Dávid Kunszenti-Kovács, László Lovász, Balázs Szegedy
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引用次数: 0
摘要
在图元(密集图序列的极限)和图解(有界度图序列的极限)中,子图密度都已被定义并作为基本工具。虽然人们已经描述了 "中间范围 "的极限对象,但这些极限对象中的子图密度概念仍然难以捉摸。我们根据子图的适当稀疏性条件,定义了维数为 d 的单位球上正交性图中的子图密度。这些正交图展示了在 "中间 "范围内定义子图的主要困难,因此我们希望对它们的研究能成为定义更一般马尔可夫空间中子图密度的关键范例。研究将有限图 G 同构为正交性图的兴趣得益于这样一个事实,即这种同构只是互补图的正交表示。
Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the “middle ranges”, the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension d, under an appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs in the “middle” range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. Interest in studying homomorphisms of a finite graph G into orthogonality graphs is supported by the fact that such homomorphisms are just the orthonormal representations of the complementary graph.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research dealing with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series B is concerned primarily with graph theory and matroid theory and is a valuable tool for mathematicians and computer scientists.