Heat kernels on Cayley graphs of free products of finite groups
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.