A note proving the nullity of block graphs is unbounded

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics Pub Date : 2024-10-17 DOI:10.1016/j.disc.2024.114289

Michael Cary

引用次数: 0

Abstract

Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a

K_{2}

-free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of

K_{2}

-free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.

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证明块图的无效性是无界的说明

块图是大量群落检测和其他网络划分模型的重要基准结构。奇异图在物理科学中有许多重要用途。最近有人提出了一个猜想，即无 K2 的块图的无效性不可能大于 1。在本文中，我们利用实对称矩阵的考奇交错定理构建了一个反例族，从而证明该猜想是错误的。在此过程中，我们证明了更强的声明，即无 K2 块图的无效性是无界的。最后，我们讨论了这一结果对计算网络理论文献的影响。

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

Discrete Mathematics 数学-数学

CiteScore

1.50

自引率

12.50%

发文量

424

审稿时长

6 months

期刊介绍： Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory. Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.