给定度序列的随机稀疏图中的树
Trees in Random Sparse Graphs with a Given Degree Sequence
AI总结：
本文研究给定度序列及几乎给定度序列的随机稀疏图的有限树子图结构，重新推导稠密与极稀疏图的树结构并形成连续图景，还证明了无额外条件下随机二分图的对应结果。
摘要：
设$\boldsymbol{G}^D$为顶点数为$n$、度序列为$D=(d_1,d_2,\boldsymbol{\text{...}},d_n)$的图$G(V,E)$的集合。此外，对于$\frac{1}{2}<a<1$，我们将具有几乎给定度序列$D$的图集合定义为：$\boldsymbol{G}_a^D:=\bigcup\boldsymbol{G}^{\bar{D}}$，其中并集遍历所有满足$1\boldsymbol{\text{≤}}i\boldsymbol{\text{≤}}n$时$\boldsymbol{|}d_i-\bar{d}_i\boldsymbol{|}<d_i^a$的度序列$\bar{D}$。现在，若分别从集合$\boldsymbol{G}^D$和$\boldsymbol{G}_a^D$中均匀选取随机图$\boldsymbol{G}_{\boldsymbol{g}}(D)$和$\boldsymbol{G}_{\boldsymbol{a}}(D)$，它们会是什么样的？当$\boldsymbol{G}_{\boldsymbol{g}}(D)$是稠密图（即$\boldsymbol{|}E\boldsymbol{|}=\boldsymbol{\text{Θ}}(n^2)$）时，已通过图同伦（graphons）研究过该问题；当$\boldsymbol{G}_{\boldsymbol{g}}(D)$非常稀疏（即$d_n^2=o(\boldsymbol{|}E\boldsymbol{|})$）时，也有相关研究。在具有几乎给定度序列的稀疏图的情形下，我们研究该问题，并在一些温和条件下给出$\boldsymbol{G}_{\boldsymbol{a}}(D)$的有限树子图结构。对于具有给定度序列的随机图$\boldsymbol{G}_{\boldsymbol{g}}(D)$，我们重新推导其在稠密和极稀疏情形下的有限树结构，以形成连续的图景。此外，对于向量对$(D_1,D_2)\boldsymbol{\text{∈}}\boldsymbol{Z}^{n_1}\times\boldsymbol{Z}^{n_2}$，我们令$\boldsymbol{G}_{\boldsymbol{b}}(D_1,D_2)$为从集合$\boldsymbol{G}^{D_1,D_2}$中均匀选取的随机二分图，其中$\boldsymbol{G}^{D_1,D_2}$是所有度序列为$(D_1,D_2)$的二分图的集合，我们能够在无任何额外条件的情况下证明$\boldsymbol{G}_{\boldsymbol{b}}(D_1,D_2)$的相关结果。
英文摘要：
Let $\mathbb{G}^{D}$ be the set of graphs $G(V,\, E)$ with $\left|V\right|=n$, and the degree sequence equal to $D=(d_{1},\, d_{2},\,\dots,\, d_{n})$. In addition, for $\frac{1}{2}<a<1$, we define the set of graphs with an almost given degree sequence $D$ as follows, \[ \mathbb{G}_{a}^{D}:=\cup\,\mathbb{G}^{\bar{D}}, \] where the union is over all degree sequences $\bar{D}$ such that, for $1\leq i\leq n$, we have $\left|d_{i}-\bar{d}_{i}\right|<d_{i}^{a}$. Now, if we chose random graphs $\mathcal{G}_{\mathbf{g}}\left(D\right)$ and $\mathcal{G}_{\mathbf{a}}\left(D\right)$ uniformly out of the sets $\mathbb{G}^{D}$ and $\mathbb{G}_{a}^{D}$, respectively, what do they look like? This has been studied when $\mathcal{G}_{\mathbf{g}}\left(D\right)$ is a dense graph, i.e. $\left|E\right|=Θ(n^{2})$, in the sense of graphons, or when $\mathcal{G}_{\mathbf{g}}\left(D\right)$ is very sparse, i.e. $d_{n}^{2}=o(\left|E\right|)$. In the case of sparse graphs with an almost given degree sequence, we investigate this question, and give the finite tree subgraph structure of $\mathcal{G}_{\mathbf{a}}\left(D\right)$ under some mild conditions. For the random graph $\mathcal{G}_{\mathbf{g}}\left(D\right)$ with a given degree sequence, we re-derive the finite tree structure in dense and very sparse cases to give a continuous picture. Moreover, for a pair of vectors $\left(D_{1},D_{2}\right)\in\mathbb{Z}^{n_{1}}\times\mathbb{Z}^{n_{2}}$, we let $\mathcal{G}_{\mathbf{b}}\left(D_{1},D_{2}\right)$ be the random bipartite graph that is chosen uniformly out of the set $\mathbb{G}^{D_{1},D_{2}}$, where $\mathbb{G}^{D_{1},D_{2}}$ is the set of all bipartite graphs with the degree sequence $\left(D_{1},D_{2}\right)$. We are able to show the result for $\mathcal{G}_{\mathbf{b}}\left(D_{1},D_{2}\right)$ without any further conditions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
