Lov\'asz theta type norms and Operator Systems
Lov\'asz theta type norms and Operator Systems
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Lovasz theta 类型规范和操作系统
DOI：
10.1016/j.laa.2015.03.022
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
V. Paulsen
中科院分区：
文献类型：
--
作者：
Carlos Ortiz Marrero;V. Paulsen
对于每一个n阶图，都有一个n× n矩阵的子空间，称为图的算子系统。证明了两个图同构当且仅当它们对应的算子系统是酉完全序同构。这意味着研究图等价于研究这些特殊算子系统，直到它们范畴中同构的自然概念。我们定义新的图论参数，通过这种识别。在研究图的算子系统时得到的某些商范数会产生图的一类新的参数族。然后，我们展示了这些参数的基本性质，并明确地写下如何通过半定程序计算它们，并讨论了它们与Lovász theta函数的相似之处。最后，我们探讨了这个家庭中的一个特殊的参数，并建立了三明治定理，适用于一些图。
To each graph on n vertices there is an associated subspace of the n× n matrices called the operator system of the graph. We prove that two graphs are isomorphic if and only if their corresponding operator systems are unitally completely order isomorphic. This means that the study of graphs is equivalent to the study of these special operator systems up to the natural notion of isomorphism in their category. We define new graph theory parameters via this identification. Certain quotient norms that arise from studying the operator system of a graph give rise to a new family of parameters of a graph. We then show basic properties about these parameters and write down explicitly how to compute them via a semidefinite program, and discuss their similarities to the Lovász theta function. Finally, we explore a particular parameter in this family and establish a sandwich theorem that holds for some graphs.