Talk:Graph (mathematics)

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. needs more on motivation and applications. Tompw (talk) 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. Geometry guy 16:09, 9 June 2007 (UTC)

Why are we duplicating the stuff at Graph theory? Dysprosia 23:29, 23 Sep 2003 (UTC)

Oh ok, I see now. Someone coulda told me ;) Dysprosia 23:37, 23 Sep 2003 (UTC)

[edit] Generalizations section

The previous text was this: 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. 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.Wandrer2 09:28, 26 January 2007 (UTC)

[edit] Merged graph (mathematics) into graph theory

I merged the two articles. See Wikipedia_talk:WikiProject_Mathematics#Graph_.28mathematics.29_vs_Graph_theory for a discussion. MathMartin 16:22, 9 Jan 2005 (UTC) after that he did another theory —Preceding unsigned comment added by 217.38.127.254 (talk) 17:44, 4 September 2007 (UTC)

[edit] Loops?

The current definition of "directed graph" allows loops. (And it did before I edited it!) I suggest that by default, a directed graph should not allow loops; that possibility should only be mentioned in the alternate definitions section. I'll make the change eventually unless there are objections. Dbenbenn 14:48, 13 Jan 2005 (UTC)

In my opinion it is easier to call a graph without loops simple graph and a graph with loops graph then the other way round. But you are free to change the definition. MathMartin 19:55, 18 Jan 2005 (UTC)

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 allow loops. Anyone else have an opinion? dbenbenn | talk 02:47, 19 Jan 2005 (UTC)

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 using adjectives like simple. 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. MathMartin 13:47, 19 Jan 2005 (UTC)

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. --Zero 09:21, 19 Jan 2005 (UTC)

[edit] Oriented graphs

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 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". --Zero 00:25, 17 August 2005 (UTC)

[edit] Variorum

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 Graph theory talk page. Jon Awbrey 19:10, 18 March 2006 (UTC)

[edit] Articles for edge and node

The articles edge (graph theory), node (graph theory) and node (computer science) are essentially impossible to write since you cannot explain edge without explaining node and graph (graph theory) and vice-versa. Therefore, I made these articles redirect to graph (graph theory). ylloh 01:01, 15 April 2006 (UTC)

[edit] Citation Style

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 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. Jon Awbrey 13:30, 27 June 2006 (UTC)

[edit] unordered pair?

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 Axiom of the unordered pair, is not what we want here, is it?). --Beaumont (@) 16:04, 18 October 2006 (UTC)

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 not an ordered pair. Perhaps a definition of the term "unordered pair" should be added to the "ordered pair" article, which unordered pair could then link to. Paul August ☎ 17:15, 18 October 2006 (UTC)

Ok I've now done the above. Paul August ☎ 17:25, 18 October 2006 (UTC)

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 make it clear in the article --Beaumont (@) 22:57, 4 November 2006 (UTC)

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, a} to be a "pair" with non-distinct elements (See Balakrishnan, V. K.; Schaum's Outline of Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0070054894.) 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; Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0849339820.) Paul August ☎ 23:55, 4 November 2006 (UTC)

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 vertex but nevertheless add only 1 to the row and column sums of the adjacency matrix. McKay 03:00, 5 November 2006 (UTC)

[edit] Definition of a Graph

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. --Llygadebrill 21:11, 26 February 2007 (UTC)

Done. YOu could have done it yorself. `'mikka 21:32, 26 February 2007 (UTC)

[edit] Definition of a Vertex

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 (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? hmoraldo

You are right that the wording is sub-optimal but your suggested fix is not good. The story is, first, that V 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. McKay 03:44, 12 March 2007 (UTC)

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

[edit] Quivers vs. representations of quivers

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?

Thanks SophomoricPedant 04:19, 3 June 2007 (UTC)

Since no one responded, I will adjust the wording to reflect the above distinction SophomoricPedant 04:19, 3 June 2007 (UTC)

[edit] Implementation

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 Wikipedia article for it that could be cited from here. 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. Here you can get familiar with it.

Phegyi81 14:15, 14 June 2007 (UTC)

You dit it real nice.

?? ??

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

Phegyi81 11:14, 2 July 2007 (UTC)

[edit] Linear Graph

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? MagiMaster (talk) 22:13, 11 February 2008 (UTC)

I think graphs im Maths are too complicated and you never use them in every day life (jobs) —Preceding unsigned comment added by 88.108.216.138 (talk) 17:23, 27 February 2008 (UTC)

[edit] Include Euler and Hamiltonian graphs

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

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.

--Tangi-tamma (talk) 20:32, 6 April 2008 (UTC)

The Graph (mathematics) article covers the definition and basic properties of graphs, whereas the Graph theory 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 Wikipedia:Merging_and_moving_pages#Proposing_a_merger. The cleanup tags are not appropriate for this situation, so I have removed them from both articles. Gandalf61 (talk) 08:16, 7 April 2008 (UTC)

[edit] Definitions

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 carefully fairly confused. The main definition of a graph 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 one edge between any two different vertices"? Sounds like a tautology to me. In fact what is a self-loop in the sense of the main graph definition anyway?

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 G=(V,E,j) where j is a function that assigns to every element e of E a subset of V consisting of one or two elements.

I understand that this is a Wikipedia article, not a mathematical textbook, but shouldn't the formal general definition of a pseudo-graph be contained somewhere in the article? Nsk92 (talk) 04:11, 21 May 2008 (UTC)

I agree. I have tried to generalise the initial definition so that it defines a pseudograph (which actually redirects to multigraph in Wikipedia). This is now consistent with the definition sequence in glossary of graph theory 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 ! Gandalf61 (talk) 09:14, 21 May 2008 (UTC)

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 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 multiset 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 V and E 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 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. Nsk92 (talk) 12:09, 21 May 2008 (UTC)

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 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. Gandalf61 (talk) 13:33, 21 May 2008 (UTC)

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 actually tend to think about a graph as a 1-dimentional CW-complex. But I am certainly not going to advocate that approach here. Nsk92 (talk) 14:06, 21 May 2008 (UTC)