有限群自由积的Cayley图上热核
Heat kernels on Cayley graphs of free products of finite groups
- Université de Genève（日内瓦大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对两类有限群自由积的Cayley图，利用强正则覆盖与谱传递原理，推导出归一化拉普拉斯热核的显式公式，并刻画了相应谱结构。
AI中文摘要：
我们获得了两个无限族Cayley图的归一化拉普拉斯算子的热核的显式公式。第一个族对应于有限群$G=G_1*\dots*G_r$的自由积，其中$r\ge2$，且所有因子$G_i$具有相同的阶$L\ge2$。第二个族对应于两个阶不相等的任意非平凡有限群$G*H$的自由积。在这两个族中，生成集由每个因子中的所有非单位元组成。我们扩展了Chung--Yau的方法，证明了每个族中的Cayley图分别强正则覆盖一个加权半直线和一个加权直线。我们解决了半直线和直线上的谱问题，然后应用强覆盖和正则覆盖的谱传递原理推导出Cayley图上热核的公式。对于第一个族，拉普拉斯算子的谱由一个区间组成，当$L>r$时，还有一个额外的孤立特征值。当$L=2$时，我们恢复了正则树的已知结果。对于第二个族，拉普拉斯算子的谱由两个区间和两个特征值组成。
英文摘要：
We obtain explicit formulas for heat kernels of the normalized Laplacians for two infinite families of Cayley graphs. The first family corresponds to free products of finite groups $G=G_1*\dots*G_r$, with $r\ge2$, and where all factors $G_i$ have the same order $L\ge2$. The second family corresponds to free products of two arbitrary nontrivial finite groups $G*H$ of unequal order. In both families, the generating set consists of all non-identity elements in each factor. Extending the approach of Chung--Yau, we show that the Cayley graph in each family strongly and regularly covers a weighted half-line and a weighted line, respectively. We solve the spectral problems on the half-line and the line, and then apply spectral transfer principles for strong and regular coverings to derive formulas for the heat kernels on the Cayley graphs. For the first family, the spectrum of the Laplacian consists of a single interval together with, when $L>r$, one additional isolated eigenvalue. When $L=2$, we recover the well-known results for regular trees. For the second family, the spectrum of the Laplacian consists of two intervals and two eigenvalues.
