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Vector bundles over manifolds have fundamental importance in differential geometry, algebraic topology etc. Are there any applications of this concept (or some variation of it) for graphs (finite or infinite)?

The only place I have seen something like this is in a paper on spanning forests by Kenyon, where the application seems somewhat specialized.

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    $\begingroup$ Real n-plane bundles over a finite graph are classified by the first cohomology of the graph with $\mathbb{Z}_2$ coefficients. Complex $n$-plane bundles are all trivial. $\endgroup$ Oct 29, 2011 at 18:02
  • $\begingroup$ This is not exactly what you're looking for, but Baker and Norine, in their paper "Harmonic morphisms and hyperelliptic graphs" (arxiv.org/abs/0707.1309) define an analog of differential forms which they call harmonic forms (section 4.3 on page 17). $\endgroup$ Oct 29, 2011 at 19:11
  • $\begingroup$ @Charlie Frohman: any reference? $\endgroup$ Oct 29, 2011 at 19:38
  • $\begingroup$ I don't believe that complex bundles are trivial. For instance, the circle can be represented as a graph, and has nontrivial complex vector bundles. I would use the interpretation of vector bundles as representations of the fundamental group. The fundamental group of a graph is just a free group on (Edges-Vertices+1) generators, and so vector bundles are just the same as choosing (Edges-Vertices+1) n by n matrices. $\endgroup$
    – Will Sawin
    Oct 29, 2011 at 20:50
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    $\begingroup$ If you want similar applicability to the theory of vector bundles over manifolds, transported into graphs you're going to have to look further-afield than vector bundles. Manifolds are homogeneous, but graphs generally are not. So you'll want to look at a less homogeneous corresponding idea, like sheaves and singular fiber bundles. For example, this is what you need if you want to talk about regular neighbourhoods. $\endgroup$ Oct 29, 2011 at 21:52

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Another cool paper is http://arxiv.org/abs/1107.5588 Where they talk about line bundles on graphs (and define very nice new class of integrable systems).

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Have a look at this paper http://arxiv.org/abs/0912.4048 by Cappel and Miller, they called them "transmitions" and generalize some of the standard spectral graph theory. Well transmitions are more general than vector bundles over graphs, they somehow correspond to vector bundles over directed graphs....more or less. Have a look, they have some applications at the end of the paper.

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