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The Cellular Homology of Digraphs

Xinxing Tang, Shing-Tung Yau

TL;DR

This work builds a cellular homology theory for digraphs by restricting GLMY path theory to strongly regular minimal paths that admit singular cubical realizations. Central to the construction is the admissible-pair framework $(P, ext{Supp}(P))$, which yields a quotient cellular chain complex $C_n(G;R)$ and a boundary $partial^{ ext{cell}}$ compatible with the path-boundary on supports; the resulting homology $H^{ ext{cell}}_*(G;R)$ enjoys functoriality and, in the acyclic setting, homotopy invariance. The authors establish a Künneth-type formula for digraph products and demonstrate the theory on circulant and Johnson digraphs to illustrate computations and deformation-retraction arguments. They further discuss the relationship between cellular and singular cubical homology, conjecturing an isomorphism $H^{ ext{cell}}_*(G;R)\u2261 H^c_*(G;R)$ for acyclic graphs and interpreting admissible relations as cubical-boundary conditions. Overall, the paper provides a robust bridge between discrete digraph topology and classical topological invariants, offering concrete computation tools and a geometric interpretation via CW-type structures on digraphs.

Abstract

In \cite{TY}, we investigate the pair $(P, \Supp(P))$ of minimal path $P$ and its supporting sub-digraph $\Supp(P)$ in the path complex of a digraph $G$ under the strongly regular condition. In this paper, first, we consider the special minimal path $P$ specified by the admissible condition (Definition \ref{admpair}), which means that $(P,\Supp(P))$ admits a singular cubical realization. Based on such a subset, we systematically introduce the definitions of cellular chain complex associated to $G$ and prove the well-definedness. Then we study several properties of such cellular homologies. Finally, we present several intriguing examples as well as some important observations.

The Cellular Homology of Digraphs

TL;DR

This work builds a cellular homology theory for digraphs by restricting GLMY path theory to strongly regular minimal paths that admit singular cubical realizations. Central to the construction is the admissible-pair framework , which yields a quotient cellular chain complex and a boundary compatible with the path-boundary on supports; the resulting homology enjoys functoriality and, in the acyclic setting, homotopy invariance. The authors establish a Künneth-type formula for digraph products and demonstrate the theory on circulant and Johnson digraphs to illustrate computations and deformation-retraction arguments. They further discuss the relationship between cellular and singular cubical homology, conjecturing an isomorphism for acyclic graphs and interpreting admissible relations as cubical-boundary conditions. Overall, the paper provides a robust bridge between discrete digraph topology and classical topological invariants, offering concrete computation tools and a geometric interpretation via CW-type structures on digraphs.

Abstract

In \cite{TY}, we investigate the pair of minimal path and its supporting sub-digraph in the path complex of a digraph under the strongly regular condition. In this paper, first, we consider the special minimal path specified by the admissible condition (Definition \ref{admpair}), which means that admits a singular cubical realization. Based on such a subset, we systematically introduce the definitions of cellular chain complex associated to and prove the well-definedness. Then we study several properties of such cellular homologies. Finally, we present several intriguing examples as well as some important observations.
Paper Structure (27 sections, 18 theorems, 250 equations, 32 figures)

This paper contains 27 sections, 18 theorems, 250 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Organization
  4. Path Complex and Minimal Path
  5. Path complex
  6. Minimal path
  7. The Cellular Homology of Digraphs
  8. The singular cubical complex of digraphs
  9. Admissible pair and admissible relations
  10. Admissible pair
  11. Admissible relations
  12. CW complex of the digraph
  13. Cellular homology of a digraph
  14. Properties of cellular homology of digraphs
  15. Functoriality and Homotopy invariance
  16. ...and 12 more sections

Key Result

Theorem 2.10

Let $P\in\Omega_n(G;\mathbb Z)$ be a minimal path with the starting vertex $S$ and ending vertex $E$, $\mathop{\mathrm{Supp}}\nolimits(P)$ be its supporting digraph and $d_S$, $d_E$ be the functions defined above. (1) Let $S_1=d_S^{-1}(1)$ and $E_1=d_E^{-1}(1)$. Then $P$ is a linear combination of s where (2) For any $v\in d_E^{-1}(k)\cap\mathop{\mathrm{Supp}}\nolimits(P)$, in $\mathop{\mathrm{Su

Figures (32)

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  • ...and 27 more figures

Theorems & Definitions (94)

  • Definition 1.2: Definition \ref{['admpair']}, \ref{['admrel']}
  • Definition 1.3: Definition \ref{['cellular']}
  • Conjecture 1.4: Conjecture \ref{['conj']}
  • Conjecture 1.5: Conjecture \ref{['contractible']}
  • Definition 2.1: HY
  • Definition 2.2
  • Remark 2.3: GLMY's regular condition and GLMY's path complex
  • Definition 2.4: HY
  • Definition 2.5: TY
  • Example 2.6
  • ...and 84 more