Analytic methods for uniform hypergraphs
Analytic methods for uniform hypergraphs
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DOI：
10.1016/j.laa.2014.05.005
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发表时间：
2013-08
期刊：
影响因子：
--
通讯作者：
V. Nikiforov
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文献类型：
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作者：
V. Nikiforov
本文提出了一些研究均匀超图的分析方法。它的出发点是2-图的谱理论，特别是2-图的最大和最小特征值λ和λ min 。首先，将这两个参数扩展到加权均匀超图；其次，将特征值数 λ 和 λ min 扩展到特征值函数 λ (p) 和 λ min (p)，其中还包含其他图参数，例如拉格朗日和边数。通过这种方式，函数 λ (p) 和 λ min (p) 无缝连接超图中的谱结果和传统结果。特别是，这个新观点有助于证明谱极值和边缘极值问题是渐近等价的。当然，所有关于 λ (p) 和 λ min (p) 的结果也扩展了谱超图理论，但比以前更深入地研究了问题。事实上，即使对于 2 图，由此产生的理论也是新的，其中一些已经解决的主题再次成为研究挑战。这篇论文涵盖了多个主题，有一百多个具体陈述来支撑超图的分析理论。这些主题中最重要的是极值超图问题的 Perron-Frobenius 型理论和方法。提出了许多悬而未决的问题，并概述了可能进一步研究的方向。
This paper presents some analytic methods for studying uniform hypergraphs. Its starting point is the spectral theory of 2-graphs, in particular, the largest and the smallest eigenvalues λ and λ min of 2-graphs. First, these two parameters are extended to weighted uniform hypergraphs; second, the eigenvalues-numbers λ and λ min are extended to eigenvalues-functions λ (p) and λ min (p), which also encompass other graph parameters like the Lagrangian and the number of edges. In this way the functions λ (p) and λ min (p) seamlessly join spectral and traditional results in hypergraphs. In particular, this new viewpoint helps to show that spectral extremal and edge extremal problems are asymptotically equivalent. Naturally, all results about λ (p) and λ min (p) also extend spectral hypergraph theory, but delve into deeper problems than before. In fact, the resulting theory is new even for 2-graphs, where some well-settled topics become research challenges again. The paper covers a multitude of topics, with more than a hundred concrete statements to underpin an analytic theory for hypergraphs. Essential among these topics are a Perron–Frobenius type theory and methods for extremal hypergraph problems. Many open problems are raised and directions for possible further research are outlined.