}}(G)\leq ({\sqrt {2(d-\lambda _{2})}}+1)^{2}-1}
73:
52:
5556:{\displaystyle 2\cdot (d-\lambda _{2})\geq {\frac {h_{\text{in}}^{2}}{4}}}
4904:. But keep in mind that they all appear in the scientific publications as
4179:
3037:
the wikipedia policy against original work has a good reason for being so.
2273:
6302:
6283:
6251:
6222:
6207:
6146:
6127:
6111:
6088:
6074:
6056:
4771:
4727:
4692:
4569:
4554:
4539:
4225:
4205:
3694:
3677:
3662:
3639:
3591:
3551:
3490:
3458:
3262:
3067:
3013:
2997:
2969:
2947:
2933:
2915:
2900:
2626:
2538:
2523:
2509:
2400:
2380:{\displaystyle h_{\text{out}}(G)\leq h(G)\leq d\cdot h_{\text{out}}(G)}
2269:
previously, thanks for catching that. I corrected this mistake for now.
2155:
1943:
1887:
1710:
1172:
1157:
1100:
1085:
Same for the other suggestion for the degree that follows : d = | G |.
1067:
1053:
843:
827:
804:
776:
735:
607:
5885:{\displaystyle h_{\text{in}}(G)\leq {\sqrt {8\cdot (d-\lambda _{2})}}}
3052:
to reinvent the complex reasoning that went through your head one day.
2242:
in the notation of Bobkov et al.. I wrongly thought it to be equal to
5405:{\displaystyle \lambda _{\infty }\geq {\frac {h_{\text{in}}^{2}}{4}}}
4629:
to be the vertex set of the smaller connected component). Otherwise,
1466:{\displaystyle \lambda _{\infty }\geq {\frac {h_{\text{in}}^{2}}{4}}}
4501:{\displaystyle h_{\text{in}}(G)\leq 2\cdot {\sqrt {d-\lambda _{2}}}}
1331:{\displaystyle h_{\text{in}}(G)\leq 2\cdot {\sqrt {d-\lambda _{2}}}}
6178:
no drawings, which is quite surprizing in a text about graph theory
3055:
Also, you did not convince me that WP:CALC does not apply to maths.
2960:
that it's dangerous to let contributors infer their own results. --
1870:
These inequalities are lower bounds on the spectral gap, just like
6293:
the definition of expander is missing: I am really disappointed!--
504:
The final paras here hang by a very tenuous thread from the rest.
4963:(2) Let's also assume G is d-regular (we will need that later on)
4173:(Interestingly, the question whether these inequalities satisfy
3896:
The inequalities are now accompanied by the following footnote:
3886:{\displaystyle h_{\text{in}}(G)\leq {\sqrt {8(d-\lambda _{2})}}}
3295:
but, e.g., the expander mixing lemma breaks down if you replace
2981:
Wow. Quoting : "Routine calculations - Policy shortcut: WP:CALC
2387:. But something better could be known. Do you know a reference?
5351:{\displaystyle 2\cdot (d-\lambda _{2})\geq \lambda _{\infty }}
3207:{\displaystyle \lambda =\max\{|\lambda _{2}|,|\lambda _{n}|\}}
2952:
OK ! We got to it. We proved that your original formulas were
291:
Find pictures for the biographies of computer scientists (see
15:
5109:
be the second biggest eigenvalue of the adjacency matrix of G
4734:
to Ylloh: I noticed that the inaccuracy on matching Bobkov's
1664:
I understand it but can't see how you can say that (their)
3668:
This is not claimed in the current version of the article.
2174:
What we write in the notation of the wikipedia article as
1662:
is the second-largest eigenvalue of the adjacency matrix).
6156:. This is precisely the point where we completly diverge.
6118:
article. That's what verifiability is all about, IMHO. --
4187:
harvtxt error: no target: CITEREFBobkovHoudréTetali2000 (
4164:
harvtxt error: no target: CITEREFBobkovHoudréTetali2000 (
3905:
harvtxt error: no target: CITEREFBobkovHoudréTetali2000 (
2617:
I'd also like to thank you for your kind help on this, --
4576:
For your question about the correctness, notice that if
4065:(they explicitly state this inequality, albeit they use
4058:{\displaystyle \lambda _{\infty }\leq 2(d-\lambda _{2})}
2838:{\displaystyle \lambda _{\infty }\leq 2(d-\lambda _{2})}
6175:
Let me show you how I see the current English article:
3721:
3682:
Oh, you fixed that already. Thanks. This one is solved.
1863:
from the adjacency matrix as in the wikipedia article).
1790:
that they consider is some other notion that satisfies
6159:
I hate to say "I did better", but the French article
6015:
5898:
5827:
5736:
5667:
5569:
5499:
5418:
5366:
5306:
5263:
5233:
5173:
5118:
5088:
5040:
5006:
4972:
4946:
4910:
4883:
4856:
4829:
4802:
4740:
4635:
4615:
4582:
4449:
4355:
4307:
4267:
4240:
4140:
4098:
4071:
4016:
3989:
3953:
3917:
3831:
3733:
3561:
3521:
3501:
3417:
3397:
3375:
3348:
3321:
3301:
3274:
3223:
3145:
3119:
3086:
3080:"In some contexts, the spectral gap is defined to be
2884:{\displaystyle \lambda _{\infty }\leq d-\lambda _{2}}
2851:
2796:
2738:
2696:
2665:
2638:
2582:
2555:
2458:
2425:
2314:
2248:
2213:
2180:
2121:
2090:
2051:
2015:
1958:
1842:
1829:{\displaystyle \lambda _{\infty }\leq d-\lambda _{2}}
1796:
1769:
1743:
1723:
1670:
1641:
1608:
1595:{\displaystyle \lambda _{\infty }\leq d-\lambda _{2}}
1562:
1535:
1514:{\displaystyle \lambda _{\infty }\leq d-\lambda _{2}}
1481:
1427:
1348:
1279:
1185:
1009:
951:
907:
887:
867:
710:
651:
620:
544:
519:
2002:{\displaystyle h(G)\geq {\frac {d-\lambda _{2}}{2}}}
443:, a collaborative effort to improve the coverage of
85:, a collaborative effort to improve the coverage of
6274:. I think we can do better than that. Suggestions?
3575:is defined in the first sentence of Section 2.4 in
3253:
This article definitly needs clarification IMHO. --
2146:
eigenvalue of the Laplacian (pages 153 and 156). --
1691:
is the second-smallest eigenvalue of the Laplacian.
789:Isn't the Reingold paper from 2004 and not 2008 ?
697:{\displaystyle \max\{\lambda _{2},|\lambda _{n}|\}}
6028:
5980:
5884:
5811:
5722:
5651:
5555:
5482:
5404:
5350:
5289:
5246:
5217:
5150:
5101:
5072:
5019:
4985:
4952:
4923:
4896:
4869:
4842:
4815:
4789:The academic paper these formulas are based on is
4753:
4675:
4621:
4601:
4500:
4434:
4320:
4280:
4253:
4183:
4160:
4146:
4126:
4084:
4057:
4002:
3975:
3939:
3901:
3885:
3816:
3567:
3534:
3507:
3430:
3403:
3381:
3361:
3334:
3307:
3287:
3242:
3206:
3131:
3105:
3021:I think these formulas qualify as original work :
2883:
2837:
2779:
2722:
2678:
2651:
2606:
2568:
2478:
2444:
2379:
2261:
2234:
2199:
2134:
2103:
2073:
2037:
2001:
1855:
1828:
1782:
1755:
1737:of the adjacency matrix correspond to eigenvalues
1729:
1683:
1654:
1627:
1594:
1548:
1513:
1465:
1412:
1330:
1264:
1030:
995:
937:
893:
873:
723:
696:
633:
588:
530:
197:Computer science articles needing expert attention
3441:adjacency matrix vs. normalized adjacency matrix.
5218:{\displaystyle \mu _{2}=2\cdot (d-\lambda _{2})}
3152:
2732:Because this is a regular graph, we know that ;
652:
5290:{\displaystyle \mu _{2}\geq \lambda _{\infty }}
2723:{\displaystyle \lambda _{\infty }\leq \mu _{2}}
2549:I don't think there is a better lower vaue for
2276:is better (it's newer and gives better bounds).
1129:What is in Bobkov's paper is the definition of
337:WikiProject Computer science/Unreferenced BLPs
1952:What I am trying to focus on is the relation
996:{\displaystyle |\partial (W)|\geq \gamma |W|}
641:, but in terms of the 2nd largest eigenvalue
8:
3720:) added the "Original research" template in
3201:
3155:
691:
655:
254:Computer science articles without infoboxes
192:Computer science articles needing attention
4339:During more than one year, this so-called
3495:Ah, as I understood it, in Hoory's paper,
2780:{\displaystyle \mu _{2}=2(d-\lambda _{2})}
2084:Where is it stated in Bobkov that (their)
389:
158:Here are some tasks awaiting attention:
132:
47:
6020:
6014:
5966:
5945:
5924:
5903:
5897:
5871:
5850:
5832:
5826:
5787:
5775:
5772:
5758:
5737:
5735:
5709:
5703:
5689:
5668:
5666:
5637:
5619:
5607:
5601:
5589:
5568:
5542:
5537:
5531:
5519:
5498:
5468:
5450:
5438:
5432:
5423:
5417:
5391:
5386:
5380:
5371:
5365:
5342:
5326:
5305:
5281:
5268:
5262:
5254:be the Poincaré-type constant ( page 155)
5238:
5232:
5206:
5178:
5172:
5142:
5123:
5117:
5093:
5087:
5064:
5045:
5039:
5011:
5005:
4977:
4971:
4945:
4915:
4909:
4888:
4882:
4861:
4855:
4834:
4828:
4807:
4801:
4791:λ∞, vertex isoperimetry and concentration
4745:
4739:
4676:{\displaystyle \mu _{2}/2=d-\lambda _{2}}
4667:
4646:
4640:
4634:
4614:
4593:
4581:
4490:
4478:
4454:
4448:
4420:
4399:
4381:
4360:
4354:
4312:
4306:
4272:
4266:
4245:
4239:
4139:
4115:
4097:
4076:
4070:
4046:
4021:
4015:
3994:
3988:
3958:
3952:
3922:
3916:
3872:
3854:
3836:
3830:
3802:
3780:
3762:
3738:
3732:
3560:
3526:
3520:
3500:
3422:
3416:
3396:
3374:
3353:
3347:
3326:
3320:
3300:
3279:
3273:
3234:
3222:
3196:
3190:
3181:
3173:
3167:
3158:
3144:
3118:
3113:. In other contexts, the spectral gap is
3097:
3085:
2875:
2856:
2850:
2826:
2801:
2795:
2768:
2743:
2737:
2714:
2701:
2695:
2670:
2664:
2643:
2637:
2583:
2581:
2560:
2554:
2479:{\displaystyle {\frac {\lambda _{2}}{2}}}
2465:
2459:
2457:
2436:
2424:
2362:
2319:
2313:
2253:
2247:
2224:
2218:
2212:
2191:
2179:
2126:
2120:
2095:
2089:
2056:
2050:
2020:
2014:
1987:
1974:
1957:
1926:, the upper bound is from some othe guy.
1847:
1841:
1820:
1801:
1795:
1774:
1768:
1742:
1722:
1675:
1669:
1646:
1640:
1619:
1607:
1586:
1567:
1561:
1540:
1534:
1505:
1486:
1480:
1452:
1447:
1441:
1432:
1426:
1398:
1380:
1368:
1362:
1353:
1347:
1320:
1308:
1284:
1278:
1250:
1229:
1211:
1190:
1184:
1008:
988:
980:
969:
952:
950:
927:
916:
908:
906:
886:
866:
715:
709:
686:
680:
671:
662:
650:
625:
619:
581:
573:
568:
557:
543:
520:
518:
6322:Mid-importance Computer science articles
5073:{\displaystyle \mu _{2}=2\cdot \nu _{2}}
4997:of the Laplacian matrix of G ( page 153)
1527:My second problem is your explanation :
731:is always d, and is always the largest)
6152:articles aren't meant to be by experts
5151:{\displaystyle \nu _{2}=d-\lambda _{2}}
2391:Thanks again for sharing your opinion!
1872:Cheeger_constant#Cheeger.27s_inequality
1110:The formulas giving an upper bound for
1106:Upper bounds for vertex expansion rates
391:
49:
19:
3025:that would enable them to do the same.
1148:Sorry for the bad news, once again. --
99:Knowledge:WikiProject Computer science
6327:WikiProject Computer science articles
4010:that they use in the paper satisfies
2686:, for the sake of our mental sanity.
881:ratio when, for each vertices subset
102:Template:WikiProject Computer science
7:
5112:(7) Since the graph G is d-regular,
3058:Thank you for your understanding. --
437:This article is within the scope of
79:This article is within the scope of
6295:2A00:1398:9:FB03:226:BBFF:FE0E:AC42
6172:can be explained with simple words.
5167:(8) From (5) and (7), we can write
3044:Do you realize that you introduced
1003:is verified. One might notice that
38:It is of interest to the following
6213:you're finally willing to drop. --
6029:{\displaystyle \lambda _{\infty }}
6021:
5424:
5372:
5343:
5282:
5247:{\displaystyle \lambda _{\infty }}
5239:
4254:{\displaystyle \lambda _{\infty }}
4246:
4184:Bobkov, Houdré & Tetali (2000)
4161:Bobkov, Houdré & Tetali (2000)
4022:
4003:{\displaystyle \lambda _{\infty }}
3995:
3902:Bobkov, Houdré & Tetali (2000)
2857:
2802:
2702:
1802:
1783:{\displaystyle \lambda _{\infty }}
1775:
1568:
1549:{\displaystyle \lambda _{\infty }}
1541:
1487:
1433:
1354:
957:
273:Timeline of computing 2020–present
14:
6337:Mid-priority mathematics articles
6317:C-Class Computer science articles
4193:with the more recent reference.)
4127:{\displaystyle 2(d-\lambda _{2})}
3940:{\displaystyle h_{\text{out}}(G)}
2607:{\displaystyle {\frac {h(G)}{d}}}
2500:E.g., Theorem 4.8 in Hoory et al.
2074:{\displaystyle h_{\text{out}}(G)}
457:Knowledge:WikiProject Mathematics
299:Computing articles needing images
3983:. Then notice that the quantity
3976:{\displaystyle h_{\text{in}}(G)}
3515:is just a commodity writing for
2038:{\displaystyle h_{\text{in}}(G)}
1031:{\displaystyle \gamma \leq h(G)}
460:Template:WikiProject Mathematics
424:
414:
393:
149:
72:
51:
20:
3900:This follows from Theorem 1 in
3444:adjacency matrix vs. Laplacian.
1908:for the vertex expansion rates
1475:This boils down to the same if
477:This article has been rated as
119:This article has been rated as
5963:
5951:
5932:
5921:
5915:
5909:
5877:
5858:
5844:
5838:
5764:
5745:
5695:
5676:
5634:
5604:
5595:
5576:
5525:
5506:
5465:
5435:
5332:
5313:
5212:
5193:
4602:{\displaystyle d=\lambda _{2}}
4466:
4460:
4417:
4405:
4386:
4378:
4372:
4366:
4182:, and I'm inclined to replace
4121:
4102:
4052:
4033:
3970:
3964:
3934:
3928:
3878:
3859:
3848:
3842:
3786:
3767:
3750:
3744:
3243:{\displaystyle d-\lambda _{2}}
3197:
3182:
3174:
3159:
3106:{\displaystyle d-\lambda _{2}}
2832:
2813:
2774:
2755:
2595:
2589:
2569:{\displaystyle h_{\text{out}}}
2445:{\displaystyle d-\lambda _{2}}
2374:
2368:
2346:
2340:
2331:
2325:
2235:{\displaystyle \lambda _{2}/2}
2200:{\displaystyle d-\lambda _{2}}
2068:
2062:
2032:
2026:
1968:
1962:
1899:Knowledge:No_original_research
1628:{\displaystyle d-\lambda _{2}}
1395:
1365:
1296:
1290:
1247:
1235:
1216:
1208:
1202:
1196:
1025:
1019:
989:
981:
970:
966:
960:
953:
917:
909:
687:
672:
582:
574:
565:
548:
1:
6238:Sorry if I am irritating you.
6187:no vulgarization explanations
2790:From that, we can infer that
451:and see a list of open tasks.
353:Tag all relevant articles in
93:and see a list of open tasks.
6332:C-Class mathematics articles
6190:prove-it-yourself assertions
5102:{\displaystyle \lambda _{2}}
5031:of the Laplacian matrix of G
4995:smallest non-zero eigenvalue
4924:{\displaystyle \lambda _{2}}
4843:{\displaystyle \lambda _{2}}
4816:{\displaystyle \lambda _{2}}
4754:{\displaystyle \lambda _{2}}
4704:with the inequality between
4343:transformation led to these
4321:{\displaystyle \lambda _{2}}
4281:{\displaystyle \lambda _{2}}
4085:{\displaystyle \lambda _{2}}
3703:"Original research" template
3535:{\displaystyle \lambda _{2}}
3431:{\displaystyle \lambda _{2}}
3362:{\displaystyle \lambda _{2}}
3335:{\displaystyle \lambda _{2}}
3288:{\displaystyle \lambda _{2}}
2679:{\displaystyle \lambda _{2}}
2262:{\displaystyle \lambda _{2}}
2135:{\displaystyle \lambda _{2}}
2104:{\displaystyle \lambda _{2}}
1856:{\displaystyle \lambda _{2}}
1684:{\displaystyle \lambda _{2}}
1655:{\displaystyle \lambda _{2}}
1068:20:57, 6 February 2024 (UTC)
736:03:25, 12 October 2005 (UTC)
724:{\displaystyle \lambda _{1}}
634:{\displaystyle \lambda _{2}}
362:WikiProject Computer science
138:WikiProject Computer science
82:WikiProject Computer science
6303:16:03, 29 August 2017 (UTC)
3608:, and therefore a big d - λ
3542:. Did I miss some point? --
2845:, but we cannot infer that
1529:noticing that the quantity
938:{\displaystyle |W|\leq n/2}
757:Directed graph as bipartite
608:19:04, 10 August 2005 (UTC)
589:{\displaystyle E(S,/S)/|S|}
293:List of computer scientists
6353:
6284:04:26, 29 March 2012 (UTC)
6252:22:30, 28 March 2012 (UTC)
6223:22:16, 28 March 2012 (UTC)
6208:21:10, 28 March 2012 (UTC)
6147:20:27, 28 March 2012 (UTC)
6128:18:24, 28 March 2012 (UTC)
6112:18:17, 28 March 2012 (UTC)
6089:20:27, 28 March 2012 (UTC)
6075:17:28, 28 March 2012 (UTC)
6057:17:20, 28 March 2012 (UTC)
5029:second smallest eigenvalue
4793:. I will refer to it as .
4772:16:55, 28 March 2012 (UTC)
4728:16:02, 28 March 2012 (UTC)
4693:14:51, 28 March 2012 (UTC)
4570:14:48, 28 March 2012 (UTC)
4555:14:25, 28 March 2012 (UTC)
4540:07:23, 28 March 2012 (UTC)
4226:02:29, 28 March 2012 (UTC)
4206:02:15, 28 March 2012 (UTC)
3695:20:14, 27 March 2012 (UTC)
3678:19:31, 27 March 2012 (UTC)
3663:17:04, 27 March 2012 (UTC)
3640:17:00, 26 March 2012 (UTC)
3592:15:16, 26 March 2012 (UTC)
3552:20:44, 25 March 2012 (UTC)
3491:20:38, 25 March 2012 (UTC)
3459:18:00, 25 March 2012 (UTC)
3263:15:37, 24 March 2012 (UTC)
3132:{\displaystyle d-\lambda }
3076:Definition of spectral gap
3068:22:12, 27 March 2012 (UTC)
3014:21:48, 27 March 2012 (UTC)
2998:21:14, 27 March 2012 (UTC)
2970:22:20, 27 March 2012 (UTC)
2948:21:48, 27 March 2012 (UTC)
2934:20:54, 27 March 2012 (UTC)
2916:20:45, 27 March 2012 (UTC)
2901:20:27, 27 March 2012 (UTC)
2627:20:04, 27 March 2012 (UTC)
2539:21:48, 27 March 2012 (UTC)
2524:20:53, 27 March 2012 (UTC)
2510:20:45, 27 March 2012 (UTC)
2401:19:29, 27 March 2012 (UTC)
2156:17:28, 27 March 2012 (UTC)
2144:twice the smallest nonzero
1944:17:18, 27 March 2012 (UTC)
1888:16:51, 27 March 2012 (UTC)
1756:{\displaystyle d-\lambda }
1711:14:57, 27 March 2012 (UTC)
1173:22:50, 26 March 2012 (UTC)
1158:19:24, 26 March 2012 (UTC)
1101:14:13, 24 March 2012 (UTC)
1054:21:09, 19 March 2012 (UTC)
844:12:01, 19 March 2012 (UTC)
828:10:09, 19 March 2012 (UTC)
805:23:01, 19 April 2010 (UTC)
777:18:39, 8 August 2006 (UTC)
742:Spectral expansion section
125:project's importance scale
5493:(13) From (11) and (12),
4328:variable to be the first
768:cannot be interpreted as
476:
409:
355:Category:Computer science
131:
118:
105:Computer science articles
67:
46:
5995:Remarks from MathsPoetry
5300:(11) From (8) and (10),
5020:{\displaystyle \nu _{2}}
4986:{\displaystyle \mu _{2}}
4897:{\displaystyle \nu _{2}}
4870:{\displaystyle \mu _{2}}
3568:{\displaystyle \lambda }
3508:{\displaystyle \lambda }
3404:{\displaystyle \lambda }
3382:{\displaystyle \lambda }
3308:{\displaystyle \lambda }
2659:what Bobkov et al. call
2652:{\displaystyle \mu _{2}}
1730:{\displaystyle \lambda }
600:23:55, 5 Jan 2005 (UTC)
510:18:29, 6 Sep 2004 (UTC)
483:project's priority scale
357:and sub-categories with
1340:Bobkov' theorem 1 says:
874:{\displaystyle \gamma }
440:WikiProject Mathematics
6289:definition of expander
6030:
5982:
5886:
5813:
5724:
5653:
5557:
5490:(theorem 1, page 156)
5484:
5406:
5352:
5291:
5248:
5219:
5152:
5103:
5074:
5021:
4987:
4954:
4925:
4898:
4871:
4844:
4817:
4755:
4677:
4623:
4603:
4502:
4436:
4322:
4282:
4255:
4148:
4128:
4086:
4059:
4004:
3977:
3941:
3887:
3818:
3569:
3536:
3509:
3432:
3405:
3383:
3363:
3336:
3309:
3289:
3244:
3208:
3133:
3107:
2885:
2839:
2781:
2724:
2680:
2653:
2608:
2570:
2480:
2446:
2409:Your calculations are
2381:
2263:
2236:
2201:
2136:
2105:
2075:
2039:
2003:
1857:
1830:
1784:
1763:of the Laplacian. The
1757:
1731:
1685:
1656:
1629:
1596:
1550:
1515:
1467:
1414:
1332:
1266:
1032:
997:
939:
895:
875:
725:
698:
635:
590:
532:
318:Computer science stubs
28:This article is rated
6031:
5983:
5887:
5814:
5725:
5654:
5558:
5485:
5407:
5353:
5292:
5249:
5220:
5153:
5104:
5075:
5022:
4988:
4955:
4926:
4899:
4872:
4845:
4818:
4781:
4756:
4678:
4624:
4604:
4503:
4437:
4323:
4283:
4256:
4149:
4129:
4087:
4060:
4005:
3978:
3942:
3888:
3819:
3570:
3537:
3510:
3433:
3406:
3384:
3364:
3337:
3310:
3290:
3245:
3209:
3134:
3108:
2886:
2840:
2782:
2725:
2690:Bobkov states that :
2681:
2654:
2609:
2571:
2481:
2447:
2382:
2264:
2237:
2202:
2137:
2106:
2076:
2040:
2004:
1858:
1831:
1785:
1758:
1732:
1686:
1657:
1630:
1597:
1551:
1516:
1468:
1415:
1333:
1267:
1091:I hope this helps. --
1033:
998:
940:
896:
876:
726:
699:
636:
591:
533:
6013:
6009:(13) By eliminating
5896:
5825:
5734:
5665:
5567:
5497:
5416:
5364:
5304:
5261:
5231:
5171:
5116:
5086:
5038:
5004:
4970:
4953:{\displaystyle \pi }
4944:
4908:
4881:
4854:
4827:
4800:
4738:
4633:
4613:
4580:
4447:
4353:
4305:
4265:
4238:
4175:Knowledge:Notability
4138:
4096:
4069:
4014:
3987:
3951:
3915:
3829:
3731:
3559:
3519:
3499:
3415:
3395:
3373:
3346:
3319:
3299:
3272:
3221:
3143:
3117:
3084:
2891:as you announced. --
2849:
2794:
2736:
2694:
2663:
2636:
2580:
2553:
2456:
2423:
2312:
2246:
2211:
2178:
2169:Routine calculations
2119:
2088:
2049:
2013:
1956:
1840:
1794:
1767:
1741:
1721:
1668:
1639:
1606:
1560:
1533:
1479:
1425:
1346:
1277:
1183:
1007:
949:
905:
885:
865:
764:I don't understand.
708:
649:
618:
542:
517:
463:mathematics articles
136:Things you can help
6161:fr:Graphe expanseur
6135:Kaluza-Klein_theory
6041:Remarks from Tkuvho
5547:
5396:
3621:fr:Graphe expanseur
3580:fr:Graphe expanseur
3473:fr:Graphe expanseur
1556:they use satisfies
1457:
6026:
5978:
5882:
5809:
5720:
5649:
5553:
5533:
5480:
5402:
5382:
5348:
5287:
5244:
5215:
5148:
5099:
5070:
5017:
4983:
4950:
4921:
4894:
4867:
4840:
4813:
4751:
4673:
4619:
4599:
4498:
4432:
4318:
4278:
4251:
4144:
4124:
4082:
4055:
4000:
3973:
3937:
3883:
3814:
3565:
3532:
3505:
3428:
3401:
3379:
3359:
3332:
3305:
3285:
3240:
3204:
3129:
3103:
2881:
2835:
2777:
2720:
2676:
2649:
2604:
2566:
2490:. See proof below.
2476:
2442:
2377:
2259:
2232:
2197:
2132:
2101:
2071:
2035:
1999:
1853:
1826:
1780:
1753:
1727:
1681:
1652:
1625:
1592:
1546:
1511:
1463:
1443:
1410:
1328:
1262:
1028:
993:
935:
891:
871:
721:
694:
631:
586:
531:{\displaystyle /S}
528:
432:Mathematics portal
34:content assessment
5954:
5906:
5880:
5835:
5807:
5806:
5793:
5790:
5767:
5718:
5712:
5698:
5647:
5625:
5622:
5551:
5540:
5478:
5456:
5453:
5400:
5389:
4622:{\displaystyle S}
4496:
4457:
4408:
4363:
4234:relation between
4147:{\displaystyle G}
3961:
3925:
3881:
3839:
3789:
3741:
2602:
2563:
2474:
2365:
2322:
2059:
2023:
1997:
1461:
1450:
1408:
1386:
1383:
1326:
1287:
1238:
1193:
894:{\displaystyle W}
795:comment added by
497:
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6344:
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5908:
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5904:
5891:
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5888:
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5851:
5837:
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5808:
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5714:
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5552:
5546:
5541:
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5524:
5523:
5489:
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5479:
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5457:
5455:
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5439:
5433:
5428:
5427:
5411:
5409:
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5349:
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5346:
5331:
5330:
5296:
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5288:
5286:
5285:
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5272:
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5242:
5224:
5222:
5221:
5216:
5211:
5210:
5183:
5182:
5157:
5155:
5154:
5149:
5147:
5146:
5128:
5127:
5108:
5106:
5105:
5100:
5098:
5097:
5079:
5077:
5076:
5071:
5069:
5068:
5050:
5049:
5026:
5024:
5023:
5018:
5016:
5015:
4992:
4990:
4989:
4984:
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4981:
4959:
4957:
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4930:
4928:
4927:
4922:
4920:
4919:
4903:
4901:
4900:
4895:
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4892:
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4868:
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4865:
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4846:
4841:
4839:
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4819:
4814:
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4811:
4760:
4758:
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4749:
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4680:
4679:
4674:
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4671:
4650:
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4628:
4626:
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4608:
4606:
4605:
4600:
4598:
4597:
4507:
4505:
4504:
4499:
4497:
4495:
4494:
4479:
4459:
4458:
4455:
4441:
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4438:
4433:
4425:
4424:
4409:
4404:
4403:
4382:
4365:
4364:
4361:
4327:
4325:
4324:
4319:
4317:
4316:
4287:
4285:
4284:
4279:
4277:
4276:
4260:
4258:
4257:
4252:
4250:
4249:
4216:unproblematic. —
4192:
4169:
4153:
4151:
4150:
4145:
4133:
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4130:
4125:
4120:
4119:
4091:
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4051:
4050:
4026:
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4009:
4007:
4006:
4001:
3999:
3998:
3982:
3980:
3979:
3974:
3963:
3962:
3959:
3946:
3944:
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3938:
3927:
3926:
3923:
3910:
3892:
3890:
3889:
3884:
3882:
3877:
3876:
3855:
3841:
3840:
3837:
3823:
3821:
3820:
3815:
3807:
3806:
3801:
3797:
3790:
3785:
3784:
3763:
3743:
3742:
3739:
3619:would help, see
3574:
3572:
3571:
3566:
3541:
3539:
3538:
3533:
3531:
3530:
3514:
3512:
3511:
3506:
3437:
3435:
3434:
3429:
3427:
3426:
3410:
3408:
3407:
3402:
3388:
3386:
3385:
3380:
3368:
3366:
3365:
3360:
3358:
3357:
3341:
3339:
3338:
3333:
3331:
3330:
3314:
3312:
3311:
3306:
3294:
3292:
3291:
3286:
3284:
3283:
3249:
3247:
3246:
3241:
3239:
3238:
3213:
3211:
3210:
3205:
3200:
3195:
3194:
3185:
3177:
3172:
3171:
3162:
3138:
3136:
3135:
3130:
3112:
3110:
3109:
3104:
3102:
3101:
2890:
2888:
2887:
2882:
2880:
2879:
2861:
2860:
2844:
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2841:
2836:
2831:
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2806:
2805:
2786:
2784:
2783:
2778:
2773:
2772:
2748:
2747:
2729:
2727:
2726:
2721:
2719:
2718:
2706:
2705:
2685:
2683:
2682:
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2650:
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2647:
2613:
2611:
2610:
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2603:
2598:
2584:
2575:
2573:
2572:
2567:
2565:
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2561:
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2482:
2477:
2475:
2470:
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2460:
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2378:
2367:
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2324:
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2320:
2268:
2266:
2265:
2260:
2258:
2257:
2241:
2239:
2238:
2233:
2228:
2223:
2222:
2206:
2204:
2203:
2198:
2196:
2195:
2141:
2139:
2138:
2133:
2131:
2130:
2110:
2108:
2107:
2102:
2100:
2099:
2080:
2078:
2077:
2072:
2061:
2060:
2057:
2044:
2042:
2041:
2036:
2025:
2024:
2021:
2008:
2006:
2005:
2000:
1998:
1993:
1992:
1991:
1975:
1932:Cheeger_constant
1862:
1860:
1859:
1854:
1852:
1851:
1835:
1833:
1832:
1827:
1825:
1824:
1806:
1805:
1789:
1787:
1786:
1781:
1779:
1778:
1762:
1760:
1759:
1754:
1736:
1734:
1733:
1728:
1690:
1688:
1687:
1682:
1680:
1679:
1661:
1659:
1658:
1653:
1651:
1650:
1634:
1632:
1631:
1626:
1624:
1623:
1601:
1599:
1598:
1593:
1591:
1590:
1572:
1571:
1555:
1553:
1552:
1547:
1545:
1544:
1520:
1518:
1517:
1512:
1510:
1509:
1491:
1490:
1472:
1470:
1469:
1464:
1462:
1456:
1451:
1448:
1442:
1437:
1436:
1419:
1417:
1416:
1411:
1409:
1404:
1403:
1402:
1387:
1385:
1384:
1381:
1369:
1363:
1358:
1357:
1337:
1335:
1334:
1329:
1327:
1325:
1324:
1309:
1289:
1288:
1285:
1271:
1269:
1268:
1263:
1255:
1254:
1239:
1234:
1233:
1212:
1195:
1194:
1191:
1037:
1035:
1034:
1029:
1002:
1000:
999:
994:
992:
984:
973:
956:
944:
942:
941:
936:
931:
920:
912:
900:
898:
897:
892:
880:
878:
877:
872:
807:
730:
728:
727:
722:
720:
719:
703:
701:
700:
695:
690:
685:
684:
675:
667:
666:
640:
638:
637:
632:
630:
629:
605:Vastinnocentaims
595:
593:
592:
587:
585:
577:
572:
561:
537:
535:
534:
529:
524:
513:Is the notation
508:Charles Matthews
465:
464:
461:
458:
455:
434:
429:
428:
418:
411:
410:
405:
397:
390:
366:
360:
235:Computer science
164:Article requests
153:
146:
145:
133:
107:
106:
103:
100:
97:
96:Computer science
87:Computer science
76:
69:
68:
63:
59:Computer science
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31:
25:
24:
16:
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6291:
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6099:
6043:
6016:
6011:
6010:
5997:
5962:
5941:
5899:
5894:
5893:
5867:
5828:
5823:
5822:
5783:
5774:
5754:
5732:
5731:
5705:
5685:
5663:
5662:
5633:
5615:
5603:
5585:
5565:
5564:
5515:
5495:
5494:
5464:
5446:
5434:
5419:
5414:
5413:
5367:
5362:
5361:
5338:
5322:
5302:
5301:
5277:
5264:
5259:
5258:
5234:
5229:
5228:
5202:
5174:
5169:
5168:
5138:
5119:
5114:
5113:
5089:
5084:
5083:
5060:
5041:
5036:
5035:
5007:
5002:
5001:
4973:
4968:
4967:
4942:
4941:
4937:
4911:
4906:
4905:
4884:
4879:
4878:
4857:
4852:
4851:
4830:
4825:
4824:
4803:
4798:
4797:
4784:
4741:
4736:
4735:
4717:
4710:
4703:
4663:
4636:
4631:
4630:
4611:
4610:
4589:
4578:
4577:
4486:
4450:
4445:
4444:
4416:
4395:
4356:
4351:
4350:
4308:
4303:
4302:
4268:
4263:
4262:
4241:
4236:
4235:
4186:
4163:
4136:
4135:
4111:
4094:
4093:
4072:
4067:
4066:
4042:
4017:
4012:
4011:
3990:
3985:
3984:
3954:
3949:
3948:
3918:
3913:
3912:
3911:by solving for
3904:
3868:
3832:
3827:
3826:
3776:
3761:
3757:
3756:
3734:
3729:
3728:
3705:
3611:
3607:
3557:
3556:
3522:
3517:
3516:
3497:
3496:
3418:
3413:
3412:
3393:
3392:
3371:
3370:
3349:
3344:
3343:
3322:
3317:
3316:
3297:
3296:
3275:
3270:
3269:
3230:
3219:
3218:
3186:
3163:
3141:
3140:
3115:
3114:
3093:
3082:
3081:
3078:
2871:
2852:
2847:
2846:
2822:
2797:
2792:
2791:
2764:
2739:
2734:
2733:
2710:
2697:
2692:
2691:
2666:
2661:
2660:
2639:
2634:
2633:
2585:
2578:
2577:
2556:
2551:
2550:
2461:
2454:
2453:
2432:
2421:
2420:
2358:
2315:
2310:
2309:
2306:
2296:
2289:
2249:
2244:
2243:
2214:
2209:
2208:
2187:
2176:
2175:
2122:
2117:
2116:
2113:second-smallest
2091:
2086:
2085:
2052:
2047:
2046:
2016:
2011:
2010:
1983:
1976:
1954:
1953:
1920:
1913:
1843:
1838:
1837:
1816:
1797:
1792:
1791:
1770:
1765:
1764:
1739:
1738:
1719:
1718:
1671:
1666:
1665:
1642:
1637:
1636:
1615:
1604:
1603:
1582:
1563:
1558:
1557:
1536:
1531:
1530:
1501:
1482:
1477:
1476:
1428:
1423:
1422:
1394:
1376:
1364:
1349:
1344:
1343:
1316:
1280:
1275:
1274:
1246:
1225:
1186:
1181:
1180:
1177:Knowledge says:
1142:
1134:
1123:
1115:
1108:
1080:
1060:107.159.247.194
1005:
1004:
947:
946:
903:
902:
883:
882:
863:
862:
813:
790:
787:
762:
759:
744:
711:
706:
705:
676:
658:
647:
646:
621:
616:
615:
540:
539:
515:
514:
502:
500:Various remarks
462:
459:
456:
453:
452:
430:
423:
403:
372:
369:
364:
358:
346:Project-related
341:
322:
303:
277:
258:
239:
220:
201:
177:
104:
101:
98:
95:
94:
61:
32:on Knowledge's
29:
12:
11:
5:
6350:
6348:
6340:
6339:
6334:
6329:
6324:
6319:
6309:
6308:
6290:
6287:
6267:
6264:
6263:
6262:
6261:
6260:
6259:
6258:
6257:
6256:
6255:
6254:
6239:
6236:
6232:
6228:
6196:
6193:
6192:
6191:
6188:
6185:
6182:
6179:
6173:
6169:
6157:
6104:David Eppstein
6098:
6095:
6094:
6093:
6092:
6091:
6077:
6060:
6059:
6042:
6039:
6038:
6037:
6023:
6019:
6007:
6004:
6001:
5996:
5993:
5991:
5989:
5988:
5977:
5974:
5969:
5965:
5961:
5958:
5953:
5948:
5944:
5940:
5937:
5934:
5931:
5928:
5923:
5920:
5917:
5914:
5911:
5902:
5879:
5874:
5870:
5866:
5863:
5860:
5857:
5854:
5849:
5846:
5843:
5840:
5831:
5821:(15) Finally,
5819:
5805:
5800:
5797:
5786:
5782:
5779:
5771:
5766:
5761:
5757:
5753:
5750:
5747:
5744:
5741:
5717:
5708:
5702:
5697:
5692:
5688:
5684:
5681:
5678:
5675:
5672:
5659:
5646:
5640:
5636:
5632:
5629:
5618:
5614:
5611:
5606:
5600:
5597:
5592:
5588:
5584:
5581:
5578:
5575:
5572:
5550:
5545:
5536:
5530:
5527:
5522:
5518:
5514:
5511:
5508:
5505:
5502:
5491:
5477:
5471:
5467:
5463:
5460:
5449:
5445:
5442:
5437:
5431:
5426:
5422:
5399:
5394:
5385:
5379:
5374:
5370:
5358:
5345:
5341:
5337:
5334:
5329:
5325:
5321:
5318:
5315:
5312:
5309:
5298:
5284:
5280:
5276:
5271:
5267:
5255:
5241:
5237:
5225:
5214:
5209:
5205:
5201:
5198:
5195:
5192:
5189:
5186:
5181:
5177:
5164:
5163:
5159:
5158:
5145:
5141:
5137:
5134:
5131:
5126:
5122:
5110:
5096:
5092:
5080:
5067:
5063:
5059:
5056:
5053:
5048:
5044:
5032:
5014:
5010:
4998:
4980:
4976:
4964:
4961:
4949:
4936:
4933:
4918:
4914:
4891:
4887:
4864:
4860:
4837:
4833:
4810:
4806:
4783:
4780:
4779:
4778:
4777:
4776:
4775:
4774:
4748:
4744:
4732:
4731:
4730:
4715:
4708:
4701:
4670:
4666:
4662:
4659:
4656:
4653:
4649:
4643:
4639:
4618:
4596:
4592:
4588:
4585:
4574:
4573:
4572:
4528:
4525:
4522:
4513:
4510:
4509:
4508:
4493:
4489:
4485:
4482:
4477:
4474:
4471:
4468:
4465:
4462:
4453:
4442:
4431:
4428:
4423:
4419:
4415:
4412:
4407:
4402:
4398:
4394:
4391:
4388:
4385:
4380:
4377:
4374:
4371:
4368:
4359:
4345:wrong formulas
4337:
4315:
4311:
4299:
4292:
4289:
4275:
4271:
4248:
4244:
4231:
4218:David Eppstein
4156:
4155:
4143:
4123:
4118:
4114:
4110:
4107:
4104:
4101:
4079:
4075:
4054:
4049:
4045:
4041:
4038:
4035:
4032:
4029:
4024:
4020:
3997:
3993:
3972:
3969:
3966:
3957:
3936:
3933:
3930:
3921:
3894:
3893:
3880:
3875:
3871:
3867:
3864:
3861:
3858:
3853:
3850:
3847:
3844:
3835:
3824:
3813:
3810:
3805:
3800:
3796:
3793:
3788:
3783:
3779:
3775:
3772:
3769:
3766:
3760:
3755:
3752:
3749:
3746:
3737:
3704:
3701:
3700:
3699:
3698:
3697:
3683:
3651:
3650:
3649:
3648:
3647:
3646:
3645:
3644:
3643:
3642:
3628:
3624:
3613:
3609:
3605:
3564:
3529:
3525:
3504:
3479:
3476:
3465:
3447:
3446:
3445:
3442:
3439:
3425:
3421:
3400:
3378:
3356:
3352:
3329:
3325:
3304:
3282:
3278:
3237:
3233:
3229:
3226:
3203:
3199:
3193:
3189:
3184:
3180:
3176:
3170:
3166:
3161:
3157:
3154:
3151:
3148:
3128:
3125:
3122:
3100:
3096:
3092:
3089:
3077:
3074:
3073:
3072:
3071:
3070:
3056:
3053:
3049:
3042:
3038:
3034:
3031:
3030:
3029:
3026:
3019:
2979:
2978:
2977:
2976:
2975:
2974:
2973:
2972:
2878:
2874:
2870:
2867:
2864:
2859:
2855:
2834:
2829:
2825:
2821:
2818:
2815:
2812:
2809:
2804:
2800:
2788:
2787:
2776:
2771:
2767:
2763:
2760:
2757:
2754:
2751:
2746:
2742:
2730:
2717:
2713:
2709:
2704:
2700:
2673:
2669:
2646:
2642:
2630:
2629:
2615:
2601:
2597:
2594:
2591:
2588:
2559:
2547:
2546:
2545:
2544:
2543:
2542:
2541:
2495:
2491:
2473:
2468:
2464:
2452:(ours) equals
2439:
2435:
2431:
2428:
2417:
2414:
2407:
2389:
2388:
2376:
2373:
2370:
2361:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2318:
2304:
2299:
2294:
2287:
2277:
2270:
2256:
2252:
2231:
2227:
2221:
2217:
2194:
2190:
2186:
2183:
2172:
2161:
2160:
2159:
2158:
2129:
2125:
2098:
2094:
2082:
2070:
2067:
2064:
2055:
2034:
2031:
2028:
2019:
1996:
1990:
1986:
1982:
1979:
1973:
1970:
1967:
1964:
1961:
1947:
1946:
1918:
1911:
1895:
1894:
1893:
1892:
1891:
1890:
1877:
1876:
1875:
1868:
1864:
1850:
1846:
1823:
1819:
1815:
1812:
1809:
1804:
1800:
1777:
1773:
1752:
1749:
1746:
1726:
1699:
1692:
1678:
1674:
1649:
1645:
1622:
1618:
1614:
1611:
1589:
1585:
1581:
1578:
1575:
1570:
1566:
1543:
1539:
1525:
1522:
1508:
1504:
1500:
1497:
1494:
1489:
1485:
1473:
1460:
1455:
1446:
1440:
1435:
1431:
1420:
1407:
1401:
1397:
1393:
1390:
1379:
1375:
1372:
1367:
1361:
1356:
1352:
1341:
1338:
1323:
1319:
1315:
1312:
1307:
1304:
1301:
1298:
1295:
1292:
1283:
1272:
1261:
1258:
1253:
1249:
1245:
1242:
1237:
1232:
1228:
1224:
1221:
1218:
1215:
1210:
1207:
1204:
1201:
1198:
1189:
1178:
1140:
1132:
1121:
1113:
1107:
1104:
1079:
1076:
1075:
1074:
1073:
1072:
1071:
1070:
1042:
1039:
1027:
1024:
1021:
1018:
1015:
1012:
991:
987:
983:
979:
976:
972:
968:
965:
962:
959:
955:
934:
930:
926:
923:
919:
915:
911:
890:
870:
859:graph expander
852:
849:
812:
809:
786:
783:
781:
760:
758:
755:
743:
740:
739:
738:
718:
714:
693:
689:
683:
679:
674:
670:
665:
661:
657:
654:
628:
624:
584:
580:
576:
571:
567:
564:
560:
556:
553:
550:
547:
527:
523:
501:
498:
495:
494:
491:
490:
487:
486:
475:
469:
468:
466:
449:the discussion
436:
435:
419:
407:
406:
398:
386:
385:
382:
381:
378:
377:
374:
373:
371:
370:
368:
367:
350:
342:
340:
339:
333:
323:
321:
320:
314:
304:
302:
301:
296:
288:
278:
276:
275:
269:
259:
257:
256:
250:
240:
238:
237:
231:
221:
219:
218:
212:
202:
200:
199:
194:
188:
178:
176:
175:
169:
157:
155:
154:
142:
141:
129:
128:
121:Mid-importance
117:
111:
110:
108:
91:the discussion
77:
65:
64:
62:Mid‑importance
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
6349:
6338:
6335:
6333:
6330:
6328:
6325:
6323:
6320:
6318:
6315:
6314:
6312:
6305:
6304:
6300:
6296:
6288:
6286:
6285:
6281:
6277:
6273:
6265:
6253:
6249:
6245:
6240:
6237:
6233:
6229:
6226:
6225:
6224:
6220:
6216:
6215:Joel B. Lewis
6211:
6210:
6209:
6205:
6201:
6197:
6194:
6189:
6186:
6183:
6180:
6177:
6176:
6174:
6170:
6167:
6162:
6158:
6155:
6150:
6149:
6148:
6144:
6140:
6136:
6131:
6130:
6129:
6125:
6121:
6116:
6115:
6114:
6113:
6109:
6105:
6096:
6090:
6086:
6082:
6078:
6076:
6072:
6068:
6064:
6063:
6062:
6061:
6058:
6054:
6050:
6045:
6044:
6040:
6017:
6008:
6005:
6002:
5999:
5998:
5994:
5992:
5975:
5972:
5967:
5959:
5956:
5946:
5942:
5938:
5935:
5929:
5926:
5918:
5912:
5900:
5872:
5868:
5864:
5861:
5855:
5852:
5847:
5841:
5829:
5820:
5803:
5798:
5795:
5784:
5780:
5777:
5769:
5759:
5755:
5751:
5748:
5742:
5739:
5715:
5706:
5700:
5690:
5686:
5682:
5679:
5673:
5670:
5660:
5644:
5638:
5630:
5627:
5616:
5612:
5609:
5598:
5590:
5586:
5582:
5579:
5573:
5570:
5548:
5543:
5534:
5528:
5520:
5516:
5512:
5509:
5503:
5500:
5492:
5475:
5469:
5461:
5458:
5447:
5443:
5440:
5429:
5420:
5397:
5392:
5383:
5377:
5368:
5359:
5339:
5335:
5327:
5323:
5319:
5316:
5310:
5307:
5299:
5278:
5274:
5269:
5265:
5256:
5235:
5226:
5207:
5203:
5199:
5196:
5190:
5187:
5184:
5179:
5175:
5166:
5165:
5161:
5160:
5143:
5139:
5135:
5132:
5129:
5124:
5120:
5111:
5094:
5090:
5081:
5065:
5061:
5057:
5054:
5051:
5046:
5042:
5033:
5030:
5012:
5008:
4999:
4996:
4993:be twice the
4978:
4974:
4965:
4962:
4947:
4939:
4938:
4934:
4932:
4916:
4912:
4889:
4885:
4862:
4858:
4835:
4831:
4808:
4804:
4794:
4792:
4787:
4782:Ylloh's proof
4773:
4769:
4765:
4746:
4742:
4733:
4729:
4725:
4721:
4714:
4707:
4700:
4696:
4695:
4694:
4690:
4686:
4668:
4664:
4660:
4657:
4654:
4651:
4647:
4641:
4637:
4616:
4594:
4590:
4586:
4583:
4575:
4571:
4567:
4563:
4562:Joel B. Lewis
4558:
4557:
4556:
4552:
4548:
4543:
4542:
4541:
4537:
4533:
4529:
4526:
4523:
4519:
4514:
4511:
4491:
4487:
4483:
4480:
4475:
4472:
4469:
4463:
4451:
4443:
4429:
4426:
4421:
4413:
4410:
4400:
4396:
4392:
4389:
4383:
4375:
4369:
4357:
4349:
4348:
4346:
4342:
4338:
4335:
4331:
4313:
4309:
4300:
4297:
4293:
4290:
4273:
4269:
4242:
4232:
4229:
4228:
4227:
4223:
4219:
4214:
4210:
4209:
4208:
4207:
4203:
4199:
4194:
4190:
4185:
4181:
4176:
4171:
4167:
4162:
4141:
4116:
4112:
4108:
4105:
4099:
4077:
4073:
4047:
4043:
4039:
4036:
4030:
4027:
4018:
3991:
3967:
3955:
3931:
3919:
3908:
3903:
3899:
3898:
3897:
3873:
3869:
3865:
3862:
3856:
3851:
3845:
3833:
3825:
3811:
3808:
3803:
3798:
3794:
3791:
3781:
3777:
3773:
3770:
3764:
3758:
3753:
3747:
3735:
3727:
3726:
3725:
3723:
3719:
3716:
3713:
3709:
3702:
3696:
3692:
3688:
3684:
3681:
3680:
3679:
3675:
3671:
3667:
3666:
3665:
3664:
3660:
3656:
3641:
3637:
3633:
3629:
3625:
3622:
3618:
3614:
3603:
3599:
3595:
3594:
3593:
3589:
3585:
3581:
3577:
3562:
3555:
3554:
3553:
3549:
3545:
3527:
3523:
3502:
3494:
3493:
3492:
3488:
3484:
3480:
3477:
3474:
3470:
3466:
3462:
3461:
3460:
3456:
3452:
3448:
3443:
3440:
3423:
3419:
3398:
3391:
3390:
3376:
3354:
3350:
3327:
3323:
3302:
3280:
3276:
3267:
3266:
3265:
3264:
3260:
3256:
3251:
3235:
3231:
3227:
3224:
3215:
3191:
3187:
3178:
3168:
3164:
3149:
3146:
3126:
3123:
3120:
3098:
3094:
3090:
3087:
3075:
3069:
3065:
3061:
3057:
3054:
3050:
3047:
3043:
3039:
3035:
3032:
3027:
3023:
3022:
3020:
3017:
3016:
3015:
3011:
3007:
3002:
3001:
3000:
2999:
2995:
2991:
2986:
2982:
2971:
2967:
2963:
2959:
2955:
2951:
2950:
2949:
2945:
2941:
2937:
2936:
2935:
2931:
2927:
2923:
2919:
2918:
2917:
2913:
2909:
2905:
2904:
2903:
2902:
2898:
2894:
2876:
2872:
2868:
2865:
2862:
2853:
2827:
2823:
2819:
2816:
2810:
2807:
2798:
2769:
2765:
2761:
2758:
2752:
2749:
2744:
2740:
2731:
2715:
2711:
2707:
2698:
2689:
2688:
2687:
2671:
2667:
2644:
2640:
2628:
2624:
2620:
2616:
2599:
2592:
2586:
2557:
2548:
2540:
2536:
2532:
2527:
2526:
2525:
2521:
2517:
2513:
2512:
2511:
2507:
2503:
2499:
2498:
2496:
2492:
2489:
2471:
2466:
2462:
2437:
2433:
2429:
2426:
2418:
2415:
2412:
2408:
2405:
2404:
2403:
2402:
2398:
2394:
2371:
2359:
2355:
2352:
2349:
2343:
2337:
2334:
2328:
2316:
2307:
2300:
2297:
2290:
2283:
2278:
2275:
2271:
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1984:
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1977:
1971:
1965:
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1881:
1878:
1873:
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1865:
1848:
1844:
1836:(where I use
1821:
1817:
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1103:
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1069:
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784:
782:
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774:MotherFunctor
771:
767:
756:
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712:
681:
677:
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2292:
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769:
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642:
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512:
506:
503:
479:Mid-priority
478:
438:
404:Mid‑priority
345:
344:
328:Unreferenced
326:
325:
307:
306:
281:
280:
262:
261:
243:
242:
224:
223:
205:
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181:
180:
162:
161:
120:
80:
40:WikiProjects
6244:MathsPoetry
6235:principles.
6200:MathsPoetry
6181:no examples
6154:for experts
6120:MathsPoetry
6067:MathsPoetry
5661:(14) Hence
5297:( page 156)
4960:( page 153)
4764:MathsPoetry
4762:fragile. --
4532:MathsPoetry
4518:immediately
4092:instead of
3708:MathsPoetry
3687:MathsPoetry
3655:MathsPoetry
3632:MathsPoetry
3544:MathsPoetry
3483:MathsPoetry
3255:MathsPoetry
3060:MathsPoetry
2990:MathsPoetry
2962:MathsPoetry
2926:MathsPoetry
2893:MathsPoetry
2632:Let's call
2619:MathsPoetry
2516:MathsPoetry
2308:because of
2148:MathsPoetry
1906:lower bound
1703:MathsPoetry
1150:MathsPoetry
1093:MathsPoetry
1046:MathsPoetry
836:JackSchmidt
820:MathsPoetry
811:Definition?
791:—Preceding
753:Claude-Guy
614:eigenvalue
454:Mathematics
445:mathematics
401:Mathematics
6311:Categories
6266:definition
6097:Discussion
4180:this paper
2274:this paper
4521:document.
3722:this edit
3627:graphs...
3604:a small λ
643:magnitude
216:Computing
5227:(9) Let
5082:(6) Let
5000:(4) Let
4966:(3) Let
4196:Thanks!
3718:contribs
3617:examples
3481:Best, --
3139:, where
3041:article.
2494:through.
1904:I meant
1867:needed).
1044:Best, --
793:unsigned
733:Blokhead
264:Maintain
207:Copyedit
5027:be the
4330:nonzero
3602:implies
2956:. This
2111:is the
1701:Best --
848:Agreed.
645:, i.e,
481:on the
245:Infobox
183:Cleanup
123:on the
30:C-class
6231:maths.
6049:Tkuvho
4877:, and
4341:simple
4294:It is
2958:proves
226:Expand
36:scale.
6276:Sasha
6166:found
6139:ylloh
6081:ylloh
5360:(12)
5257:(10)
4935:Proof
4720:Sasha
4685:ylloh
4547:Sasha
4296:wrong
4198:ylloh
3670:ylloh
3584:ylloh
3451:ylloh
3315:with
3046:wrong
3006:ylloh
2954:wrong
2940:ylloh
2908:ylloh
2531:ylloh
2529:them.
2502:ylloh
2488:wrong
2393:ylloh
1936:ylloh
1880:ylloh
1696:lower
1635:when
1165:ylloh
861:by a
857:is a
598:Gauge
309:Stubs
283:Photo
140:with:
6299:talk
6280:talk
6248:talk
6219:talk
6204:talk
6143:talk
6124:talk
6108:talk
6085:talk
6071:talk
6053:talk
5892:and
5730:and
5563:and
5412:and
4931:...
4768:talk
4724:talk
4711:and
4689:talk
4566:talk
4551:talk
4536:talk
4261:and
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4202:talk
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4166:help
3947:and
3907:help
3712:talk
3691:talk
3674:talk
3659:talk
3636:talk
3588:talk
3548:talk
3487:talk
3455:talk
3411:vs.
3259:talk
3064:talk
3010:talk
2994:talk
2966:talk
2944:talk
2930:talk
2912:talk
2897:talk
2623:talk
2535:talk
2520:talk
2506:talk
2397:talk
2291:and
2282:Alon
2152:talk
2045:and
1940:talk
1924:h(G)
1915:and
1897:See
1884:talk
1707:talk
1169:talk
1154:talk
1137:and
1118:and
1097:talk
1064:talk
1050:talk
840:talk
824:talk
801:talk
785:SL=L
746:Hi,
5905:out
5789:out
5621:out
5452:out
4362:out
3924:out
3740:out
3598:not
3469:one
3369:to
3214:."
3153:max
2562:out
2419:if
2411:not
2364:out
2321:out
2305:out
2295:out
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1192:out
1143:(G)
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945:,
653:max
473:Mid
115:Mid
6313::
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365:}}
359:{{
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210::
186::
167::
127:.
42::
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