论文标题
在Digraphs的Laplacian上
On the Laplacian spread of digraphs
论文作者
论文摘要
在本文中，我们使用受限的数值范围将拉普拉斯散布的概念扩展到简单的有向图（Digraphs）。首先，我们为几个digraphs家族提供拉普拉斯散布价值。然后，我们证明了所有多边形和平衡的挖掘者的拉普拉斯传播上的尖锐上边界。特别是，我们表明，用于平衡挖掘的拉普拉斯散布的有效性等效于简单无向图的拉普拉斯（Laplacian）扩散猜想，该图表是在2011年猜想，并在2021年被证明。最后，我们陈述了几个由经验数据激发的开放猜想。
In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validity of the Laplacian spread bound for balanced digraphs is equivalent to the Laplacian spread conjecture for simple undirected graphs, which was conjectured in 2011 and proven in 2021. Moreover, we prove an equivalent statement for weighted balanced digraphs with weights between $0$ and $1$. Finally, we state several open conjectures that are motivated by empirical data.
