Path homology of circulant digraphs

TL;DR

This work develops a Fourier-flavored framework for GLMY path homology on circulant digraphs $\vec{C}_n^S$, reducing computations to finite-dimensional $\tau$-eigenspace blocks and exposing how arithmetic of $n$ and the additive structure of $S$ governs low-dimensional chains and Betti numbers. It provides explicit bases and symbol-matrix descriptions for key cases (notably $\vec{C}_5^{1,2}$) and proves a stability phenomenon in the no-wrap-around regime: beyond a finite cyclotomic set $\mathcal{Q}_+(S)$, nontrivial Fourier modes vanish and Betti numbers stabilize for almost all primes $p$. The paper also constructs chain maps and homotopies to relate larger connection sets to canonical small ones, establishing a deformation-retraction style viewpoint that explains when $H_*^{\mathrm{path}}$ agrees with simpler models. Finally, it connects the directed circulant picture to the undirected circulant graphs via discrete tori, providing concrete Betti-number computations for several families and conjecturing a general zero-for-$m\ge3$ pattern in the no-wrap regime.

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 $τ$ and a Fourier decomposition, we reduce many rank computations for the GLMY boundary maps to finite-dimensional $τ$-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.

Figures (3)

• Figure 1: circulant digraph $G=\vec{C}_5^{1,2}$
• Figure 2: circulant digraph $\vec{C}_7^{1,3}$
• Figure 3: circulant graph $C_7^{1,3}$ as a symmetric digraph

Theorems & Definitions (58)

• Theorem 1.1: Lemma \ref{['basis']}, Theorem \ref{['thm1']}
• Theorem 1.2: Theorem \ref{['thm:1s']}, Theorem \ref{['thm:S12d']}
• Proposition 1.3: Proposition \ref{['prop:Fourier']}
• Theorem 1.4: Theorem \ref{['thm:strong-stability']}
• Proposition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof