图幂的零强迫数

中文摘要 AI 辅助

简单图 $G$ 的 $k$ 次幂，记作 $G^k$，是以 $V(G)$ 为顶点集的图，其中两个顶点相邻当且仅当它们在 $G$ 中的距离不超过 $k$。我们研究图幂的零强迫数。图的幂比原图稠密得多，且通常不包含在现有的关于图零强迫的结果中，因此对其研究需要不同的方法。与通常的零强迫行为在删边下的表现相反，我们证明零强迫参数（及其变体）关于取幂是单调的。我们还确定了路径和圈的幂的零强迫数，并给出了蜘蛛图和网格图的幂的上界。接着，我们提出了图幂的零强迫数的谱下界，该下界仅使用基图的谱，并利用线性规划方法计算这些界。最后，我们研究了所得界的紧性。为推导我们的结果，我们使用了图论、线性代数和多项式优化中的技术。

英文摘要

The $k$-th power of a simple graph $G$, denoted $G^k$, is the graph with vertex set $V(G)$ where two vertices are adjacent if they are within distance $k$ in $G$. We investigate the zero forcing number of graph powers. Powers of graphs are much denser and generally not encompassed by existing results on zero forcing of graphs, hence their study requires a different approach. In contrast with the usual zero forcing behavior under edge deletion, we show that the zero forcing parameter (and variations of it) is monotone with respect to taking powers. We also determine the zero forcing number of powers of paths and cycles, together with upper bounds for powers of spiders and of grids. We then present spectral lower bounds on the zero forcing number of graph powers which uniquely use the spectrum of the base graph, as well as linear programming methods to compute these bounds. Finally, we study the sharpness of the derived bounds. To derive our results we use techniques ranging from graph theory, linear algebra and polynomial optimization.

发表机构

• Eindhoven University of Technology（埃因霍温理工大学）
• Vrije Universiteit Brussel（布鲁塞尔自由大学）
• University of St. Thomas（圣托马斯大学）
• Vrije Universiteit Amsterdam（阿姆斯特丹自由大学）
• University of Amsterdam（阿姆斯特丹大学）

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