Eigenvalue Approach to Dense Clusters in Hypergraphs

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2025-03-02 DOI:10.1002/jgt.23218

Yuly Billig

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Abstract

In this article, we investigate the problem of finding in a given weighted hypergraph a subhypergraph with the maximum possible density. Using the notion of a support matrix we prove that the density of an optimal subhypergraph is equal to $∥ A^{T} A ∥$ for an optimal support matrix $A$ . Alternatively, the maximum density of a subhypergraph is equal to the solution of a minimax problem for column sums of support matrices. We study the density decomposition of a hypergraph and show that it is a significant refinement of the Dulmage–Mendelsohn decomposition. Our theoretical results yield an efficient algorithm for finding the maximum density subhypergraph and more generally, the density decomposition for a given weighted hypergraph.

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超图中密集聚类的特征值方法

在本文中，我们研究了在给定的加权超图中寻找具有最大可能密度的子超图的问题。利用支持矩阵的概念证明了最优子超图的密度等于∥a→T→a→∥求最优支持矩阵A。或者，子超图的最大密度等于支持矩阵列和的极大极小问题的解。我们研究了超图的密度分解，并证明了它是Dulmage-Mendelsohn分解的一个重要改进。我们的理论结果产生了一种有效的算法，用于寻找最大密度子超图，更一般地说，用于给定加权超图的密度分解。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .