Path homology of circulant digraphs

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.