论文标题
Cayley图中的常规套装的注释
A note on regular sets in Cayley graphs
论文作者
论文摘要
图$γ$的顶点的子集$ r $据说为$（κ，τ）$ - 如果$ r $诱导$κ$ regratular subgraph，并且$ r $以外的每个顶点与$ r $ in $ r $ in $ r $的每个顶点恰好相邻。特别是，如果$ r $是$（κ，τ）$ - 有限的$ g $上的一些cayley图，则$ r $称为$（κ，τ）$ - 常规套件$ g $。令$ h $为$ g $的非平凡的普通亚组，$κ$和$τ$每对满足$ 0 \leqκ\ leq | h | -1 $，$ 1 \ leq leq leq leq leq | h | $和$ \ gcd（2，| h | -1）\midκ$。证明（i）如果$τ$甚至是$ h $，则$ h $是$（κ，τ）$ - 常规$ g $; （ii）如果$τ$是奇数的，则$ h $是$（κ，τ）$ - 常规集合$ g $时，并且仅当它是$（0,1）$ - 常规套件的$ g $。
A subset $R$ of the vertex set of a graph $Γ$ is said to be $(κ,τ)$-regular if $R$ induces a $κ$-regular subgraph and every vertex outside $R$ is adjacent to exactly $τ$ vertices in $R$. In particular, if $R$ is a $(κ,τ)$-regular set of some Cayley graph on a finite group $G$, then $R$ is called a $(κ,τ)$-regular set of $G$. Let $H$ be a non-trivial normal subgroup of $G$, and $κ$ and $τ$ a pair of integers satisfying $0\leqκ\leq|H|-1$, $1\leqτ\leq|H|$ and $\gcd(2,|H|-1)\midκ$. It is proved that (i) if $τ$ is even, then $H$ is a $(κ,τ)$-regular set of $G$; (ii) if $τ$ is odd, then $H$ is a $(κ,τ)$-regular set of $G$ if and only if it is a $(0,1)$-regular set of $G$.
