Fractional strong matching preclusion for Cartesian product graphs

图的整数 k-匹配排除

Integer k-matching preclusion of graphs

本文推广匹配排除数，定义整数 k-匹配排除数与强整数 k-匹配排除数，证明偶数 k 时其与分数匹配排除数的关系，并求出多类图的对应数值。

forts、(分数)零强迫数以及图的Cartesian乘积

Forts, (fractional) zero forcing, and Cartesian products of graphs

本文引入了fort数和分数零强迫数，并利用超图理论为Cartesian乘积图的零强迫数建立了下界，同时展示了达到该下界的图类。

凸集笛卡尔积的分数 Helly 定理

Fractional Helly theorem for Cartesian products of convex sets

该论文将 Eckhoff 关于轴对齐盒子的最优分数 Helly 定理推广到任意维度凸集的笛卡尔积，证明了一个最优定量界，并构造了极值例子，表明附加 (p,d+1)-条件对改善界无实质影响，且最优界是稳定的。

On the fractional metric dimension of graphs

In [S. Arumugam, V. Mathew and J. Shen, On fractional metric dimension of graphs, preprint], Arumugam et al. studied the fractional metric dimension of the cartesian product of two graphs, and proposed four open problems. In this paper, we determine the fractional metric dimension of vertex-transitive graphs, in particular, the fractional metric dimension of a vertex-transitive distance-regular graph is expressed in terms of its intersection numbers. As an application, we calculate the fractional metric dimension of Hamming graphs and Johnson graphs, respectively. Moreover, we give an inequali…

The Fractional Local Metric Dimension of Graphs

The fractional versions of graph theoretic-invariants multiply the range of applications in scheduling, assignment and operational research problems. In this paper, we introduce the fractional version of local metric dimension of graphs. The local resolving neighborhood $L(xy)$ of an edge $xy$ of a graph $G$ is the set of those vertices in $G$ which resolve the vertices $x$ and $y$. A function $f:V(G)\rightarrow[0, 1]$ is a local resolving function of $G$ if $f(L(xy))\geq1$ for all edges $xy$ in $G$. The minimum value of $f(V(G))$ among all local resolving functions $f$ of $G$ is the fractiona…

The fractional strong metric dimension in three graph products

For any two distinct vertices $x$ and $y$ of a graph $G$, let $S\{x, y\}$ denote the set of vertices $z$ such that either $x$ lies on a $y-z$ geodesic or $y$ lies on an $x-z$ geodesic. Let $g: V(G) \rightarrow [0,1]$ be a real valued function and, for any $U \subseteq V(G)$, let $g(U)=\sum_{v \in U}g(v)$. The function $g$ is a strong resolving function of $G$ if $g(S\{x, y\}) \ge 1$ for every pair of distinct vertices $x, y$ of $G$. The fractional strong metric dimension, $sdim_f(G)$, of a graph $G$ is $\min\{g(V(G)): g \mbox{ is a strong resolving function of }G\}$. In this paper, after obtai…

A note on lower bounds for boxicity of graphs

The boxicity of a graph $G$ is the minimum non-negative integer $k$ such that $G$ can be isomorphic to the intersection graph of a family of boxes in Euclidean $k$-space, where a box in Euclidean $k$-space is the Cartesian product of $k$ closed intervals on the real line. In this note, we define the fractional boxicity of a graph as the optimum value of the linear relaxation of a covering problem with respect to boxicity, which gives a lower bound for its boxicity. We show that the fractional boxicity of a graph is at least the lower bounds for boxicity given by Adiga et al. in 2014. We also p…

Laplacian Fractional Revival on Graphs

We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are doub…

网格图上的2-限制广播支配

2-limited broadcast domination on grid graphs

本研究针对路径与环构成的各类网格图，通过显式构造和线性规划对偶方法，分别推导其2-限制广播支配数的上下界，为该图类的广播支配问题提供了定量边界结果。

超立方体图的堡垒及相关参数

On the forts and related parameters of the hypercube graph

本文完整刻画了超立方体图中的极小堡垒，推导了分数零强迫数与堡垒数界限，并发现维数为2的幂时多个图参数相等，同时给出了笛卡尔积中极小堡垒的一般构造。

分层图的分数列表打包

Fractional list packing for layered graphs

研究分层图的分数列表打包数，提出路径宽度加一的上界，并改进灵活列表着色定理。

量子对态传输的推广

A generalization of quantum pair state transfer

本文推广了连续量子游走中的量子对态传输理论，在哈密顿量取邻接、Laplacian 或无符号 Laplacian 矩阵下，刻画了多类图中的完美 $s$-对态传输，并利用商图与分数复苏构造了无限图族，揭示了线图与原图边态传输的等价条件。