论文标题
光谱半径和边缘跨树木
Spectral radius and edge-disjoint spanning trees
论文作者
论文摘要
储层计算是预测湍流的有力工具，其简单的架构具有处理大型系统的计算效率。然而，其实现通常需要完整的状态向量测量和系统非线性知识。我们使用非线性投影函数将系统测量扩展到高维空间，然后将其输入到储层中以获得预测。我们展示了这种储层计算网络在时空混沌系统上的应用，该系统模拟了湍流的若干特征。我们表明，使用径向基函数作为非线性投影器，即使只有部分观测并且不知道控制方程，也能稳健地捕捉复杂的系统非线性。最后，我们表明，当测量稀疏、不完整且带有噪声，甚至控制方程变得不准确时，我们的网络仍然可以产生相当准确的预测，从而为实际湍流系统的无模型预测铺平了道路。
The spanning tree packing number of a graph $G$, denoted by $τ(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $τ(G)$ is one of the classic problems in graph theory. Cioabă and Wong initiated to investigate $τ(G)$ from spectral perspectives in 2012 and since then, $τ(G)$ has been well studied using the second largest eigenvalue of the adjacency matrix in the past decade. In this paper, we further extend the results in terms of the number of edges and the spectral radius, respectively; and prove tight sufficient conditions to guarantee $τ(G)\geq k$ with extremal graphs characterized. Moreover, we confirm a conjecture of Ning, Lu and Wang on characterizing graphs with the maximum spectral radius among all graphs with a given order as well as fixed minimum degree and fixed edge connectivity. Our results have important applications in rigidity and nowhere-zero flows. We conclude with some open problems in the end.
