Classification of Cayley Rose Window Graphs

玫瑰窗图的稳定性

Stability of Rose Window graphs

本文研究玫瑰窗图的稳定性，证明 Wilson 于2008年提出的猜想：其所列九族不稳定玫瑰窗图恰好穷尽所有非平凡不稳定情形。

Arc Transitive Maps with underlying Rose Window Graphs

Let ${\cal M}$ be a map with the underlying graph $Γ$. The automorphism group $Aut({\cal M})$ induces a natural action on the set of all vertex-edge-face incident triples, called {\em flags} of ${\cal M}$. The map ${\cal M}$ is said to be a {\em $k$-orbit} map if $Aut({\cal M})$ has $k$ orbits on the set of all flags of ${\cal M}$. It is known that there are seven different classes of $2$-orbit maps, with only four of them corresponding to arc-transitive maps, that is maps for which $Aut{\cal M}$ acts arc-transitively on the underlying graph $Γ$. The Petrie dual operator links these four class…

所有广义玫瑰窗图都是哈密顿图

All generalized rose window graphs are hamiltonian

本文针对连通双循环图的哈密顿性提出猜想，并证明该猜想对所有四次、s=2的广义玫瑰窗图成立。

Cayley图中的信息与局部性

Information and Locality in Cayley Graphs

该研究围绕Cayley图观测问题中信息与局部性的张力，定义三类分离参数，通过碳环面、Heawood环面及有限单群示例验证参数差异与构造方法。

奇阶阿贝尔群上随机Cayley图的精确直径窗口

Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups

本文确定了奇阶阿贝尔群上随机Cayley图的直径临界窗口，证明覆盖时间收敛到泊松过程，直径转变呈Gumbel分布，并给出精确阈值常数\\(d!/2^d\\)及\\(d=2\\)时的路径-环分解。

Thompson群$F$的局部密度改进的代数枚举

Cayley-graph density of Thompson's group F: local deletions and finite-window bounds

本文通过代数枚举局部删除规则，证明Thompson群$F$的密度下界大于3.50074529，并证明固定窗口最优规则等价于加权最密子图问题。