分裂图中的准核

遗传类中的拟核及其在Break图中的应用

Quasi-kernels in Hereditary Classes and Applications to Break

本文引入加权半邻域性质，证明break图存在大小不超过2n/3的小拟核，并给出两类特殊break图的n/2界，同时证明任意n顶点有向图存在大小至少为√n的拟核。

Towards the Small Quasi-Kernel Conjecture

Let $D=(V,A)$ be a digraph. A vertex set $K\subseteq V$ is a quasi-kernel of $D$ if $K$ is an independent set in $D$ and for every vertex $v\in V\setminus K$, $v$ is at most distance 2 from $K$. In 1974, Chvátal and Lovász proved that every digraph has a quasi-kernel. P. L. Erdős and L. A. Székely in 1976 conjectured that if every vertex of $D$ has a positive indegree, then $D$ has a quasi-kernel of size at most $|V|/2$. This conjecture is only confirmed for narrow classes of digraphs, such as semicomplete multipartite, quasi-transitive, or locally demicomplete digraphs. In this note, we state…

The kernel of chromatic quasisymmetric functions on graphs and hypergraphic polytopes

We study the chromatic symmetric function on graphs, and show that its kernel is spanned by the modular relations. We generalize this result to the chromatic quasisymmetric function on hypergraphic polytopes, a family of generalized permutahedra. We use this description of the kernel of the chromatic symmetric function to find other graph invariants that may help us tackle the tree conjecture.

贪心构造小准核

Greedily Constructing Small Quasi-Kernels

本文提出一种贪心算法，用于构造小准核，并在不同条件下给出了准核阶数的上界。

拟核猜想的一个可变版本

A variable version of the quasi-kernel conjecture

本文研究有向图拟核的大、小 QK 猜想之间的关系，证明二者在尖锐形式下的等价性及带较弱常数的蕴含关系，并对核完美数为 $k$ 的有向图给出大小至多 $(1-1/k)n$ 的拟核。

A Result on the Small Quasi-Kernel Conjecture

Any directed graph $D=(V(D),A(D))$ in this work is assumed to be finite and without self-loops. A source in a directed graph is a vertex having at least one ingoing arc. A quasi-kernel $Q\subseteq V(D)$ is an independent set in $D$ such that every vertex in $V(D)$ can be reached in at most two steps from a vertex in $Q$. It is an open problem whether every source-free directed graph has a quasi-kernel of size at most $|V(D)|/2$, a problem known as the small quasi-kernel conjecture (SQKC). The aim of this paper is to prove the SQKC under the assumption of a structural property of directed graph…

单调图极限与拟单调图

Monotone graph limits and quasimonotone graphs

最近的图极限理论给出了一个强有力的框架，用于理解图的合适的（收敛的）序列$(G_n)$的性质，该限制对象可以由$[0,1]$上的对称函数$W$表示，即，内核或graphon。在这种情况下，很自然地希望将序列的特定属性与内核的特定属性相关联。在这里，我们证明了内核是单调的（即，两个变量都增加）当且仅当序列满足由某个趋于零的泛函定义的“准单调性”性质。作为一种工具，我们证明了一个不等式有关的削减和$L^1$规范的内核的形式$W_1-W_2$与$W_1$和$W_2$单调，可能是感兴趣的本身的权利，没有这样的不等式举行一般的内核。

关于分割图的零空间

On the nullspace of split graphs

研究分割图邻接矩阵的零空间，引入 clique-kernel 并证明其维度不超过一，推导零空间公式并分析不平衡分割图的结构。

加热 Quasi-Monte Carlo 图随机特征：一个 Diffusion 核的视角

Heating Up Quasi-Monte Carlo Graph Random Features: A Diffusion Kernel Perspective

本研究扩展了 q-GRFs 方法，发现其在 Ladder graphs 上能为 Diffusion 核实现更低方差估计，为基于核的学习算法及实际应用奠定基础。

Crystal bases of modified $\imath$quantum groups of certain quasi-split types

In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why $\imath$canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of $\imath$crystals whose projective limit can be thought of as the $\imath$canonical basis of the modified $\imath$quantum group at $q = \infty$.