Transfer Operators and Independence Polynomials for Strong Powers of Circulant Graphs

TL;DR

This paper analyzes independent sets in strong powers of circulant graphs using transfer operators, revealing spectral properties and exact independence polynomial computations for specific graph structures.

Contribution

It introduces a transfer matrix approach that decomposes the spectral analysis into trivial and cyclotomic components, enabling exact calculations of independence polynomials.

Findings

Spectral radius is achieved in the trivial isotypic component.

Exact independence polynomials are computed for strong cylinders and tori.

Cyclotomic sector contributes sparse corrections to the independence polynomial.

Abstract

We study independent sets in strong powers of circulant graphs using a transfer matrix formulation. The compatibility constraints separate into intra-layer and inter-layer components, yielding a transfer operator that is equivariant under the dihedral group action. The characteristic polynomial of the transfer operator factors into an \emph{anomalous} component (arising from the trivial isotypic component, with rational coefficients) and a \emph{cyclotomic} component (arising from nontrivial Fourier modes, splitting over the maximal real cyclotomic subfield). We show that the spectral radius is attained in the trivial isotypic component, so the dominant exponential growth is governed by a low-dimensional orbit-compressed operator. The independence polynomial is computed exactly for strong cylinders and tori, with the cyclotomic sector contributing a sparse correction confined to high-weight coefficients. All results are verified for $C_7$.