张成 PG(7,2) 的线四元组

spanning line configurations 的变种的 Grothendieck 组

The Grothendieck Group of the Variety of Spanning Line Configurations

研究 spanning line configurations 变种的 Grothendieck 组，证明其与广义 coinvariant 代数同构，并扩展了经典 flag 变种的同构结果。

Formulas counting spanning trees in line graphs and their extensions

For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $τ_G(M)$ is obtained, where $τ_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanni…

有向线图的沙堆群和生成树

Sandpile groups and spanning trees of directed line graphs

推广了Knuth关于有向图G及其有向线图LG的有向生成树的一个定理.沙堆群是与有向图相联系的阿贝尔群，其阶是以固定顶点为根的有向生成树的个数.当G是k次正则群时，证明了G的沙堆群同构于LG的沙堆群的k-挠子群的商.作为一个推论，我们计算的沙堆群的两个家庭的图广泛研究在计算机科学中，德Bruijn图和Kautz图。

The Turán Number for Spanning Linear Forests

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on $n$ vertices, then we often call the problem a spanning Turán problem. A linear forest is a graph whose connected components are all paths and isolated vertices. In this paper, we let $\mathcal{L}_n^k$ be the set of all linear forests of order $n$ with at least $n-k+1$ edges. We prove that when $n\geq 3k$ and $k\geq 2$, \[ ex(n;\mathcal{L}_n^k)=\binom{n-k+1}{2…

有向线图的生成树计数

Counting the spanning trees of a directed line graph

有向图G的线图LG对G的每条边都有一个顶点，对G中每条长度为2的路都有一条边。1967年，克努特利用矩阵树定理证明了LG的生成树个数的一个公式，并要求给出双射证明。本文给出了Levine推广Knuth公式的生成函数恒等式的一个双射证明。作为这个证明的结果，我们找到了一个双射之间的二进制de Bruijn序列的程度n和二进制序列的长度为2^{n-1}。最后，我们确定了所有Kautz图和de Bruijn图的临界群，推广了Levine的一个结果。

非本原线性群的基尺寸与一般线性群在生成元组上的轨道

Base sizes of imprimitive linear groups and orbits of general linear groups on spanning tuples

对于对称群$S_\ell$的子群$L$，我们确定了作用在$V_d(q)^\ell$上的非本原线性群$GL_d(q)\wr L$的最小基尺寸.这是通过计算生成$m$元组上的$GL_d(q)$的轨道数来实现的，它原来是$V_m(q)$的$d$维子空间的数目.然后，我们使用这些结果证明，对于某些家庭的子群$L$，仿射群的稳定剂是大子群的$GL_d(q)\wr L$满足猜想的Pyber有关基地。

任意连通图的线图生成树个数的表达式

Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

对于任何图$G$，设$t(G)$是$G$的生成树的数目，$L(G)$是$G$的线图，并且对于任何非负整数$r$，$S_r(G)$是通过将每个边$e$替换为连接$e$两端的长度为$r+1$的路而从$G$获得的图。本文用组合方法得到了$t(L(S_r(G)))$在$G$生成树中的表达式。这个结果推广了关于$t(L(S_r(G)))$与$t(G)$之间关系的一些已知结果，并给出了一个显式表达式$t(L(S_r(G)))=k^{m+s-n-1}(rk+2)^{m-n+1}t(G)$，如果$G$的阶为$n+s$，大小为$m+s$，其中$s$的顶点的度为$1$，其它顶点的度为$k$ .因此，我们证明了一个猜想，$t(L(S_1(G)))$，这样的图$G$。

Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $Γ$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$, the ratio between the minimum length of any path from $u$ to $v$ and the Euclidean distance between $u$ and $v$ is small. The maximum such ratio, over all pairs of vertices of $G$, is the spanning ratio of $Γ$. First, we show that deciding whether a graph admits a straight-line drawing with spanning ratio $1$, a proper straight-line drawing with spanning ratio $1$,…

The active bijection for graphs

The active bijection forms a package of results studied by the authors in a series of papers in oriented matroids. The present paper is intended to state the main results in the particular case, and more widespread language, of graphs. We associate any directed graph, defined on a linearly ordered set of edges, with one particular of its spanning trees, which we call its active spanning tree. For any graph on a linearly ordered set of edges, this yields a surjective mapping from orientations onto spanning trees, which preserves activities (for orientations in the sense of Las Vergnas, for span…

行列式线性生成森林

Determinantal linear spanning forests

本文提出行列式线性生成森林（DLSF），将其推广到含规范场的图场景，研究其支撑集、配分函数及无限图上的性质，明确有线与自由DLSF在常返图上一致、高维Z^d上不一致的结论。

平面图计数：完美匹配、生成圈和Kasteleyn技术

Counting Plane Graphs: Perfect Matchings, Spanning Cycles, and Kasteleyn's Technique

我们推导出改进的上界的交叉自由直边生成循环（也称为汉密尔顿旅游和简单的pathizations），可以嵌入在任何特定的一组$N$点在平面上的数量。更具体地说，我们限制了可以嵌入点集的生成环（或完美匹配）的数量与可以嵌入点集的三角剖分的数量之间的比率。相应的界限是循环的$O(1.8181^N)$和匹配的$O(1.1067^N)$。这意味着一个新的上限$O(54.543^N)$的交叉自由直边跨越循环的数量，可以嵌入在任何特定的一组$N$点在平面上（改进以前的最佳上限$O(68.664^N)$）。我们的分析是基于Kasteleyn的线性代数技术。

关于任意维点的有界度支撑树的改进界

On Improved Bounds on Bounded Degree Spanning Trees for Points in Arbitrary Dimension

在任意维数的欧氏空间中，给定点，证明了存在一棵没有度大于3的顶点的生成树，其权至多为最小生成树权的1.559倍.我们还证明了，有一组点，使得没有最大程度的3的生成树存在，有这个比例小于1.447。我们的中心结果是基于以下主张的证明：给定欧氏空间中的$n$点和一个特殊点$V$，存在端点在$V$的Hamilton路径，该路径至多比点到$V$的距离之和长1.559倍。这些证明也导致了一种方法来找到树在线性时间给定的最小生成树。