Path homology of circulant digraphs

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 $\ddot{I}„$ and a Fourier decomposition, we reduce many rank computations for the GLMY boundary maps to finite-dimensional $\ddot{I}„$-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.

30 pages, 3 figures