论文标题
几乎可控图的广义光谱特征
Generalized spectral characterizations of almost controllable graphs
论文作者
论文摘要
通过其光谱表征图是光谱图理论的一个重要主题，近年来吸引了研究人员的很多关注。通常，显示一个给定的图表要由其频谱确定，这通常是非常困难的。在王〜[j。组合。理论，Ser。 B，122（2017）：438-451]，作者给出了一个简单的算术条件，以通过其广义光谱确定一系列图。但是，该方法仅适用于所谓的\ emph {可控图}的家族；当图形不可控制时，它会失败。 在本文中，我们介绍了一类不可控制的图表，称为\ emph {几乎可控的图}，并证明，对于任何一对几乎可控制的图表$ g $和$ h $，它们都是广义上的，恰好存在两个合理的正交矩阵$ q $，constant of constant conts constant contance convants contance constant convant convant contance $ q^$ a $ a $ a $ \ rm rm t a（g） $ a（h）$分别是$ g $和$ h $的邻接矩阵。证明的主要成分是使用Binet-Cauchy公式。作为一个应用程序，我们获得了一个简单的标准，该标准是由其广义频谱确定的几乎可控的图形$ g $，在某种意义上说，它扩展了可控图的相应结果。
Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined by its spectrum. In Wang~[J. Combin. Theory, Ser. B, 122 (2017):438-451], the author gave a simple arithmetic condition for a family of graphs being determined by their generalized spectra. However, the method applies only to a family of the so called \emph{controllable graphs}; it fails when the graphs are non-controllable. In this paper, we introduce a class of non-controllable graphs, called \emph{almost controllable graphs}, and prove that, for any pair of almost controllable graphs $G$ and $H$ that are generalized cospectral, there exist exactly two rational orthogonal matrices $Q$ with constant row sums such that $Q^{\rm T}A(G)Q=A(H)$, where $A(G)$ and $A(H)$ are the adjacency matrices of $G$ and $H$, respectively. The main ingredient of the proof is a use of the Binet-Cauchy formula. As an application, we obtain a simple criterion for an almost controllable graph $G$ to be determined by its generalized spectrum, which in some sense extends the corresponding result for controllable graphs.
