Equivariant Analytic Spectral Invariants of Açıkmeşe Lifts of Graphs with Self-Loops
Abstract
We introduce and study equivariant analytic spectral invariants associated with the Açıkmeşe lift of a graph with self-loops GS. The canonical ℤ2-action yields an orthogonal isotypic decomposition of the lifted Laplacian into the anti-symmetric part ℒ(GS) and the symmetric part Msym, from which we prove that the spectrum of ℒ(GS) 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 ℤ2-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+ℒ(GS))−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.
Related
BibTeX
@misc{lim2026equivariant,
title = {{Equivariant Analytic Spectral Invariants of Açıkmeşe Lifts of Graphs with Self-Loops}},
author = {Johnny Lim},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29982},
eprint = {2608.29982},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29982},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivariant Analytic Spectral Invariants of Açıkmeşe Lifts of Graphs with Self-Loops
AU - Johnny Lim
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29982
UR - https://arxiv.org/abs/2608.29982
AB - We introduce and study equivariant analytic spectral invariants associated with the Açıkmeşe lift of a graph with self-loops G_S. The canonical ℤ_2-action yields an orthogonal isotypic decomposition of the lifted Laplacian into the anti-symmetric part ℒ(G_S) and the symmetric part M_sym, from which we prove that the spectrum of 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 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+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.
ER -