论文标题
稀疏随机图中的单环组件中给定长度的循环
Cycles of given lengths in unicyclic components in sparse random graphs
论文作者
论文摘要
令$ l $为$ \ {3,4，\ dots \} $的子集，让$ x_ {n，m}^{（l）} $是属于单轮的组件的周期数，其长度在随机图$ g（n，m）$中的长度为$ l $。我们发现$ x_ {n，m}^{（l）} $的限制分布在$ c <1/2 $和关键方案$ m = \ frac {n} {n} {2} {2} \ left（1+μn^{ - 1+μn^{ - 1/3} \ right）$（1）$（1）$（1）$（1）$（1）$（1）。取决于涉及系列$ \ sum_ {l \ in L} \ frac {z^l} {2l} $的条件，我们以poisson或正态分布为$ n \ to \ infty $。
Let $L$ be subset of $\{3,4,\dots\}$ and let $X_{n,M}^{(L)}$ be the number of cycles belonging to unicyclic components whose length is in $L$ in the random graph $G(n,M)$. We find the limiting distribution of $X_{n,M}^{(L)}$ in the subcritical regime $M=cn$ with $c<1/2$ and the critical regime $M=\frac{n}{2}\left(1+μn^{-1/3}\right)$ with $μ=O(1)$. Depending on the regime and a condition involving the series $\sum_{l \in L} \frac{z^l}{2l}$, we obtain in the limit either a Poisson or a normal distribution as $n\to\infty$.
