正则图中两两不同最长圈的最小数量的改进渐近上界

Locally s-distance transitive graphs and pairwise transitive designs

The study of locally s-distance transitive graphs initiated by the authors in previous work, identified that graphs with a star quotient are of particular interest. This paper shows that the study of locally s-distance transitive graphs with a star quotient is equivalent to the study of a particular family of designs with strong symmetry properties that we call nicely affine and pairwise transitive. We show that a group acting regularly on the points of such a design must be abelian and give a general construction for this case.

正则有向图中的零和环

Zero-Sum Cycles in Regular Digraphs

本文研究正则有向图中的零和环，证明了零和环存在的阈值，给出了顶点不交与边不交零和环的数量下界，结果渐近最优，相关结论可推广至欧拉有向图，所用技术基于Friedland的行列式-永久式论证。

高度边连通的正则图，不含大可分解子图

Highly edge-connected regular graphs without large factorizable subgraphs

本文构造了高度边连通的 r-正则图，证明其中不存在 r-2 个两两不交的完美匹配，部分回答了 Thomassen 关于正则图因子分解的问题。

Shortest-weight paths in random regular graphs

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hatα\log n$ edges where $\hatα$ is the unique solution of the equation $α\log(\frac{d-2}{d-1}α) - α= \frac{d-3}{d-2}$, for $α> \frac{d-1}{d-2}$.

On orders of automorphisms of vertex-transitive graphs

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $Γ$, $Γ\not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\la…

Cartesian Products of Regular Graphs are Antimagic

An \emph{antimagic labeling} of a finite undirected simple graph with $m$ edges and $n$ vertices is a bijection from the set of edges to the integers $1,...,m$ such that all $n$ vertex sums are pairwise distinct, where a vertex sum is the sum of labels of all edges incident with the same vertex. A graph is called \emph{antimagic} if it has an antimagic labeling. In 1990, Hartsfield and Ringel \cite{HaRi} conjectured that every simple connected graph, but $K_2$, is antimagic. In this article, we prove that a new class of Cartesian product graphs are antimagic. In addition, by combining this res…

伪几何强正则图与一个规则点

Pseudo-Geometric Strongly Regular Graphs with a Regular Point

本文研究了一类伪几何强正则图，其第二子构成是某个强正则图或完全图的覆盖。通过分析此类图的结构，作者找到了许多新的强正则图，并回答了Gardiner等人提出的问题。

某些实现二项式边理想正则性界限的图的特征化

Characterization of Some Graphs Realizing Regularity Bounds for Binomial Edge Ideals

本文研究了二项式边理想正则性界限的图特征化问题，探讨了正则性在不同区间内的可能取值。