Talk:Graph (discrete mathematics)

3944:: we should work to keep our articles as readable to as general an audience as possible, and that means keeping the definitions simple, clear, and focused on the most important properties of the object in question. For instance (using a cut-based definiition of connectivity) we could define a path as a connected subgraph with no multiple adjacencieis in which, for every triple of distinct vertices, one of the three vertices disconnects the other two (puts them on opposite sides of a cut), but I think that would be unnecessarily indirect and confusing. For the same reason, I think defining subgraphs by their vertex degrees is also unnecessarily indirect and confusing (not to mention that it leads to messy special cases since it is usually a good idea to allow one vertex and no edges to count as a path). The definition I prefer is an alternating sequence of vertices and edges, starting and ending at a vertex, with an incidence between each consecutive vertex-edge pair in the sequence. (You may prefer to call this a walk, and add a condition that there be no repeated vertex in the sequence). We need both vertices and edges in the sequence: vertices because the length-zero paths have no edges, and edges to cleanly handle multigraphs. — 1641:

article itself, it discusses graphs in general, including various models of them (simple graphs, multigraphs, directed graphs etc). That is perfectly appropriate and in that context it is useful to have a general definition which covers, as subcases, several different models. The substance of my previous comment was simply to make the definitions mathematically correct. If you want to have a separate article about simple graphs, I'd be fine with that even if it introduces a bit of duplication. I am not particularly interested in philosophical debates about terminology. I will say, however, that graphs with loop-edges and with multiple edges are very useful and natural objects in, for example, topology and group theory. (In fact, I think that many topologist define a graph, at least for themselves, as a one-dimensional

1941:. But this article is of rather general nature, where the main issues are those of style, presentation and organization, not of some deeply technical knowledge. As I said, if the article starts with a lead section discussing the concept of a graph informally and in some generality covering most basic graph models used in mathematics, followed by a more detailed list list of such models with more formal definitions, I'll be quite satisfied. There may be good reasons (such as tradition and perhaps simplicity of definition) to start with simple graphs and make something of an emphasis on them, I don't have strong feelings here. In fact, I have to say that I don't have a particularly strong interest in this article. I only commented here back in May because at the time the presentation was mathematically inconsistent. 4601:. I can assure you this was not my intention, and I am not linked to any of these references whatsoever. I guess I am simply used to using much more references because I have an academic background, and the references I added are classical references used in graph theory. Would you consider reverting these modifications? I feel that they made a useful addition to the article because the link between graph and network is barely evoked even though these two concepts are practically equivalent. Similarly, I feel like mentioning random graphs could be very useful, as they are an important subfield of graph theory. I am sorry if that violated the best practices on Knowledge: as you can see, I am fairly new to editing and any advice on how to avoid this king of misunderstanding would be very welcome! 4532:: I would like to discuss this. Thank you for giving the reason; without that, it's impossible to know what to think (that's why I reverted your first reversion and probably why you made that first reversion). You wrote, '"graph" is a countable noun, the plural construction is much more natural (and certainly grammatically correct)'. I disagree with the conclusion. The plural should appear in "types", not (necessarily) "graph". It is good standard English to not pluralize the object of the preposition in this context. The issue is not that "graph" is "countable" or that "graphs" is ungrammatical; it is that the plural belongs on "type" and is unnecessary on "graph", and it is good style not to pluralize twice. 102: 92: 64: 31: 3871:

fact that the definition added was wrong repeatedly). Instead, I prefer the previous definition, which explicitly gives the structure in vertices and edges. I would like to invite other editors to give their opinions. (Of course, both definitions offered are equivalent: they define the finite paths (or cycles). The question is, which one is better for a broad article of this sort?) I am also open to constructive suggestions for improving the wording of the structural definition. --

2930:

connectivity are even worse because not all graphs are connected. I agree there is a problem (the ambiguity between node-link graphs and function domain-range graphs) but I don't think these solutions have been adequately thought through. We should have a proper discussion before any vote on whether we like the new name, not voting first (with random interspersed alternative names confusing the vote). My own preference would be to replace this article with a proper article on

4565:

the normal way to ask a dealership whether they have sedans and hatchbacks and two-seaters is to ask what "types of cars" are available; "types of car" sounds stilted. If you had some concrete evidence that in some other variety of English, this is an accepted rule, I suppose we could discuss what dialect the article is written in. (Maybe "type of" / "kind of" / "sort of" are different from other superficially similar constructions?) --

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article we will draw flak from folks who will say we have made it too complicated for the general reader. Your {V,E,A}-triple approach is interesting - the assignment function A would assign each edge in E to an unordered pair of vertices, right ? But it is heading still further into abstract territory, which some editors may have objections to. Anyway, I'll leave further improvements up to you.

2126: 4639:. Wedging in the application of graph theory to network science doesn't really work, from that perspective. Moreover, the Albert–Barabási review article is from 2002, pretty much the height of the "scale-free networks" hype era and before the people in that field were adequately scrupulous about things like testing whether a straight-ish line on a log-log plot really is a power law. 2768:, because I think that graphs are heavily used throughout this large area of mathematics, not only in graph theory. This title also makes it clear that the concept belongs to mathematics, and only to some specific part of it. So the user is forced to check whether the concept, that he is thinking about, matches the article. Discrete functions, like sequences, belong rather to 22: 3249:: Professionally I am considered a "discrete mathematician", but I have no idea of what that actually means. However, as the main function of this disambiguator is to differentiate this topic from the graphs of functions, it does the job in the least objectionable way and the general reader doesn't have to know what this "field" is in order to get that point. 168: 4261:) makes the definitions harder to read, as they do not stand out. If the definitions were less subtle it would be fine, but here this is not the case (simple graphs, multigraphs, simple graphs permitting loops, multiple graphs permitting loops, and likewise for directed graphs). So I think we should keep the bulleted version for the reader. 78: 3816:: Highly confusing title: The aim of the term between parentheses is to disambiguate. Not only "connection" does not disambiguate anything, but it is so ambiguous that even an experimented mathematician (as I am) needs explanations or context for imagining that "Graph (connections)" refers to graphs of graph theory. 861:

maintain when there gets to be more than a few of them, they lead to information loss when the source data is too complex for the Ref format, and all in all they make the article look very amateurish and antiquated, as footnotes for citations were already being phased out in the late 60's, just from what I recall.

665:. Another reason for the broader definition is to define a broader notion of graph homomorphism (as you have already noticed). But this is just my opinion. At the moment I am a bit busy so I am unable to work on either article. If you think you can make the definitions more consistent or clearer go ahead. 4564:

You are asserting a general principle which is not valid in my dialect of English (which is odd because you and I grew up in the same place, albeit a few decades apart). For an uncountable noun I would certainly not pluralize ("types of milk" means "whole, or two percent, or skim, or goat milk") but

4292:

By the way, your edits are difficult to review because, firstly you do not provide edit summaries that explain the reasons of your change, secondly because you mix non-controversial minor edits (such fixing grammar and typos) with potentially controversial edits). So, please, edit only one section at

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for the vertices, joined by lines or curves for the edges." I would be more inclined to challenge this/ask for a reliable source if (1) there were any discussion of this point in the body of the article, rather than in the lead, and (2) if that discussion had any reliable sourcing. Presumably graph

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Our team has created a very effective graph and network optimization tool: an open source library written in C++ language. We think that our project is matured enough to be mentioned here in external links section. Or we could create a brand new Knowledge article for it that could be cited from here.

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You're welcome. You are correct that with the definition given in the article, a non-directed graph is not allowed to have loops. If you want to alow a non-directed graph to have loops, then you can simple drop the word "distinct" in the definition of edge given in the the article, taking the set {a,

4616:

The references you added are not good choices for the rigorous study of graph theory. The description of random graphs is wrong: there is no sense in which random graphs are antithetical to regular graphs; random regular graphs are studied all the time. It is also incorrect to say that random graphs

4122:

In addition to the widespread introduction of the nonstandard (in the context of graph theory) term "arrows" for directed edges, Maggyero's edits also widely introduced the word "link". IANAGT, but this also seems like a nonstandard usage to me. (It was mentioned in the article previously, but not

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what is a path or a cycle and why they have this name (and this is the origin of the error that I have quoted in this definition). Personally, I do not like the definition in terms of "there exists a indexing of the nodes such that", as it suggests wrongly that the verification of the definition may

2929:

for graphs that have much more to do with Euclidean geometry, group theory, and model theory than they do with combinatorics). And if "Graph (graph theory)" is good, why isn't "Graph (graph (graph theory) theory)" better? They both explain equally well what a graph is. Boris's suggestions involving

2344:

unfortunately. While there is general agreement that the current title is problematic, there is clearly no consensus as to how exactly this article should be renamed. Seeing as the discussion here has died, I'm closing this discussion because I think the chances of it reaching a consensus for any of

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I added an explicit qualifier that the vertices in an unordered pair do not have to be distinct. I'll think some more about the organization of this article, but, to be honest, I am not sure if I want to work on it seriously. As I said, I am much more of a geometer/topologist myself and I personally

1033:

I find strange that a formal definition might depend on sets of some kind of object that is never defined. What are vertices? I guess some description on what kind of object is required to be a vertex or node would add to the article. But as I understand, there is no requirement for vertices objects

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I question this: "A more fundamental difference is that, in a directed graph (or multigraph), the directions are fixed, but in an oriented graph (or multigraph), only the underlying graph is fixed, while the orientation may vary." I don't know what "may vary" means. It seems to me that an oriented

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Yes I would change the definition of graph to allow loops (my main concern is consistency too). I think it is easier having a broad definition for graphs and digraphs which includes loops and multiple edges than giving separate definitions. The broad definition can be narrowed down, if necessary, by

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I don't have particularly strong preferences about the order in which various notions are presented in the main body of the article. However, I do think that the lead section there should be a fairly general discussion, of general and not of very formal nature, about graph as a concept (nodes, arcs

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and end-point assignment function. (My personal feeling is that the insistence on a graph being an ordered pair rather than an ordered triple is somewhat misguided, although it seems to be customary.) This might require a fairly substantial rewrite of the entire article which may not be that great

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given in the article is already that of a simple graph. If edges are defined as two-element subsets of the vertex set, then you automatically exclude loop-edges and multiple edges. So what does it mean then to say that a "A simple graph is an undirected graph that has no self-loops and no more than

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There is an inane edit war going on about the definitions of path graph and cycle graph given in this article. One editor prefers a definition in terms of connectedness and degree sequence. In my opinion, this definition actively obscures the structure of the graph (and this is reinforced by the

936:

Thanks. Actually, my post was motivated by the question "how do we define a loop in not directed graph". Formally, the set {v,v} collapses to the singleton {v} and it's no longer "a pair". Well, I do not think it's a big problem here, but perhaps someone knows a standard solution so that we could

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model for the notion of a graph (a set of vertices, a set of edges, an involution on the set of edges corresponding to taking a formal inverse of an edge, plus an initial vertex function from the set of edges to the set of vertices) which also does not require the graph to be simple. But I am not

1640:

I don't particularly appreciate references to starting the "war over ownership", "religious debate" and "the food fight". If you are looking to start any of these, you should do that elsewhere, not on Knowledge. On the substance of your edit, I don't actually have a problem with it. Regarding the

1625:

In fact, on further reflection, I think I'll massage the entry to restore the simple, undirected, loopless notion as the first definition, label it as merely the most common, and then retain the current (more general) definition. I feel that this approach is likely the best compromise between my

3085:

Is "discrete structure" something that a random non-mathematically-educated Knowledge reader who gets to the disambiguation page and wants to know where to go will recognize? And isn't that reader the primary target for these disambiguators? "Discrete mathematics" at least has mathematics in it.

2258:

Please explain in the article why the word "edge" is used. If an "edge" is just a line, why isn't it called a line? If there is some subtle distinction that makes an "edge" not a line, it might be helpful to explain that. Alternately, if the meaning is identical but the term lives on for merely

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Yeah, there isn't a single way of doing this that makes everyone happy. The most common approach is to hide the issue in sloppy terminology. Usually confusion is avoided. One place where the distinction matters is in enumeration, where loops are usually regarded as adding 2 to the degree of a

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JA: My recommendation is that we continue to use the citation styles that are standard in math journals, and avoid the use of footnotes. These things make the line spacing on the face article very jagged, make the text in the edit window very hard to proofread, they become almost impossible to

3197:, because I think that graphs are heavily used throughout this large area of mathematics, not only in graph theory. This title also makes it clear that the concept belongs to mathematics. I think that confusion with discrete functions, like sequences, is unlikely, because they belong rather to 1376:

I deliberately took the word "distinct" out of the definition, so that the two vertices that define an edge can be the same, which allows for loops (in my world, an 'unordered pair' is a multiset, so {X,X} is a perfectly good unordered pair). If we load too many subsidiary definitions into the

1339:

In terms of formalism, it is better now. However, you are still excluding loop-edges in the main definition, right? I understood the phrase "unordered pairs of vertices" to mean "unordered pairs of distinct vertices", although maybe this should be mentioned explicitly. In terms of clarity, the

1262:

As a mathematician working mostly in geometry and topology, I always find the "definitions" section in graph theory textbooks rather annoying since they often avoid defining the object at the proper level of generality. The same is true here which should make anyone who is reading this article

3215:: Graph are very commonly used outside discrete mathematics (and even outside mathematics). Nevertheless, this has the advantage to clearly state that it is mathematics and that it is not about graph of functions (for almost everybody, graph of functions are graphs of continuous functions). 1034:(they need not properties other than being comparable for equality, which is always the case for math objects). So the only modification I suppose necessary is to tell this in the text: "V is a set of mathematical objects of any kind; they are called vertices (or nodes)". Am I wrong? 1703:(OK, guess I shoulda known my attempt at light-heartedness might come across wrong, so for that I apologize.) All I'm really proposing is to put the simple, loopless definition first, labeled as the most widely used,and then follow it with the other, labeled as the most general. 1206:

The articles Graph (mathematics) and Graph Theory overlap several things. Why did someone write like this? I recommend reworking both of them and combinig as a single article. In summary, we are trying to dump duplicate materials at different places. They are not well organized.

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article covers the area of mathematics called graph theory, its history, current problems and applications. I think they are separate topics, although there is bound to be some overlap between the articles. However, if you want to propose a merger, follow the instructions at

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If I were you I would leave your motherhood and apple pie comments for your family. I resent the suggestion that I disagree with the consensus policy. I certainly agree with it. Excessive quoting of wikipedia policy I think is also looked down upon in wikipedia policy.

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article and they do have a correct formal definition there. Still, there is something to be said for giving a self-contained presentation in this article. One possibility is to rewrite the main definition completely an define a graph as a triple consisting of two sets

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Is seems like someone has written the definition of a linear graph based on the wrong definition of a graph. Can anyone verify this? If no one says otherwise, or beats me to it, I'll rewrite the definition later. BTW, would a linear graph be, by definition, connected?

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since I don't think that discrete mathematics is a well defined area and I've always felt that the essential part of graph theory has been a subtopic of combinatorics. I would want to avoid getting too fancy with the disambiguators and let the disambiguation page

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a move and this made a disambiguation. And in fact 'Graph of a function' is probably the wrong name for that and it should be widened slightly as for instance in a relation the graph is not necessarily a functional one. Can't think what better name to use though.

3232:. Nobody really knows what discrete mathematics is or whether it differs from combinatorics, but graphs are in some sense discrete so this seems vague enough to cover most of the bases and yet specific enough to disambiguate these from graphs of functions. — 1682:

I think it makes perfect sense to have an article that discusses various models of graphs together, under one roof (and I think that the present article serves that purpose), and also to have a number of subarticles that are devoted to particular submodels.

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Woa, take it easy. My opinion certainly should not count more than anybody else's opinion here. People should listen to what others say and then make up their own minds and express their opinions. I am also not any kind of an expert in graph theory: I am a

1707:

does raise a good point about the advisability of breaking out the bulk of the discussion on each flavor of object into separate articles that are pointed to by the master article, which summarizes the distinctions. Is that what I hear you to support as

2074:

I'd sure like to know why we have graphs. Just starting them in discrete math class, I find them fascinating, and I can't help but imagine they have countless real world applications, but there's no discussion of this on the page...could someone help?

3833:. The purpose of a disambiguator is to allow readers who don't know the subject well to figure out which of several links with similar names they should go to. This one fails that test: it is not going to work even for readers who do know the subject. — 1072:

My understanding of quivers is that a quiver _is_ just a directed graph, and that a representation of a quiver attaches a vector space to each vertex and linear mappings to each edge. Could someone please correct me if I am wrong or otherwise comment?

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I would not call this an ambiguity. I think that you meant that a reader used to e.g. the term "vertices" may not recognize the title with "nodes" as a match. On the other hand, I would expect "nodes" and "links" to be in every commoner's vocabulary.

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Something the article should make more clear is which of the two intuitive concepts of loop it is referring to. It's one thing to have a line coming out from a node and back around directly in to the same node again. It's another to connect A-B-C-A..

1673:. I'd say the formal definition of multigraphs is simpler, which is nice for giving the initial formal definition, but for introductory examples, simple graphs are more appropriate. Loops and multiple edges can be introduced in separate examples. 2051:

I have no problem if the bold highlighting of defined terms is changed to italic highlighting. Note, however, that the original edit that I reverted had removed the highlighting of terms altogether. I oppose the complete removal of highlighting.

3738:: Readers interested in graphs do not necessarily know if their graphs are directed or not (in many texts, the distinction is only done by the context). Moreover, "undirected graph" is not proposed by our search engine when asking for "graph". 3890:

and in the litterature as well (e.g., Gross, J. T. and Yellen, J. Graph Theory and Its Applications, 2nd ed. Boca Raton, FL: CRC Press, 2006). I have never seen your definition, but you are right, they are equivalent. I suggest we use both of

1606:

Since tons of mathematicians would be in the same pew with me in this religious debate, I've adjusted the blurb acknowledging our usage. In particular, where it used to attribute the usage to "some authors," it now refers to "many authors."

1645:). From the point of view of covering theory it is completely unnatural to exclude graphs with loop-edges and multiple edges from consideration. Same in group theory when Cayley graphs and graphs of groups are considered. I personally like 4288:

If the definitions are hard to read, it is not because of the use or not of bulleted lists, be because a poor and confusing style. I'll begin to fix that, but only for one section, because this is a lot of work, and I have other things to

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Fellows, when defining connected graphs, woudn`t be useful to define what is a "path" (in opposition to "edge")? I mean, a path of length 1 is merely an edge - and that worth mentioning to clarify someone who is unfamiliar with this theme

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Is the concept of a connected graph too simple to be mentioned? Either the "Graph Types" or the "Properties of Graphs" section should mention "Connected Graph" or "a graph is called connected", if even a "finite graph" is explained. --

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any decisioin on this until we discuss this more and come up with a better choice of names. The "discrete mathematics" and "combinatorics" disambiguators are inaccurate as some graphs are far from discrete or combinatorial (see the

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graph consists of a pair (undirected graph, orientation). A given undirected graph can be oriented in different ways but these lead to different oriented graphs, not to the same oriented graph where the orientation has "varied". --

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I'm a computer science student working in a computer networks division, and I'm using graphs in everyday life. So do traffic engineers, city engineers and a whole lot of people building the infrastructure you take for granted. --

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I'm wary of definitions involving connectivity because connectivity is often defined in terms of the existence of paths (although one can instead use cuts) and we're in danger of circular reasoning. Also, we should bear in mind

3925:) seems therefore much better (its writing could certainly be improved). Therefore, I suggest to begin by a definition in terms of graph traversal, and to continue with the characterization in terms of connectedness and degree. 2026:

with an edit summary of: (rv - bold text highlights defined terms, which is necessary in this overview article). Is this true? Is this the article where bolding can be used 74 times? (Yes, I did count them.) Any thoughts?

3357:: Narrower than "discrete mathematics"; not clear that graph theory is completely included in combinatorics (topology of graphs, for example); not clear that every potentially interested user knows what is combinatorics. 2730:. I think the phrase "discrete mathematics" is less ambiguous than "graph theory", if you don't already know what graph theory is. Also, the English word "discrete" is helpful, even without knowledge of the mathematics. 1797:

Neither of us has the monopoly on setting wiki policy. What I wrote specifically is that I think "it would make sense" to defer to his opinion, based both on his expertise AND the quality of his contributions to date.

1664:

I think he was being facetious - he just wanted to start a calm discussion on the topic. The essential organization question here is whether we want this article to discuss simple graphs and to discuss multigraphs at

1462:", it may be both a metric for the network considered as a graph, as well as have other meanings. I will try to write something here, but "as a computer science student" why don't you do this yourselves as well? 3116:, but more discussion is needed before choosing the final title. Please put each proposed new title into a separate subsection to allow clean argumentation and voting. Please review the arguments (see also the 2235:

Why is there figure of something that is *distinctly not a graph*, but a pseudograph, next to the definition? Would it not be much better to have a *graph* illustrating, you know, the definition of a *graph*?

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of an idea. Another possibility is, after the main definition is stated, to give a more expanded commentary on what it actually means and how to think about a multi-set of pairs of vertices more formally.

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I am surprised why a directed graph should _not_ allow self loops. While literature may offer mutually exclusive definitions, at least the articles on Knowledge should not contradict one another. Compare

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I'm not sure why there would be confusion. A loop is defined in the article to be an edge from a vertex to itself. The other notion is a cycle, and it's not really defined on this article (surprisingly

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Everybody who gets familiar with graphs wants to use them. If our library was mentioned here, the next step towards using graphs would be presented here. The name of our open source library is LEMON.

2152:; a discussion will now take place over on Commons about whether to remove the file. This gives you an opportunity to contest the deletion, although please review Commons guidelines before doing so. 680:

The graph theory literature is divided on whether "digraph" allows loops. Whichever we give as the primary definition, the section on variations of definitions should mention the inconsistency. --

3776:: I initially thought this was a good idea, but on reconsideration it raises too many problems. Chief of which is that, to be useful, a reader would have to know enough about the subject already. 2857: 2554:

defined over a discrete domain makes sense and is widely used in discrete mathematics. Moreover, while "Graph theory" is clear for everybody, "discrete mathematics" is not. For example I see that

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The current definition of degree says that a loop is counted twice. A bit later in the page it says the maximum degree for a graph of n vertices (with loops) is n. I believe this should be n+1.

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theorists comment on this in their textbooks sometimes? Maybe it (the diagrammatic/pictorial representation of graphs) even deserves a small section in this article, if it can be supported. --

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Hmmm. It seems to me that your statements "Nsk92 is the most serious expert in the field. It would make sense to defer to his opinion" and "I certainly agree with <Knowledge:Consensus: -->

341: 4635:(which is a good place to ask questions!), I think the material you added was off-topic for an article that focuses on graph theory as pure mathematics. The opening paragraphs of an article 158: 4311:

The current definitions are pretty clear, I put a lot of work and time into it. If you don't have time, please do nothing instead of messing with the articles. Okay for the edit summaries.

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I think the 3rd bullet point of the definition of a graph should be moved into the paragraph below, it's not part of the definition of a graph, it just defines some extra terminology. --

4699: 3300:: Yes, it is circular, but one may suppose that everybody who may be interested in this article has heard of the term "graph theory" and knows that it is not about graphs of functions. 1275:

The actual general definition of a graph (where both loop-edges and multiple edges are allowed; if I remember correctly, this is called a pseudo-graph) is something like this: a triple

481: 3618:. Contradicts the more standard but more technical mathematical vocabulary of vertices and edges. Also, a node-link diagram is a visual representation of a graph, not itself a graph. — 4653:

Okay I understand your perspective much better now. Sorry for the inconvenience, I was genuinely trying to contribute to the subject and I think I was biased towards network science.

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problem with the new version is that it refers to the notion of a multi-set, which is defined in another article and with which most people are probably not familiar. I looked up the

3587:. Ambiguous, as nodes are often called vertices or points and links are often called edges, lines or arcs. One cannot guess that readers are aware of such variations of terminology. 1298:

I understand that this is a Knowledge article, not a mathematical textbook, but shouldn't the formal general definition of a pseudo-graph be contained somewhere in the article?

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Of course not all matroids correspond to graphs - that's the whole point of a generalization. The same applies to hypergraphs and any other generalization. I removed the remark.

398: 336: 2806:? Anyway, in the end, the actual title almost does not matter. The most important thing is that the user is reminded that mathematics has more than one concept called graph. 728: 3391:: Although my personal favorite, I'll concede to the above points and the fact that a large segment of readers looking for this topic would have no idea of what that meant. 3516:. In the intended usage, these terms are synonymous, but "network" carries the connotation of non-mathematical applications of graphs only, which I think we should avoid. — 1603:

Edges of cardinality not equal to 2 are perfectly respectable entities and are emminently worth studying, they merely merit (to our way of thinking) different terminology.

4729: 259: 2619:-- the latter usage is only clear to people who already know what the object in question is, who won't be making this mistake; there is no possibility of confusion about 269: 4719: 4689: 1626:

wish to avoid confusing the typical reader ("Whoa, that's not the way I've seen it!") and yours to avoid allowing too narrow a conception to set in. Any objections?—

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The characterizations in terms of connectedness and degrees are the best for verifying that a (sub)graph is a path or a cycle. However they are of little help for

3374:. Too restrictive. There are too many other uses of graphs in areas of mathematics beyond pure combinatorics (e.g. in model theory, topology, group theory, etc). — 1820:

a Knowledge policy. You are, of course, free to express your disagreement with that policy, but that doesn't alter the fact that deferring to experts' opinions is

1558:

Under important graphs, should we add the n-cube? i.e. the graph of joining the vertices of a hypercube that differ by 1 bit when the vertices are in a bit stream?

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which gives a general definition of graph first, then defines simple graphs and multigraphs. I just hope I haven't achieved precision at the expense of clarity !

1745:

My impression is that among the participants in this discussion, Nsk92 is the most serious expert in the field. It would make sense to defer to his opinion.

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Well, I guess I mainly care about consistency between "graph" and "directed graph". It sounds like you think the definition of "graph" should be changed to

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allow loops; that possibility should only be mentioned in the alternate definitions section. I'll make the change eventually unless there are objections.

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here to start a "food fight", as you put it, and I think that people are perfectly entitled to their own little preferences and prejudices in this regard.

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An "unordered pair" is just a set with two elements. Otherwise simply called a "pair". The adjective "unordered" is sometime added to emphasize that it is

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JA: I will be making some comments on the variations in definitions and terminology, but just to keep my headings together I will locate everything on the

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Indeed, we have several possible moves, each being better than nothing; however, we do nothing, since we do not know which possibility is optimal; just

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Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.

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between them, with or without arrows, etc) that in some reasonable way covers most of the models of graphs that are later on discussed in the article.

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Tee-hee. Your earlier suggestion "Graph (mathematical structure)" seems reasonable to me, although I still prefer "Graph (discrete mathematics)". --

727:: "For a directed graph, a loop adds one to the in degree and one to the out degree." – This directly contradicts the definition on this page (x/=y). 875:

Could one explain what means the unordered pair used in the definition of the graph? It seems that there is no such an entry in wikipedia (and the

4694: 3435:: not really ambiguous, but "mathematical structure" is a concept, which seems to be understood only by people having good skills in mathematics. 2345:

the proposed variants to be almost nonexistent. No prejudice against any new RMs if anyone can think up a title they think will gain a consensus.

2042:

I agree that most of these term definitions should be italics, per standard article style. Bold is only acceptable for terms which redirect here.

1771:

Katzmik - one of the problems with the "defer to an expert's opinion" approach is that it is not how Knowledge works - what we should do is reach

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The current definition of "directed graph" allows loops. (And it did before I edited it!) I suggest that by default, a directed graph should

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Hamiltonian and Euler graphs are more important than others. Needs grouping graphs according to their applications- for a non-mathematician.

506: 2260: 2028: 2016:"Italics are used sparingly to emphasize words in sentences (bolding is normally not used at all for this purpose).") About an hour later, 1167: 587: 2331:

Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a

4293:

a time, and explain your edit in an edit summary. If an edit would need several different explanations, then split it in several edits.

2242: 2216: 2082: 1431: 965:.) Or more explicitly, you can define each edge to be either a one or two element subset of V, (See Gross, Jonathon L, and Yellen, Jay; 389: 1045:

is a set (any set will do). And, second, that the elements of the set will henceforth be called "vertices". I'll try making a change.

1484: 1235: 1184: 768: 752: 4648: 370: 115: 69: 2820:. A hard case! Maybe "Graph (a network of lines connecting points)" versus "Graph (a set that represents a function or relation)"? 2596:

is not a very good article. By contrast, Category:coding theory is contained in Category:discrete mathematics, as it should be. --

1987:

Would it make sense to have a minimal discussion of graph invariants, such as diameter, valence, girth, perhaps eigenvalue, etc?

2659: 2571: 2402: 2179: 4077:, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section. 3499:: In computer science, a network is essentially a graph. Thus this title strongly suggest that this is about computer networks. 4679: 4091:) 17:20, 3 January 2007 (UTC) Agreed. And also an extra image or two (an image is used twice at the moment) and a longer lead. 462: 973: 962: 732: 3755:. My preferred option. The links intended for readers who don't know or care what kind of graph they want can be changed to 3533:: While I am sure that there is a segment of editors that would have no problems with this, it just rubs me the wrong way. 1906:" are somewhat contradictory. However, as you have now descended to making derogatory personal comments (which contravenes 2512: 2468: 4402: 4338: 4032: 3992: 3973: 3187: 3113: 2976: 2764: 2727: 2677: 2645: 2620: 2612: 2543: 2365: 427: 308: 44: 2918: 2593: 2563: 2175: 876: 437: 351: 4333:

Thank you, this time your edits are more respectful than a mere revert, and I like the fact that you redirect to the

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In case of simple graphs, the two definitions of a multigraph coincide. So, there is no need to discuss this here.

3679:. Confusing: May be understood as being about plotting triangles and polygons, which have also vertices and edges. 2735: 472: 234:

related articles on Knowledge. If you would like to participate, please visit the project page, where you can join

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agreed and changed it. Now, I see things differently. For me, and for lots of other practicing mathematicians,

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Graph (insert this entire present discussion here, but first wait until it exceeds one-hundred-thousand words)

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You are right that the wording is sub-optimal but your suggested fix is not good. The story is, first, that

4159: 4140: 3876: 2879: 2865: 2839: 2628: 2601: 2520: 2440: 2246: 2212: 2086: 1972: 1713: 1631: 1615: 1563: 1488: 1446: 1188: 825: 3333: 2972: 2780: 2681: 2508: 2381: 2106: 1930: 3792: 2417:– the extra disambiguation is not warranted. How some editors mislink is not really a title criterion. 906:. Perhaps a definition of the term "unordered pair" should be added to the "ordered pair" article, which 4644: 3198: 3156: 3013: 2769: 2731: 2504: 2478: 2464: 2332: 1215: 1086: 1077: 980: 926: 914: 408: 50: 1009: 101: 4658: 4606: 3276: 2774: 2673: 2656: 2649: 2616: 2486: 2460: 2399: 2238: 2204: 2132: 2078: 1536: 1520: 1459: 1427: 1422:

As a computer science student heavily working on computer networks, we are constantly talking about "

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Thank you for the corrections. But in this case, is the edge set of a directed graph still denoted

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would be better, since that topic is discussed more than discrete mathematics, and the page is in

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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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page. It is most appropriate to have an applications section there and not here (in my opinion)

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You're right -- there are many real world applications. There is an applications section on the

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Another thing I noticed is the following addition (my bold): "Typically, a graph is depicted in

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and my perspective is colored accordingly. There are real graph theory experts here, including

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one edge between any two different vertices"? Sounds like a tautology to me. In fact what is a

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Thank you for your fast reply. Now I signed both messages with my brand new user account name.

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I think graphs im Maths are too complicated and you never use them in every day life (jobs)

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is a sort of circular reference and it does not make clear that it belongs to mathematics.

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but the word does not occur in the article. Just mentioning. — Is there no bot for this?

3028: 4592: 4566: 4529: 3702: 3343:. I think that graphs are used in a much wider part of mathematics than combinatorics. 2935: 2753: 2157: 2053: 2017: 1911: 1825: 1784: 1586: 1544: 1503: 1467: 1378: 1326: 1243: 989:

vertex but nevertheless add only 1 to the row and column sums of the adjacency matrix.

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Requested articles/Applied arts and sciences/Computer science, computing, and Internet

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I've asked Google image for "graph connections" and feel satisfied with the result.

2899:. Then, indeed, "Graph (connected nodes)" versus "Graph (of function or relation)". 1824:

how Knowledge works. It makes sense to me that editors should think for themselves.

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The definition in terms of connectedness and degree is used in the main article on

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I have been bold and moved the article to the title that has the least opposition.

2943: 2687: 2623:, and graphs are studied in a larger area of mathematics than just graph theory. -- 2559: 2346: 2299: 2197:

Missing: "Two graphs are said to be equal iff ...". Should follow each definition.

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Since no one responded, I will adjust the wording to reflect the above distinction

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I decided to just edit the page. I'm pretty confident the existing text was wrong.

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article for simple graphs (currently we only introduce edges without own identity

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move. You are right that the current disambiguation is insufficient, but I think

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Well, I'm not in love, but it does seem better than the current alternatives. —

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I agree. I have tried to generalise the initial definition so that it defines a

120: 3720:. I think we should have an article which discusses all basic types of graphs. 4654: 4602: 4364: 4334: 4084: 3887: 3012:

Subsequent comments should be made in a new section on this talk page or in a

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than to discrete mathematics, so I think the probability of confusion is low.

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implies simple, and if you want your edges to be multisets, you're welcome to

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Wikipedia_talk:WikiProject_Mathematics#Graph_.28mathematics.29_vs_Graph_theory

97: 77: 4420:), for completeness and consistency between both articles, what do you think? 4367:

article introduces 2 definitions for multigraphs (edges without own identity

2160:

then you may need to upload it to Knowledge (Commons does not allow fair use)

3160: 2749: 1540: 1499: 1463: 360: 2800:, but maybe that could be confused with computer engineering. Or how about 1242:

appropriate for this situation, so I have removed them from both articles.

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article covers the definition and basic properties of graphs, whereas the

4636: 2709:. Maybe "Graph (mathematical structure)" versus "Graph (of a function)"? 1775:

through discussion, and then implement that consensus. Having said that,

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were first studied by Erdős and Rényi without also mentioning Gilbert. —

2298:." I guess you don't have to draw edges "straight" for one distinction. 1430:". Is there a similar concept in the article? Should it be included? -- 4501:

It may be not colloquial, but I meant "exactly the same in this case".

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https://en.wikipedia.org/search/?title=Graph_theory&oldid=896700053

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Since no one responded, I refer our library in External links section.

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a} to be a "pair" with non-distinct elements (See Balakrishnan, V. K.;

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in Knowledge). This is now consistent with the definition sequence in

3086:"Discreet structure" could be a private room in a Tokyo love hotel. — 626:

then the other way round. But you are free to change the definition.

2644:, but I'm relisting so we can get consensus between the two options 3112:

The title ambiguity problem has been solved by moving the page to

2981: 167: 4213:? (You have renamed the words "arrow" to "edge" but not the sets 2963:. I essentially agree with David's comments above and think that 2874:

If we do the insertion recursively, we don't have to wait ;). --

2680:

is problematic for the reasons stated by D.Lazard. Shall we try

1101: 2566:. Myself, I am unable to say which parts of my research domain ( 4589:

The modifications I recently made to this page were revoked by

3286:. It is a circular reference and does not mention mathematics. 2672:. A move is probably a good idea if we can agree on a title. 1396:. But I am certainly not going to advocate that approach here. 436:

Find pictures for the biographies of computer scientists (see

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and vice-versa. Therefore, I made these articles redirect to

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are essentially impossible to write since you cannot explain

3850:: Meaningless. Sounds more like a social network buzz-word. 3701:

Narrow the scope of this article to complement the articles

3644:. Just a standard maths vocabulary variant of the previous. 166: 4176:

for the node-link diagrams described in your other thing. —

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It would be easy to start a war over ownership of the term

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have an exponential complexity. The definition in term of

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16:22, 9 Jan 2005 (UTC) after that he did another theory

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seem to have almost reached a workable consensus anyway.

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In my opinion it is easier to call a graph without loops

2274:

It is convention, but you'll note that the intro says: "

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historical reasons, that would also be helpful to know.

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actually tend to think about a graph as a 1-dimentional

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are supposed to summarize the material that comes after

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I've done as the consensus seemed to indicate. Enjoy.—

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asserted above that the definition was too narrow, and

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What do you think of "Graph (discrete structure)"? —

3002:

The above discussion is preserved as an archive of a

2640:. It sounds like we have consensus to move away from 1236:

Knowledge:Merging_and_moving_pages#Proposing_a_merger

4150:(The same thing is true of the edits by Maggyero to 1263:

carefully fairly confused. The main definition of a

230:, a collaborative effort to improve the coverage of 119:, a collaborative effort to improve the coverage of 2335:. No further edits should be made to this section. 2163:If the image isn't freely licensed and there is no 725:

https://en.wikipedia.org/Loop_(graph_theory)#Degree

3317:, redundantly redundant and circularly circular. — 3117: 3016:. No further edits should be made to this section. 1272:in the sense of the main graph definition anyway? 342:Computer science articles needing expert attention 4033:Graph (discrete mathematics)#Properties of graphs 3993:Graph (discrete mathematics)#Properties of graphs 3573:. Simple and precise, although a bit unorthodox. 2942:, and move the discussion of types of graph into 2592:Largely your examples highlight that the article 1910:) I see no point in continuing this discussion. 4700:Knowledge level-4 vital articles in Mathematics 4073:, and are posted here for posterity. Following 3662:to be a good disambiguator for naive readers. — 2676:is problematic for the reasons stated by JBL. 4547:To me, "types of graphs" sounds more natural. 1498:Done. Thanks for bringing it to an attention. 482:WikiProject Computer science/Unreferenced BLPs 4067:The comment(s) below were originally left at 958:, McGraw-Hill; 1 edition (February 1, 1997). 8: 3031:. Could we choose one possibility randomly? 1283:is a function that assigns to every element 4633:the WikiProject Mathematics discussion page 4083:needs more on motivation and applications. 3724:should focus on applications and problems. 1537:Category talk:Graphs#Subcategory suggestion 399:Computer science articles without infoboxes 337:Computer science articles needing attention 4070:Talk:Graph (discrete mathematics)/Comments 2321:The following is a closed discussion of a 1518: 742: 303:Here are some tasks awaiting attention: 277: 192: 58: 4730:High-importance Computer science articles 4103:Substituted at 18:19, 17 July 2016 (UTC) 2439:is ambiguous and none of its meanings is 2200:Also a word on isomorphy would be nice. 1581:, but I'll refrain—for now, anyway :-). 4257:Your last edit Applying MOS:LISTBULLET ( 3709:. Move the discussion of graph types to 2117:File:Directed.svg Nominated for Deletion 1018:Done. YOu could have done it yorself. `' 4720:Featured articles on Mathematics Portal 4690:Knowledge vital articles in Mathematics 879:, is not what we want here, is it?). -- 194: 60: 19: 2971:is clearly a poor choice and I prefer 2172:This notification is provided by a Bot 1068:Quivers vs. representations of quivers 765:2601:1C2:1B7F:9FA0:441D:A012:C730:FECF 749:2601:1C2:1B7F:9FA0:441D:A012:C730:FECF 641:loops. Anyone else have an opinion? 244:Knowledge:WikiProject Computer science 4735:WikiProject Computer science articles 4705:C-Class vital articles in Mathematics 3417:. Almost equivalent to the ambiguous 2135:, has been nominated for deletion at 247:Template:WikiProject Computer science 7: 2340:The result of the move request was: 1295:consisting of one or two elements. 1199:Include Euler and Hamiltonian graphs 224:This article is within the scope of 113:This article is within the scope of 4341:article can focus on simple graphs. 2368:– Users often incorrectly point to 2167:then it cannot be uploaded or used. 49:It is of interest to the following 4401:), we may also do the same in the 2574:) belong to discrete mathematics. 2382:Graph (disambiguation)#Mathematics 418:Timeline of computing 2020–present 14: 4725:C-Class Computer science articles 4715:Top-priority mathematics articles 4581:Revoked modifications (suspected 4075:several discussions in past years 4039:, which is already the target of 4037:Glossary of graph theory#adjacent 1599:for this different beast, namely 969:, CRC Press (December 30, 1998). 967:Graph Theory and Its Applications 444:Computing articles needing images 133:Knowledge:WikiProject Mathematics 4685:Knowledge level-4 vital articles 3997:Two edges of a graph are called 3922:that I have tried (in this edit 2784:appears a bit too narrow to me. 2572:Computational algebraic geometry 2141:Deletion requests September 2011 2124: 956:Schaum's Outline of Graph Theory 294: 217: 196: 136:Template:WikiProject Mathematics 100: 90: 76: 62: 29: 20: 4485:What do you mean they coincide? 2131:An image used in this article, 937:make it clear in the article -- 572:I merged the two articles. See 264:This article has been rated as 153:This article has been rated as 4695:C-Class level-4 vital articles 4663:12:43, 24 September 2024 (UTC) 4649:22:44, 23 September 2024 (UTC) 4627:17:15, 23 September 2024 (UTC) 4611:12:21, 23 September 2024 (UTC) 4003:" I guess one could just add " 4001:if they share a common vertex. 3408:Graph (mathematical structure) 3201:than to discrete mathematics. 2787:Graph (mathematical structure) 2443:, it should be disambiguated. 2012:with an edit summary of: (See 1104:you can get familiar with it. 1: 2513:Category:discrete mathematics 2469:Category:discrete mathematics 2314:Requested move 5 January 2016 2184:19:45, 6 September 2011 (UTC) 2062:10:19, 29 November 2008 (UTC) 2047:08:44, 29 November 2008 (UTC) 2037:19:50, 28 November 2008 (UTC) 1317:(which actually redirects to 1176:17:23, 27 February 2008 (UTC) 1151:22:13, 11 February 2008 (UTC) 1023:21:32, 26 February 2007 (UTC) 1013:21:11, 26 February 2007 (UTC) 773:03:59, 29 November 2022 (UTC) 757:03:44, 29 November 2022 (UTC) 729:2601:647:5C00:4460:0:0:0:AA8E 596:17:44, 4 September 2007 (UTC) 498:Tag all relevant articles in 238:and see a list of open tasks. 127:and see a list of open tasks. 4710:C-Class mathematics articles 4575:22:48, 4 November 2023 (UTC) 4560:22:29, 4 November 2023 (UTC) 4542:22:17, 4 November 2023 (UTC) 4403:Graph (discrete mathematics) 4382:and edges with own identity 4339:Graph (discrete mathematics) 3954:18:23, 22 January 2016 (UTC) 3935:16:42, 22 January 2016 (UTC) 3904:14:56, 22 January 2016 (UTC) 3881:14:22, 22 January 2016 (UTC) 3860:19:20, 30 January 2016 (UTC) 3843:18:15, 30 January 2016 (UTC) 3826:16:26, 30 January 2016 (UTC) 3808:15:22, 30 January 2016 (UTC) 3786:19:20, 30 January 2016 (UTC) 3769:18:15, 30 January 2016 (UTC) 3748:14:40, 30 January 2016 (UTC) 3731:09:37, 30 January 2016 (UTC) 3689:21:52, 30 January 2016 (UTC) 3672:20:59, 30 January 2016 (UTC) 3651:20:23, 30 January 2016 (UTC) 3628:18:15, 30 January 2016 (UTC) 3609:17:53, 30 January 2016 (UTC) 3597:14:46, 30 January 2016 (UTC) 3580:09:37, 30 January 2016 (UTC) 3566:09:17, 30 January 2016 (UTC) 3543:19:20, 30 January 2016 (UTC) 3526:18:15, 30 January 2016 (UTC) 3509:14:33, 30 January 2016 (UTC) 3483:19:20, 30 January 2016 (UTC) 3466:18:15, 30 January 2016 (UTC) 3445:14:28, 30 January 2016 (UTC) 3428:09:37, 30 January 2016 (UTC) 3401:19:20, 30 January 2016 (UTC) 3384:18:15, 30 January 2016 (UTC) 3367:14:24, 30 January 2016 (UTC) 3350:09:37, 30 January 2016 (UTC) 3327:18:15, 30 January 2016 (UTC) 3310:14:18, 30 January 2016 (UTC) 3293:09:37, 30 January 2016 (UTC) 3270:19:22, 9 February 2016 (UTC) 3259:19:20, 30 January 2016 (UTC) 3242:18:15, 30 January 2016 (UTC) 3225:14:13, 30 January 2016 (UTC) 3208:09:37, 30 January 2016 (UTC) 3188:Graph (discrete mathematics) 3182:09:13, 30 January 2016 (UTC) 3127:09:12, 30 January 2016 (UTC) 3114:Graph (discrete mathematics) 3096:07:45, 30 January 2016 (UTC) 3081:06:47, 30 January 2016 (UTC) 3067:00:00, 30 January 2016 (UTC) 3056:12:22, 25 January 2016 (UTC) 3041:12:11, 25 January 2016 (UTC) 2994:04:57, 14 January 2016 (UTC) 2977:Graph (discrete mathematics) 2956:01:28, 14 January 2016 (UTC) 2909:20:00, 13 January 2016 (UTC) 2884:22:30, 13 January 2016 (UTC) 2870:22:17, 13 January 2016 (UTC) 2844:19:18, 13 January 2016 (UTC) 2830:19:14, 13 January 2016 (UTC) 2813:19:20, 13 January 2016 (UTC) 2765:Graph (discrete mathematics) 2758:17:19, 13 January 2016 (UTC) 2740:16:55, 13 January 2016 (UTC) 2728:Graph (discrete mathematics) 2719:16:28, 13 January 2016 (UTC) 2702:16:05, 13 January 2016 (UTC) 2678:Graph (discrete mathematics) 2665:14:54, 13 January 2016 (UTC) 2646:Graph (discrete mathematics) 2621:graph (discrete mathematics) 2613:Graph (discrete mathematics) 2544:Graph (discrete mathematics) 2525:15:46, 13 January 2016 (UTC) 2408:14:54, 13 January 2016 (UTC) 2366:Graph (discrete mathematics) 2355:09:46, 25 January 2016 (UTC) 1997:10:11, 24 October 2008 (UTC) 1977:14:34, 24 October 2008 (UTC) 1951:13:47, 24 October 2008 (UTC) 1920:11:18, 24 October 2008 (UTC) 1873:10:59, 24 October 2008 (UTC) 1834:10:52, 24 October 2008 (UTC) 1808:10:23, 24 October 2008 (UTC) 1793:10:20, 24 October 2008 (UTC) 1755:10:07, 24 October 2008 (UTC) 1733:00:46, 24 October 2008 (UTC) 1718:22:46, 23 October 2008 (UTC) 1693:22:28, 23 October 2008 (UTC) 1678:22:19, 23 October 2008 (UTC) 1660:22:16, 23 October 2008 (UTC) 1636:21:41, 23 October 2008 (UTC) 1620:21:28, 23 October 2008 (UTC) 1568:06:55, 14 October 2008 (UTC) 1549:17:21, 10 October 2008 (UTC) 1508:23:16, 10 October 2008 (UTC) 1493:12:29, 10 October 2008 (UTC) 1472:23:26, 10 October 2008 (UTC) 1193:12:27, 10 October 2008 (UTC) 994:03:00, 5 November 2006 (UTC) 984:23:55, 4 November 2006 (UTC) 949:22:57, 4 November 2006 (UTC) 930:17:25, 18 October 2006 (UTC) 922:Ok I've now done the above. 918:17:15, 18 October 2006 (UTC) 893:16:04, 18 October 2006 (UTC) 556:09:28, 26 January 2007 (UTC) 547:The previous text was this: 507:WikiProject Computer science 283:WikiProject Computer science 227:WikiProject Computer science 4440:End of the moved discussion 3964:redirect from »nonadjacent« 2633:16:29, 6 January 2016 (UTC) 2594:discrete mathematics topics 2584:14:45, 6 January 2016 (UTC) 2564:Discrete mathematics topics 2496:13:24, 6 January 2016 (UTC) 2482:05:50, 6 January 2016 (UTC) 2450:13:24, 6 January 2016 (UTC) 2427:05:48, 6 January 2016 (UTC) 2390:11:48, 5 January 2016 (UTC) 2221:22:47, 11 August 2012 (UTC) 2139:in the following category: 1525:01:26, 4 October 2016 (UTC) 1440:13:39, 5 October 2008 (UTC) 877:Axiom of the unordered pair 789:00:25, 17 August 2005 (UTC) 737:17:31, 4 October 2021 (UTC) 717:15:18, 6 October 2010 (UTC) 701:05:49, 1 October 2010 (UTC) 438:List of computer scientists 4751: 4234:17:27, 16 April 2019 (UTC) 4186:15:26, 16 April 2019 (UTC) 4164:12:21, 16 April 2019 (UTC) 4145:12:19, 16 April 2019 (UTC) 4035:, I have redirected it to 3635:Graph (vertices and edges) 2984:do what it is meant to do. 2546:is almost as ambiguous as 2091:20:49, 31 March 2009 (UTC) 2014:wp:manual of style#Italics 1610:Let the food fight begin!— 1050:03:44, 12 March 2007 (UTC) 851:01:01, 15 April 2006 (UTC) 812:Articles for edge and node 807:19:10, 18 March 2006 (UTC) 270:project's importance scale 4082: 3968:There is a redirect from 2308:20:30, 2 April 2014 (UTC) 2269:20:05, 2 April 2014 (UTC) 1252:08:16, 7 April 2008 (UTC) 1220:20:32, 6 April 2008 (UTC) 1111:14:15, 14 June 2007 (UTC) 866:13:30, 27 June 2006 (UTC) 614:14:48, 13 Jan 2005 (UTC) 500:Category:Computer science 276: 263: 250:Computer science articles 212: 174: 152: 85: 57: 4511:19:05, 12 May 2019 (UTC) 4497:17:14, 12 May 2019 (UTC) 4469:16:34, 12 May 2019 (UTC) 4432:15:29, 12 May 2019 (UTC) 4353:15:18, 12 May 2019 (UTC) 4323:12:38, 12 May 2019 (UTC) 4303:12:22, 12 May 2019 (UTC) 4274:09:35, 12 May 2019 (UTC) 4096:16:09, 9 June 2007 (UTC) 4055:14:51, 6 June 2016 (UTC) 4021:14:15, 6 June 2016 (UTC) 3986:13:43, 6 June 2016 (UTC) 3009:Please do not modify it. 2967:might be the way to go. 2934:just as we have one for 2790:is almost equivalent to 2380:should be redirected to 2372:when they actually mean 2328:Please do not modify it. 2111:16:30, 21 May 2009 (UTC) 1931:geometric group theorist 1908:another Knowledge policy 1406:14:06, 21 May 2008 (UTC) 1387:13:33, 21 May 2008 (UTC) 1364:12:09, 21 May 2008 (UTC) 1335:09:14, 21 May 2008 (UTC) 1323:glossary of graph theory 1308:04:11, 21 May 2008 (UTC) 1131:11:14, 2 July 2007 (UTC) 1090:04:19, 3 June 2007 (UTC) 1081:04:19, 3 June 2007 (UTC) 684:09:21, 19 Jan 2005 (UTC) 669:13:47, 19 Jan 2005 (UTC) 649:02:47, 19 Jan 2005 (UTC) 630:19:55, 18 Jan 2005 (UTC) 502:and sub-categories with 159:project's priority scale 4042:Adjacent (graph theory) 3550:Graph (nodes and links) 2919:Hadwiger–Nelson problem 2803:Graph (connected nodes) 2002:To Bold or not to Bold? 1447:Distance (graph theory) 1238:. The cleanup tags are 826:node (computer science) 622:and a graph with loops 543:Generalizations section 116:WikiProject Mathematics 4680:C-Class vital articles 4363:By the way, since the 4249:Discussion moved from 2509:Category:combinatorics 2489:is acceptable for me. 2176:CommonsNotificationBot 1812:Katzmik - I linked to 1531:Subcategory suggestion 1156:Graphs too complicated 1115:You dit it real nice. 1029:Definition of a Vertex 661:using adjectives like 463:Computer science stubs 171: 4597:because of suspected 3334:Graph (combinatorics) 3199:mathematical analysis 2973:Graph (combinatorics) 2781:Graph (combinatorics) 2770:mathematical analysis 2682:Graph (combinatorics) 2505:Category:graph theory 2465:Category:graph theory 2193:Equality vs Isomorphy 2006:I recently made this 1004:Definition of a Graph 170: 36:level-4 vital article 4337:article so that the 4025:Instead to redirect 3277:Graph (graph theory) 3023:We are Buridan's ass 2794:. I also considered 2775:Graph (graph theory) 2674:Graph (graph theory) 2650:Graph (graph theory) 2617:Graph (graph theory) 2539:Graph (graph theory) 2487:Graph (graph theory) 2461:Graph (graph theory) 2020:reverted my edit to 910:could then link to. 844:graph (graph theory) 839:graph (graph theory) 281:Things you can help 139:mathematics articles 4631:As I wrote over at 3793:Graph (connections) 3642:Support as proposer 3571:Support as proposer 3454:Graph (mathematics) 3419:Graph (mathematics) 3118:previous discussion 3022: 2969:Graph (mathematics) 2792:Graph (mathematics) 2763:Weakly in favor of 2642:Graph (mathematics) 2611:I strongly support 2552:graph of a function 2548:Graph (mathematics) 2511:, which belongs to 2507:, which belongs to 2501:Graph (mathematics) 2437:Graph (mathematics) 2378:Graph (mathematics) 2374:Graph of a function 2370:Graph (mathematics) 2362:Graph (mathematics) 1814:Knowledge:Consensus 1227:Graph (mathematics) 832:without explaining 822:node (graph theory) 818:edge (graph theory) 563:graph (mathematics) 4251:User talk:D.Lazard 4123:so heavily used.) 4063:Assessment comment 2562:are not listed in 2165:fair use rationale 576:for a discussion. 181:Mathematics Portal 172: 108:Mathematics portal 45:content assessment 4549: 4525:Types of graph(s) 4452: 4441: 4281: 4254: 4130:as a set of dots 4128:diagrammatic form 4101: 4100: 3120:) before voting. 3071:Indeed, why not? 2700: 2662: 2410: 2405: 2241:comment added by 2224: 2207:comment added by 2190: 2189: 2145:What should I do? 2137:Wikimedia Commons 2133:File:Directed.svg 2081:comment added by 1573:Multi- or simple? 1527: 1178: 1166:comment added by 946: 890: 883: 759: 747:comment added by 598: 586:comment added by 537: 536: 533: 532: 529: 528: 525: 524: 521: 520: 191: 190: 187: 186: 4742: 4596: 4548: 4458: 4450: 4439: 4419: 4400: 4381: 4287: 4279: 4248: 4208: 4121: 4080: 4079: 4072: 4044: 4030: 3852:Bill Cherowitzo 3778:Bill Cherowitzo 3729: 3696:Undirected graph 3649: 3607: 3578: 3535:Bill Cherowitzo 3475:Bill Cherowitzo 3426: 3393:Bill Cherowitzo 3348: 3291: 3267: 3251:Bill Cherowitzo 3206: 3180: 3157:Mark L MacDonald 3125: 3064: 3011: 2986:Bill Cherowitzo 2965:undirected graph 2932:undirected graph 2923:Hyperbolic group 2811: 2699: 2697: 2685: 2660: 2615:, weakly oppose 2568:computer algebra 2541: 2494: 2448: 2441:WP:primary topic 2403: 2391: 2389: 2330: 2290:are also called 2278:are also called 2250: 2223: 2201: 2156:If the image is 2128: 2121: 2120: 2093: 2025: 2011: 1596:a different term 1161: 1087:SophomoricPedant 1078:SophomoricPedant 945: 942: 889: 886: 881: 581: 511: 505: 380:Computer science 309:Article requests 298: 291: 290: 278: 252: 251: 248: 245: 242: 241:Computer science 232:Computer science 221: 214: 213: 208: 204:Computer science 200: 193: 177:selected article 141: 140: 137: 134: 131: 110: 105: 104: 94: 87: 86: 81: 80: 79: 74: 66: 59: 42: 33: 32: 25: 24: 16: 4750: 4749: 4745: 4744: 4743: 4741: 4740: 4739: 4670: 4669: 4590: 4587: 4527: 4449: 4406: 4383: 4368: 4278: 4246: 4198: 4111: 4109: 4068: 4065: 4040: 4026: 3966: 3920:graph traversal 3868: 3814:Strongly oppose 3800:Boris Tsirelson 3796: 3725: 3699: 3645: 3638: 3603: 3585:Strongly oppose 3574: 3558:Boris Tsirelson 3553: 3497:Strongly oppose 3493: 3490:Graph (network) 3422: 3411: 3344: 3337: 3287: 3280: 3265: 3202: 3191: 3176: 3121: 3110: 3073:Boris Tsirelson 3062: 3033:Boris Tsirelson 3025: 3020: 3007: 2901:Boris Tsirelson 2822:Boris Tsirelson 2807: 2797:Graph (network) 2711:Boris Tsirelson 2695: 2690: 2686: 2537: 2490: 2444: 2385: 2326: 2316: 2256: 2236: 2233: 2202: 2195: 2119: 2076: 2072: 2021: 2007: 2004: 1985: 1575: 1556: 1533: 1480: 1478:Connected Graph 1460:network metrics 1428:Network metrics 1420: 1260: 1201: 1158: 1138: 1097: 1070: 1031: 1006: 943: 887: 873: 871:unordered pair? 858: 814: 796: 781: 779:Oriented graphs 604: 570: 545: 517: 514: 509: 503: 491:Project-related 486: 467: 448: 422: 403: 384: 365: 346: 322: 266:High-importance 249: 246: 243: 240: 239: 207:High‑importance 206: 138: 135: 132: 129: 128: 106: 99: 75: 72: 43:on Knowledge's 40: 30: 12: 11: 5: 4748: 4746: 4738: 4737: 4732: 4727: 4722: 4717: 4712: 4707: 4702: 4697: 4692: 4687: 4682: 4672: 4671: 4668: 4667: 4666: 4665: 4629: 4619:David Eppstein 4586: 4579: 4578: 4577: 4562: 4552:David Eppstein 4530:User:JayBeeEll 4526: 4523: 4522: 4521: 4520: 4519: 4518: 4517: 4516: 4515: 4514: 4513: 4486: 4476: 4475: 4474: 4473: 4472: 4471: 4437: 4436: 4435: 4434: 4421: 4358: 4357: 4356: 4355: 4342: 4328: 4327: 4326: 4325: 4312: 4306: 4305: 4290: 4276: 4245: 4242: 4241: 4240: 4239: 4238: 4237: 4236: 4222: 4205:David Eppstein 4191: 4190: 4189: 4188: 4178:David Eppstein 4167: 4166: 4114:David Eppstein 4108: 4105: 4099: 4098: 4064: 4061: 4060: 4059: 4058: 4057: 3965: 3962: 3961: 3960: 3959: 3958: 3957: 3956: 3946:David Eppstein 3908: 3907: 3892: 3867: 3866:Path and cycle 3864: 3863: 3862: 3845: 3835:David Eppstein 3828: 3795: 3790: 3789: 3788: 3771: 3761:David Eppstein 3750: 3733: 3703:Directed graph 3698: 3693: 3692: 3691: 3674: 3664:David Eppstein 3653: 3637: 3632: 3631: 3630: 3620:David Eppstein 3613: 3612: 3611: 3582: 3568: 3552: 3547: 3546: 3545: 3528: 3518:David Eppstein 3511: 3492: 3487: 3486: 3485: 3468: 3458:David Eppstein 3447: 3430: 3410: 3405: 3404: 3403: 3386: 3376:David Eppstein 3369: 3352: 3336: 3331: 3330: 3329: 3319:David Eppstein 3312: 3298:Weakly support 3295: 3279: 3274: 3273: 3272: 3261: 3244: 3234:David Eppstein 3227: 3213:Weakly support 3210: 3190: 3185: 3165:David Eppstein 3109: 3106: 3105: 3104: 3103: 3102: 3101: 3100: 3099: 3098: 3088:David Eppstein 3024: 3021: 3019: 3018: 3004:requested move 2998: 2997: 2996: 2958: 2948:David Eppstein 2936:directed graph 2911: 2893: 2892: 2891: 2890: 2889: 2888: 2887: 2886: 2849: 2848: 2847: 2846: 2815: 2760: 2742: 2721: 2704: 2693: 2688: 2667: 2635: 2608: 2607: 2606: 2605: 2587: 2586: 2550:, because the 2530: 2529: 2528: 2527: 2498: 2454: 2453: 2452: 2360: 2358: 2338: 2337: 2323:requested move 2317: 2315: 2312: 2311: 2310: 2261:129.219.155.89 2255: 2252: 2232: 2229: 2227: 2194: 2191: 2188: 2187: 2169: 2168: 2161: 2147: 2146: 2129: 2118: 2115: 2114: 2113: 2071: 2068: 2067: 2066: 2065: 2064: 2029:98.166.139.216 2018:User:Gandalf61 2003: 2000: 1984: 1981: 1980: 1979: 1964: 1963: 1962: 1961: 1960: 1959: 1958: 1957: 1956: 1955: 1954: 1953: 1939:David Eppstein 1926: 1925: 1924: 1923: 1922: 1888: 1887: 1886: 1885: 1884: 1883: 1882: 1881: 1880: 1879: 1878: 1877: 1876: 1875: 1847: 1846: 1845: 1844: 1843: 1842: 1841: 1840: 1839: 1838: 1837: 1836: 1762: 1761: 1760: 1759: 1758: 1757: 1738: 1737: 1736: 1735: 1701: 1700: 1699: 1698: 1697: 1696: 1695: 1574: 1571: 1555: 1552: 1532: 1529: 1523:comment added 1511: 1510: 1479: 1476: 1475: 1474: 1419: 1416: 1415: 1414: 1413: 1412: 1411: 1410: 1409: 1408: 1369: 1368: 1367: 1366: 1259: 1256: 1255: 1254: 1200: 1197: 1196: 1195: 1168:88.108.216.138 1157: 1154: 1137: 1134: 1096: 1095:Implementation 1093: 1069: 1066: 1065: 1064: 1063: 1062: 1053: 1052: 1030: 1027: 1026: 1025: 1005: 1002: 1001: 1000: 999: 998: 997: 996: 934: 933: 932: 908:unordered pair 872: 869: 857: 856:Citation Style 854: 813: 810: 795: 792: 780: 777: 776: 775: 720: 719: 688: 687: 686: 685: 675: 674: 673: 672: 671: 670: 653: 652: 651: 650: 632: 631: 603: 600: 588:217.38.127.254 569: 559: 544: 541: 539: 535: 534: 531: 530: 527: 526: 523: 522: 519: 518: 516: 515: 513: 512: 495: 487: 485: 484: 478: 468: 466: 465: 459: 449: 447: 446: 441: 433: 423: 421: 420: 414: 404: 402: 401: 395: 385: 383: 382: 376: 366: 364: 363: 357: 347: 345: 344: 339: 333: 323: 321: 320: 314: 302: 300: 299: 287: 286: 274: 273: 262: 256: 255: 253: 236:the discussion 222: 210: 209: 201: 189: 188: 185: 184: 173: 163: 162: 151: 145: 144: 142: 125:the discussion 112: 111: 95: 83: 82: 67: 55: 54: 48: 26: 13: 10: 9: 6: 4: 3: 2: 4747: 4736: 4733: 4731: 4728: 4726: 4723: 4721: 4718: 4716: 4713: 4711: 4708: 4706: 4703: 4701: 4698: 4696: 4693: 4691: 4688: 4686: 4683: 4681: 4678: 4677: 4675: 4664: 4660: 4656: 4652: 4651: 4650: 4646: 4642: 4638: 4634: 4630: 4628: 4624: 4620: 4615: 4614: 4613: 4612: 4608: 4604: 4600: 4594: 4584: 4580: 4576: 4572: 4568: 4563: 4561: 4557: 4553: 4546: 4545: 4544: 4543: 4539: 4535: 4531: 4524: 4512: 4508: 4504: 4500: 4499: 4498: 4494: 4490: 4487: 4484: 4483: 4482: 4481: 4480: 4479: 4478: 4477: 4470: 4466: 4462: 4456: 4448: 4447: 4446: 4445: 4444: 4443: 4442: 4433: 4429: 4425: 4422: 4417: 4413: 4409: 4404: 4398: 4394: 4390: 4386: 4379: 4375: 4371: 4366: 4362: 4361: 4360: 4359: 4354: 4350: 4346: 4343: 4340: 4336: 4332: 4331: 4330: 4329: 4324: 4320: 4316: 4313: 4310: 4309: 4308: 4307: 4304: 4300: 4296: 4291: 4285: 4277: 4275: 4271: 4267: 4264: 4263: 4262: 4260: 4255: 4252: 4243: 4235: 4231: 4227: 4223: 4220: 4216: 4212: 4206: 4202: 4201:Joel B. Lewis 4197: 4196: 4195: 4194: 4193: 4192: 4187: 4183: 4179: 4175: 4174:graph drawing 4171: 4170: 4169: 4168: 4165: 4161: 4157: 4153: 4149: 4148: 4147: 4146: 4142: 4138: 4133: 4129: 4124: 4119: 4115: 4106: 4104: 4097: 4094: 4090: 4086: 4081: 4078: 4076: 4071: 4062: 4056: 4052: 4048: 4043: 4038: 4034: 4029: 4024: 4023: 4022: 4018: 4014: 4010: 4006: 4002: 3998: 3995:starts with " 3994: 3990: 3989: 3988: 3987: 3983: 3979: 3975: 3971: 3963: 3955: 3951: 3947: 3943: 3938: 3937: 3936: 3932: 3928: 3924: 3921: 3916: 3915:understanding 3912: 3911: 3910: 3909: 3905: 3901: 3897: 3893: 3889: 3885: 3884: 3883: 3882: 3878: 3874: 3865: 3861: 3857: 3853: 3849: 3846: 3844: 3840: 3836: 3832: 3829: 3827: 3823: 3819: 3815: 3812: 3811: 3810: 3809: 3805: 3801: 3794: 3791: 3787: 3783: 3779: 3775: 3774:Weakly oppose 3772: 3770: 3766: 3762: 3758: 3754: 3751: 3749: 3745: 3741: 3737: 3734: 3732: 3728: 3723: 3719: 3716: 3715: 3714: 3712: 3708: 3704: 3697: 3694: 3690: 3686: 3682: 3678: 3675: 3673: 3669: 3665: 3661: 3657: 3654: 3652: 3648: 3643: 3640: 3639: 3636: 3633: 3629: 3625: 3621: 3617: 3614: 3610: 3606: 3600: 3599: 3598: 3594: 3590: 3586: 3583: 3581: 3577: 3572: 3569: 3567: 3563: 3559: 3555: 3554: 3551: 3548: 3544: 3540: 3536: 3532: 3529: 3527: 3523: 3519: 3515: 3512: 3510: 3506: 3502: 3498: 3495: 3494: 3491: 3488: 3484: 3480: 3476: 3472: 3469: 3467: 3463: 3459: 3455: 3451: 3448: 3446: 3442: 3438: 3434: 3433:Weakly oppose 3431: 3429: 3425: 3420: 3416: 3413: 3412: 3409: 3406: 3402: 3398: 3394: 3390: 3389:Weakly oppose 3387: 3385: 3381: 3377: 3373: 3370: 3368: 3364: 3360: 3356: 3353: 3351: 3347: 3342: 3339: 3338: 3335: 3332: 3328: 3324: 3320: 3316: 3313: 3311: 3307: 3303: 3299: 3296: 3294: 3290: 3285: 3284:Weakly oppose 3282: 3281: 3278: 3275: 3271: 3268: 3262: 3260: 3256: 3252: 3248: 3245: 3243: 3239: 3235: 3231: 3228: 3226: 3222: 3218: 3214: 3211: 3209: 3205: 3200: 3196: 3193: 3192: 3189: 3186: 3184: 3183: 3179: 3174: 3170: 3166: 3162: 3158: 3154: 3150: 3146: 3142: 3141:Joel B. Lewis 3138: 3134: 3129: 3128: 3124: 3119: 3115: 3107: 3097: 3093: 3089: 3084: 3083: 3082: 3078: 3074: 3070: 3069: 3068: 3065: 3059: 3058: 3057: 3053: 3049: 3045: 3044: 3043: 3042: 3038: 3034: 3030: 3029:Buridan's ass 3017: 3015: 3010: 3005: 3000: 2999: 2995: 2991: 2987: 2983: 2978: 2974: 2970: 2966: 2962: 2959: 2957: 2953: 2949: 2945: 2941: 2937: 2933: 2928: 2924: 2920: 2915: 2912: 2910: 2906: 2902: 2898: 2895: 2894: 2885: 2881: 2877: 2873: 2872: 2871: 2867: 2863: 2862:Michael Hardy 2859: 2855: 2854: 2853: 2852: 2851: 2850: 2845: 2841: 2837: 2833: 2832: 2831: 2827: 2823: 2819: 2816: 2814: 2810: 2805: 2804: 2799: 2798: 2793: 2789: 2788: 2783: 2782: 2777: 2776: 2771: 2767: 2766: 2761: 2759: 2755: 2751: 2746: 2743: 2741: 2737: 2733: 2729: 2725: 2722: 2720: 2716: 2712: 2708: 2705: 2703: 2698: 2692: 2683: 2679: 2675: 2671: 2668: 2666: 2663: 2657: 2655: 2651: 2647: 2643: 2639: 2636: 2634: 2630: 2626: 2622: 2618: 2614: 2610: 2609: 2603: 2599: 2595: 2591: 2590: 2589: 2588: 2585: 2581: 2577: 2573: 2569: 2565: 2561: 2557: 2556:coding theory 2553: 2549: 2545: 2540: 2535: 2532: 2531: 2526: 2522: 2518: 2514: 2510: 2506: 2502: 2499: 2497: 2493: 2488: 2485: 2484: 2483: 2480: 2476: 2475: 2470: 2466: 2462: 2458: 2455: 2451: 2447: 2442: 2438: 2434: 2430: 2429: 2428: 2424: 2420: 2416: 2413: 2412: 2411: 2409: 2406: 2400: 2398: 2395: 2388: 2383: 2379: 2375: 2371: 2367: 2363: 2357: 2356: 2352: 2348: 2343: 2336: 2334: 2329: 2324: 2319: 2318: 2313: 2309: 2305: 2301: 2297: 2293: 2289: 2285: 2281: 2277: 2273: 2272: 2271: 2270: 2266: 2262: 2253: 2251: 2248: 2244: 2243:130.225.212.4 2240: 2230: 2228: 2225: 2222: 2218: 2214: 2210: 2209:Modelpractice 2206: 2198: 2192: 2186: 2185: 2181: 2177: 2173: 2166: 2162: 2159: 2155: 2154: 2153: 2151: 2144: 2143: 2142: 2138: 2134: 2130: 2127: 2123: 2122: 2116: 2112: 2108: 2104: 2100: 2096: 2095: 2094: 2092: 2088: 2084: 2083:74.10.227.130 2080: 2070:Applications? 2069: 2063: 2059: 2055: 2050: 2049: 2048: 2045: 2041: 2040: 2039: 2038: 2034: 2030: 2024: 2019: 2015: 2010: 2001: 1999: 1998: 1994: 1990: 1982: 1978: 1974: 1970: 1969:PaulTanenbaum 1966: 1965: 1952: 1948: 1944: 1940: 1936: 1935:PaulTanenbaum 1932: 1927: 1921: 1917: 1913: 1909: 1904: 1903: 1902: 1901: 1900: 1899: 1898: 1897: 1896: 1895: 1894: 1893: 1892: 1891: 1890: 1889: 1874: 1870: 1866: 1861: 1860: 1859: 1858: 1857: 1856: 1855: 1854: 1853: 1852: 1851: 1850: 1849: 1848: 1835: 1831: 1827: 1823: 1819: 1815: 1811: 1810: 1809: 1805: 1801: 1796: 1795: 1794: 1790: 1786: 1782: 1778: 1777:PaulTanenbaum 1774: 1770: 1769: 1768: 1767: 1766: 1765: 1764: 1763: 1756: 1752: 1748: 1744: 1743: 1742: 1741: 1740: 1739: 1734: 1730: 1726: 1721: 1720: 1719: 1715: 1711: 1710:PaulTanenbaum 1706: 1702: 1694: 1690: 1686: 1681: 1680: 1679: 1676: 1672: 1668: 1663: 1662: 1661: 1657: 1653: 1648: 1644: 1639: 1638: 1637: 1633: 1629: 1628:PaulTanenbaum 1624: 1623: 1622: 1621: 1617: 1613: 1612:PaulTanenbaum 1608: 1604: 1602: 1598: 1597: 1592: 1588: 1584: 1580: 1572: 1570: 1569: 1565: 1561: 1560:AndrewHarvey4 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2773: 2762: 2744: 2723: 2706: 2669: 2637: 2560:cryptography 2542:: in fact, 2533: 2472: 2467:rather than 2456: 2433:WP:PRECISION 2414: 2393: 2359: 2342:no consensus 2341: 2339: 2327: 2320: 2295: 2291: 2287: 2283: 2279: 2275: 2257: 2237:— Preceding 2234: 2231:Graph Figure 2226: 2203:— Preceding 2199: 2196: 2171: 2170: 2149: 2148: 2140: 2103:Adammanifold 2099:Graph theory 2073: 2023:this version 2005: 1986: 1821: 1817: 1671:simple graph 1643:cell complex 1609: 1605: 1600: 1595: 1594: 1590: 1578: 1576: 1557: 1534: 1512: 1485:80.136.83.51 1481: 1453: 1452: 1421: 1350: 1346: 1297: 1292: 1291:a subset of 1288: 1284: 1280: 1276: 1274: 1269: 1264: 1261: 1239: 1231:Graph theory 1209: 1205: 1202: 1185:80.136.83.51 1159: 1139: 1136:Linear Graph 1126: 1123: 1117: 1114: 1106: 1098: 1084: 1075: 1071: 1042: 1032: 1010:Llygadebrill 1007: 966: 955: 904:ordered pair 899: 874: 859: 837: 833: 829: 815: 802:talk page. 800:Graph theory 797: 782: 743:— Preceding 740: 721: 689: 662: 638: 623: 620:simple graph 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