Simplicial and Cellular Trees
Art M. Duval, Caroline J. Klivans, Jeremy L. Martin

TL;DR
This paper develops a higher-dimensional theory of cellular trees, extending classical graph concepts like spanning trees and enumeration formulas to cell complexes using topological and homological methods.
Contribution
It introduces a framework for cellular trees based on homology, generalizing classical graph results and incorporating torsion phenomena in higher dimensions.
Findings
Higher-dimensional analogues of Cayley's formula and the matrix-tree theorem.
Extension of algebraic graph theory concepts to cell complexes.
Inclusion of torsion homology in the enumeration of cellular trees.
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…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Data Visualization and Analytics · Homotopy and Cohomology in Algebraic Topology
