On Cayley's Enumeration of Alkanes (or 4-Valent Trees)

二面体群上的4度Cayley图的自同构群与结构

Automorphism Groups and Structure of 4-Valent Cayley Graphs on Dihedral Groups

研究二面体群上4度Cayley图的自同构群与结构，通过分析不同生成集的Cayley图性质，确定其分类与构造方法。

Arc-transitive Cayley graphs on non-ableian simple groups with soluble vertex stabilizers and valency seven

In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let $\Ga$ be such a Cayley graph on a non-abelian simple group $T$. It is proved that either $\Ga$ is a normal Cayley graph or $\Ga$ is $S$-arc-transitive, with $(S,T)=(\A_n,\A_{n-1})$ and $n=7,21,63$ or $84$; and, for each of these four values of $n$, there really exists arc-transitive $7$-valent non-normal Cayley graphs on $\A_{n-1}$ and specific examples are constructed.

Frobenius circulant graphs of valency six, Eisenstein-Jacobi networks, and hexagonal meshes

A Frobenius group is a transitive but not regular permutation group such that only the identity element can fix two points. A finite Frobenius group can be expressed as $G = K \rtimes H$ with $K$ a nilpotent normal subgroup. A first-kind $G$-Frobenius graph is a Cayley graph on $K$ with connection set $S$ an $H$-orbit on $K$ generating $K$, where $H$ is of even order or $S$ consists of involutions. We classify all 6-valent first-kind Frobenius circulant graphs such that the underlying kernel $K$ is cyclic. We give optimal gossiping and routing algorithms for such a circulant and compute its fo…

Asymptotic enumeration of vertex-transitive graphs of fixed valency

Let $G$ be a group and let $S$ be an inverse-closed and identity-free generating set of $G$. The \emph{Cayley graph} $\Cay(G,S)$ has vertex-set $G$ and two vertices $u$ and $v$ are adjacent if and only if $uv^{-1}\in S$. Let $CAY_d(n)$ be the number of isomorphism classes of $d$-valent Cayley graphs of order at most $n$. We show that $\log(CAY_d(n))\inΘ(d(\log n)^2)$, as $n\to\infty$. We also obtain some stronger results in the case $d=3$.

度数为7的非交换单群上的对称Cayley图

Symmetric Cayley graphs on non-abelian simple groups of valency 7

该研究改进了非正规7度对称Cayley图的已有刻画，证明其自同构群含特定弧传递非交换单正规子群，明确了群对结构与最大可解正规子群相关性质。