直积图和强积图的Laplacian谱的估计
Estimation of Laplacian spectra of direct and strong product graphs
摘要：
计算多个图的乘积已经在数学、工程、计算机科学以及最近的网络科学中进行了研究，特别是在多层网络的背景下。如何利用乘积图的因子图的谱性质来刻画乘积图的谱性质是这一领域中需要解决的重要问题之一。虽然一些这样的特征已经得到了解析（主要是邻接谱），拉普拉斯谱的直积和强积图的特征仍然是一个开放的问题。在这里，我们开发了实用的方法来估计拉普拉斯谱的直接和强的产品图的谱特性，他们的因子图使用一些启发式的假设。数值实验表明，所提出的方法产生合理的估计与百分比误差限制在+/-10%的范围内的大多数特征值。
英文摘要：
Calculating a product of multiple graphs has been studied in mathematics, engineering, computer science, and more recently in network science, particularly in the context of multilayer networks. One of the important questions to be addressed in this area is how to characterize spectral properties of a product graph using those of its factor graphs. While several such characterizations have already been obtained analytically (mostly for adjacency spectra), characterization of Laplacian spectra of direct product and strong product graphs has remained an open problem. Here we develop practical methods to estimate Laplacian spectra of direct and strong product graphs from spectral properties of their factor graphs using a few heuristic assumptions. Numerical experiments showed that the proposed methods produced reasonable estimation with percentage errors confined within a +/-10% range for most eigenvalues.
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