Asymptotics for the number of spanning trees in circulant graphs and degenerating d-dimensional discrete tori

A formula for the number of spanning trees in circulant graphs with non-fixed generators and discrete tori

We consider the number of spanning trees in circulant graphs of $βn$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $βn$ factors, while we derive a formula of $β-1$ factors. Using the same trick, we also derive a formula for the number of spanning trees in discrete tori. Moreover, the spanning tree entropy of circulant graphs with fixed and non-fixed generators is compared.

Zeta functions, heat kernels and spectral asymptotics on degenerating families of discrete tori

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with generating set given by choosing a generator for each cyclic factor. In this article we study the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First we show that the sequence of heat kernels corresponding to the degenerating family converges, after re-scaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of t…

Homotopy type of the neighborhood complexes of graphs of maximal degree at most $3$ and $4$-regular circulant graphs

To estimate the lower bound for the chromatic number of a graph $G$, Lovász associated a simplicial complex $\mathcal{N}(G)$ called the neighborhood complex and relates the topological connectivity of $\mathcal{N}(G)$ to the chromatic number of $G$. More generally he proved that the chromatic number of $G$ is bounded below by the topological connectivity of $\mathcal{N}(G)$ plus $3$. In this article, we consider the graphs of maximal degree at most $3$ and $4$-regular circulant graphs. We show that each connected component of the neighborhood complexes of these graphs is homotopy equivalent ei…

传递算子与强幂环形图的独立多项式

Transfer Operators and Independence Polynomials for Strong Powers of Circulant Graphs

本文研究环形图强幂中的独立集，通过传递矩阵方法，分离内层和外层约束，得到等变于二面体群作用的传递算子，并证明其特征多项式分解为异常和环状部分，通过C7验证结果。