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arXiv · 2402.05682

The Cellular Homology of Digraphs

Xinxing Tang·Shing-Tung Yau
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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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Xinxing Tang, Shing-Tung Yau. 2024-02-08. The Cellular Homology of Digraphs. https://arxiv.org/abs/2402.05682

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