Abstract
Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai, and Adin, and more recently by numerous authors, the fundamental topological properties of a tree — namely acyclicity and connectedness — can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley’s formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.
Keywords
- Tree
- Forest
- Spanning tree
- Matrix-tree theorem
- Matroid
- Simplicial complex
- Cell complex
- Combinatorial laplacian
- Critical group
Mathematics Subject Classification (2010):
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Notes
- 1.
- 2.
There is a slight notational error in [47], where the L Γ in Prop. 3.5 should refer to the reduced Laplacian formed by removing the rows and columns corresponding to cells of Γ.
- 3.
One needs the convention that w ∅ = 1, so that L −1 alg(Q n ) is just the 1 × 1 matrix with entry \(\sum _{v\in V (Q_{n})}w_{v} =\prod _{ j=1}^{n}(x_{j} + y_{j})\).
- 4.
In [44], it was erroneously stated that \(L_{k}^{\mathrm{ud}}(X) = L_{d-k}^{\mathrm{ud}}(Y )\). In fact, duality interchanges up-down and down-up Laplacians.
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We thank our colleagues Amy Wagler and Jennifer Wagner for valuable suggestions regarding exposition and clarity.
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Duval, A.M., Klivans, C.J., Martin, J.L. (2016). Simplicial and cellular trees. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_28
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