Molecules Structure as Planar Graphs?

  1. referframe

    referframe 149
    Gold Member

    I have read that the structure of almost all organic molecules can be represented visually as Planar Graphs, i.e. 2-dimensional lattice-like structures consisting of nodes (points) connected by lines in which no lines cross. From the perspective of Physical Chemistry, does anybody know why this is so? Some graphs are of such complexity that they required 3 dimensions to guarantee that no lines cross - reference "3 Utilities Puzzle".
    Last edited: Oct 30, 2013
  2. jcsd
  3. mathman

    mathman 6,518
    Science Advisor
    Gold Member

  4. referframe

    referframe 149
    Gold Member

    Yes, actually, I was referring to the "simplification for showing on a sheet of paper" rather than to the structure in real 3D space. For most organic molecules, it is possible to represent the topological relationship of the atoms and their connections via covalent bonds on a 2D "surface" - on a plane or on a sphere. From what I have read, organic chemists do not know why this is so.
  5. Because most organic molecules do not have the degree of interconnectedness required to be non-planar.

    Kuratowski's theorem says that every nonplanar graph contains either a topological K5 or a topological K3,3. In other words, for an organic molecule to be non-planar, you have to either have:

    1. Five carbon atoms such that there is a chain (of bonds and atoms) from each one to each of the others, with none of these 10 chains sharing any bonds or atoms with any of the others; or:

    2. Two sets of atoms of valence 3 or more, such that there is a chain from each atom in the first set to each atom in the second set, with none of these 9 chains sharing any bonds or atoms with any of the others.

    To satisfy either of these conditions, you need a lot of high-order carbons and a lot of different paths between them. That would require a large and highly interconnected molecule. But it's by no means beyond the realm of possibility, and I have to imagine that some proteins, with their abundant disulfide interconnections, are non-planar (containing K3,3). It also seems like some people have synthesized non-planar molecules intentionally: see this article, for example.

    EDIT: After posting I found this article which discusses the issue further. The author seems to have a bit of confusion between nonplanarity and knottedness (you can have a knotted molecule that has a planar graph), but apart from that the analysis seems good.
    Last edited: Oct 30, 2013
  6. Ygggdrasil

    Ygggdrasil 1,617
    Science Advisor

    Adamantane comes to mind as a molecule for which this does not appear to be true.
  7. DrDu

    DrDu 4,280
    Science Advisor

    Of course there are molecules where this is not true, e.g. adamantane.
    The problem with this kind of question is which metric to use:
    All organic molecules with a CAS number? All possible organic molecules with, say, up to 100 C atoms?
  8. Borek

    Staff: Mentor

    There is a grain of truth in that, while I think I can imagine some examples of molecules that we won't be able to reproduce this way, most molecules I can think of are "topologically flat" (with all possible disclaimers about them being in fact 3d).

    No idea about explanations. And I am not convinced they are strictly chemical.

    Sorry guys, that's incorrect. Adamantan can be drawn flat - hint.
    Last edited: Oct 31, 2013
  9. DrDu

    DrDu 4,280
    Science Advisor

    Gotcha! It is K4, not K5.
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