Connectivity of old and new models of friends-and-strangers graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED Advances in Applied Mathematics Pub Date : 2024-01-16 DOI:10.1016/j.aam.2023.102668

Aleksa Milojević

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Abstract

In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the threshold probability at which such graphs attain maximal connectivity. We slightly improve the lower bounds on the threshold probabilities, thus disproving two conjectures of Alon, Defant and Kravitz. We also improve the upper bound on the threshold probability in the case of random bipartite graphs, and obtain a tight bound up to a factor of $n^{o (1)}$ . Further, we introduce a generalization of the notion of friends-and-strangers graphs in which vertices of the starting graphs are allowed to have multiplicities and obtain generalizations of previous results of Wilson and of Defant and Kravitz in this new setting.

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新旧朋友和陌生人图谱模型的连接性

在本文中，我们将研究由 Defant 和 Kravitz 于 2020 年提出的朋友和陌生人图的连通性。我们首先考虑了由两个随机图产生的朋友和陌生人图，并考虑了这类图达到最大连通性的阈值概率。我们略微改进了阈值概率的下界，从而推翻了阿隆、迪凡特和克拉维茨的两个猜想。我们还改进了随机二方图情况下阈值概率的上限，并获得了一个高达 no(1) 倍的紧密约束。此外，我们还引入了 "朋友和陌生人图 "概念的广义化，其中允许起始图的顶点具有多重性，并在这一新环境中获得了威尔逊以及笛凡特和克拉维茨先前结果的广义化。

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来源期刊

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.

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