Simplicial and Cellular Trees

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 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.