盆栽图中最大 dissociation 集的最大数量

连通图在给定阶数和解离数下最小化谱半径

Connected graphs minimizing the spectral radius for given order and dissociation number

研究在给定阶数和解离数下连通图的最小谱半径问题，通过结构分析和差分方程构建，确定解离数为n-3的极值图特性。

具有给定解离数并达到最小谱半径的连通图

Connected graphs with a given dissociation number attaining the minimum spectral radius

本文研究给定解离数下谱半径最小的$n$阶连通图，刻画了解离数取若干特定值时的极值图，并证明解离数大于$\lceil2n/3\rceil$时极值图为树。

A bound on the dissociation number

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard even when restricted to subcubic bipartite graphs. For a graph $G$ with $n$ vertices, $m$ edges, $k$ components, and $c_1$ induced cycles of length $1$ modulo $3$, we show ${\rm diss}(G)\geq n-\frac{1}{3}\Big(m+k+c_1\Big)$. Furthermore, we characterize the extremal graphs in which every two cycles are vertex-disjoint.

具有达到最小谱半径的大解离数的连通图

Connected graphs with a large dissociation number attaining the minimum spectral radius

研究给定阶数\(n\)和解离数\(\psi\)的连通图的最小谱半径，对于\(\psi=n - k\)（\(k\geq4\)固定且\(n\)足够大），建立上下界并证明极值图所属类别。

树中极大解离集的最大数量

The maximum number of maximal dissociation sets in trees

本文针对树中极大解离集的最大数量问题，研究Wang等人2025年提出的猜想，给出阶数$n\boldsymbol{\text{≥}}3$的树中该数量的分段函数表达式，并刻画对应的极值树。

关于具有给定 dissociation number $φ$ 的树的 extremal Sombor index

On extremal Sombor index of trees with a given dissociation number $φ$

本文研究了具有给定 dissociation number $φ$ 的树的最大 Sombor index 问题，确定了阶为 $\mathbf{n}$ 的树的最大值，并给出了取得该最大值的唯一图。

关于连通图中第二大解离集数量的注记

A note on the second-largest number of dissociation sets in connected graphs

本文针对Yuan等人提出的关于n阶连通图中解离集第二大数量及其极图的问题，给出了肯定的解答。

关于具有给定阶数和分离数的图的$A_α$-指标

On the $A_α$-index of graphs with given order and dissociation number

本文研究了在给定阶数和分离数条件下图的$A_α$-指标极值问题，确定了具有最大和最小$A_α$-指标的连通图、二部图及树的结构。

Relating the independence number and the dissociation number

The independence number $α(G)$ and the dissociation number ${\rm diss}(G)$ of a graph $G$ are the largest orders of induced subgraphs of $G$ of maximum degree at most $0$ and at most $1$, respectively. We consider possible improvements of the obvious inequality $2α(G)\geq {\rm diss}(G)$. For connected cubic graphs $G$ distinct from $K_4$, we show $5α(G)\geq 3{\rm diss}(G)$, and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle-free graphs, we also obtain improvements. For subcubic graphs though, the inequality cann…

On the dissociation number of Kneser graphs

A set $D$ of vertices of a graph $G$ is a dissociation set if each vertex of $D$ has at most one neighbor in $D$. The dissociation number of $G$, $diss(G)$, is the cardinality of a maximum dissociation set in a graph $G$. In this paper we study dissociation in the well-known class of Kneser graphs $K_{n,k}$. In particular, we establish that the dissociation number of Kneser graphs $K_{n,2}$ equals $\max{\{n-1,6\}}$. We show that for any $k \geq 2$, there exists $n_0 \in \mathbb{N}$ such that $diss(K_{n,k})=α(K_{n,k})$ for any $n \geq n_0$. We consider the case $k=3$ in more details and prove t…

连通图中分离集的数量

The number of dissociation sets in connected graphs

本文研究了连通图中分离集的极值枚举问题，提出了增加分离集数量的有效工具，确定了阶为$n$的连通图和树中分离集的最大数量及极值图，并找出了具有第二大分离集数量的单圈图候选。

矩阵环正则图的解离数的拉普拉斯界

Laplacian Bounds for the Dissociation Number of Regular Graphs of Matrix Rings

该研究针对矩阵环正则图$\text{Γ}_n(q)$，通过嵌入全图、利用拉普拉斯不等式与矩阵环特征计算，得到其解离数的奇偶敏感上下界，并确定最小情形$\text{diss}(\text{Γ}_2(2))=3$。

图的子拟群染色

Sub-quorum colorings of graphs

本文研究图的子拟群染色数，建立结构界、关联2-独立数、分析计算复杂性，并确定网格图与超立方体的精确值，提出一般等式猜想。

连通图中的极大与最大导出匹配

Maximal and maximum induced matchings in connected graphs

本文确定了阶数为 $n$ 的连通图中极大与最大导出匹配数量的紧确界，并据此导出了 $O(1.5849^n)$ 时间的枚举算法及对极大解离集数量的估计。

矩形网格图中的上支配数及其与子法定人数染色的关系

Upper $k$-Star-Forming Sets, $k$-Independence, and Upper Domination

本文证明矩形网格图的上支配数等于其顶点数一半的上取整，并给出两个简短证明，进而建立其与2-独立数和子法定人数染色数之间的不等式链及间隙公式。