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arXiv · 2608.29982

Equivariant Analytic Spectral Invariants of A\c{c}{\i}kme\c{s}e Lifts of Graphs with Self-Loops

Johnny Lim
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Abstract

We introduce and study equivariant analytic spectral invariants associated with the A\c{c}{\i}kme\c{s}e lift of a graph with self-loops $G_S$. The canonical $\mathbb{Z}_2$-action yields an orthogonal isotypic decomposition of the lifted Laplacian into the anti-symmetric part $\mathcal{L}(G_S)$ and the symmetric part $M_{sym}$, from which we prove that the spectrum of $\mathcal{L}(G_S)$ is exactly the even-indexed spectrum of the lifted Laplacian. We express twisted moments of these blocks as traces of the generalised twisted moment operator against the corresponding $\mathbb{Z}_2$-projections, and obtain a tight upper bound under certain restriction with a characterization of the equality case. We introduce the equivariant heat character and establish a trace-norm stability estimate. For the matrix element of the resolvent $(pI+\mathcal{L}(G_S))^{-1}$ associated with the loop vector, we derive a Laplace-transform identity for the equivariant heat character, a determinant formula, a spectral representation, and a Laurent expansion. We further characterize the case where $S$ is a union of connected components through resolvent and equivariant heat-character identities, and obtain explicit equivariant heat characters for joins of full-loop graphs and the line graph of a full-loop connected regular graph. Finally, we introduce the regularized equivariant heat integral and derive spectral and trace formulas for it.

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Johnny Lim. 2026-08-30. Equivariant Analytic Spectral Invariants of A\c{c}{\i}kme\c{s}e Lifts of Graphs with Self-Loops. https://arxiv.org/abs/2608.29982

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