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arXiv · 2602.04140

Path homology of circulant digraphs

Xinxing Tang·Shing-Tung Yau
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Abstract

We organize and extend a set of computations and structural observations about the Grigoryan--Lin--Muranov--Yau (GLMY) path complex of circulant digraphs $\vec{C}_n^S$ and circulant graphs $C_n^S$. Using the shift automorphism $\tau$ and a Fourier decomposition, we reduce many rank computations for the GLMY boundary maps to finite-dimensional $\tau$-eigenspaces. This provides a reusable "symbol-matrix" recipe that highlights (i) the dependence on prime versus composite $n$ and (ii) stability phenomena for certain natural choices of connection sets $S$. Several fully worked examples are included, together with a discussion of how the additive structure of $S$ governs low-dimensional chains and Betti numbers.

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Xinxing Tang, Shing-Tung Yau. 2026-02-04. Path homology of circulant digraphs. https://arxiv.org/abs/2602.04140

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