Wikipedia:Reference desk/Archives/Mathematics/2007 February 26

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February 26

Several proofs of the Goldbach conjecture on arXiv, how come?

Nearly all litterature, websites and articles I have read have stated that as of now there is no proof of the Goldbach conjecture. However should one continue to search for "proof goldbach conjecture" on arXiv [1] 6 supposed "proofs" turn up. Most might appreciate that these are not proofs of the the GC, but merely attempts by undergraduates. But if they are not proofs why are they still listed on the site as "proofs", a person who does not know so much about the GC might come to believe that there actually is a proof. On the other hand if one of these proofs holds up to scrutiny how does one get to know about by not reading about it in a maths journal.

Basically this question is about the workings on arXiv, who checks that the files uploaded on the page are accurate? I myself am not qualified to do so as I am only an undergraduate in maths.

Although there are certain systems in place that are designed to prevent anyone uploading anything that doesn't belong, the detailed content of files uploaded to any of the different arXiv lists are not checked by anyone, and you can sometimes find all manner of "interesting" proofs and scientific theories on there. Many researchers wait until having a paper approved for publication by a refereed journal before posting a paper there, or at least until they've submitted it to one, but not all, and many independent researchers have no such option. Take a look at Arxiv#Peer-review. Caveat lector, basically. Spiral Wave 11:51, 26 February 2007 (UTC)

What is the relationship between graph theory and topology?

Graph theory is much like a 2d topology. But there is such a thing as weighted graph theory, e.g. a problem related to travel time or fiber length. Topology doesn't seem to put much focus on "distance", as it is a geometry of position. The reason I ask is that the Konigsberg bridges and n-color theorems are covered in entry level books on graph theory *and* topology.172.146.58.73 08:58, 26 February 2007 (UTC)

Sure. Many problems in graph theory deal with weighted graphs. Some examples are finding the minimum spanning tree of a graph, finding the shortest distance between two vertices in a graph (see Dijkstra's algorithm and breadth-first search), and the famous travelling salesman problem. —Bkell (talk) 09:07, 26 February 2007 (UTC)

There is an answer at topological graph theory, but I have to say I found the description at PlanetMath more focused. 84.239.129.42 21:51, 26 February 2007 (UTC)

For the most part these are two distinct fields, with different interests and different methods. Formally, we might describe a graph using the language of topology (for example, as a CW complex), but that offers little. The Euler characteristic is important in both algebraic topology and graph theory; but considering that the former discretizes topology, the overlap is not so surprising.

Consider a tetrahedron with vertices on the surface of a sphere, projected onto the sphere. As topologists, we can consider S² itself. As graph theorists we can see the four vertices and the six edges connecting them. As algebraic topologists, we can count faces, edges, and vertices to compute an Euler characteristic, or we can compute Betti numbers via homology groups to reach the same conclusion. A topologist might ask if the space is compact or connected or path connected or the same as, say, a torus. A graph theorist might ask if the graph was complete or a tree or had multiple components or cycles. It happens that planarity forces significant constraints on a graph (see planar graph), and by puncturing the sphere we can make the graph derived from the tetrahedron planar (see Schlegel diagram). --KSmrq^T 02:31, 27 February 2007 (UTC)