The Cellular Homology of Digraphs

Shing-Tung Yau; Xinxing Tang

The Cellular Homology of Digraphs

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arxiv 2402.05682 v2 pith:TJP3TH3X submitted 2024-02-08 math.CO math.AT

Xinxing Tang , Shing-Tung Yau This is my paper · ORCID

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keywords cellularpathsuppcomplexconditionminimalseveraladmissible

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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.

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Primitive path homology
math.AT 2024-11 conditional novelty 6.0 of 10

Primitive path homology is a new digraph invariant that coincides with GLMY path homology on asymmetric digraphs and differs on symmetric ones.

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