AI 中文总结
该研究针对带自环图的Açıkmeşe提升，利用$\text{Z}_2$作用分解拉普拉斯算子，推导了相关谱不变量的系列公式，包括预解式的洛朗展开、正则化热积分的谱公式等。
AI 中文摘要
我们引入并研究与带自环图$G_S$的Açıkmeşe提升相关的等变解析谱不变量。标准$\boldsymbol{\text{Z}_2}$作用将提升后的拉普拉斯算子正交同型分解为反对称部分$\boldsymbol{\text{L}(G_S)}$和对称部分$\boldsymbol{M_{\text{sym}}}$，由此证明$\boldsymbol{\text{L}(G_S)}$的谱恰好是提升后拉普拉斯算子的偶索引谱。我们将这些块的扭曲矩表示为广义扭曲矩算子与对应$\boldsymbol{\text{Z}_2}$投影的迹，并在一定限制下得到紧上界，同时刻画等号成立的情况。我们引入等变热特征并建立迹范数稳定性估计。对于预解式$\boldsymbol{(pI+\text{L}(G_S))^{-1}}$与环向量相关的矩阵元，我们推导了等变热特征的拉普拉斯变换恒等式、行列式公式、谱表示及洛朗展开。我们还通过预解式和等变热特征恒等式刻画$S$为连通分支并集的情况，并得到全环图的联图及全环连通正则图的线图的显式等变热特征。最后，我们引入正则化等变热积分并推导其谱公式和迹公式。
英文摘要
We introduce and study equivariant analytic spectral invariants associated with the Açıkmeşe 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.
Comments37 pages, 1 figure. Comments are welcome!
