代数连通性和最大连通正则图的可实现边界

IF 0.9 3区 数学 Q2 MATHEMATICS
Geoffrey Exoo, Theodore Kolokolnikov, Jeanette Janssen, Timothy Salamon
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引用次数: 0

摘要

我们用直径和周长推导出了规则图形代数连通性(谱隙)的可实现上限。对于偶数直径的图,这个界限与著名的 Alon-Boppana-Friedman 界限一致,但对于奇数直径的图,这个界限有所改进。对于周长约束,我们证明只有摩尔图才能达到,而这些摩尔图只存在于众所周知的特例中。对于直径约束,我们使用随机算法和穷举搜索相结合的方法来找到达到该约束的图。对于三规则图,我们找到了所有直径在()以下()的可达图。这些图极为罕见,而且周长也很高;例如,我们在 44 个顶点上找到了 45 个不同的立方图,它们在周长为 8 时都达到了上界。 我们还展示了几个无限族,它们在直径或周长方面都达到了上界。特别是,当是素数的幂时,我们构造了一个直径为 4、周长为 6 的不规则图,它达到了直径的上限。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Attainable bounds for algebraic connectivity and maximally connected regular graphs

Attainable bounds for algebraic connectivity and maximally connected regular graphs

We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon–Boppana–Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for well-known special cases. For the diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters D $D$ up to and including D = 9 $D=9$ (the case of D = 10 $D=10$ is open). These graphs are extremely rare and also have high girth; for example, we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when D = 7 $D=7$ ; all have girth 8. We also exhibit several infinite families attaining the upper bound with respect to diameter or girth. In particular, when d $d$ is a power of prime, we construct a d $d$ -regular graph having diameter 4 and girth 6 which attains the upper bound with respect to diameter.

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来源期刊
Journal of Graph Theory
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 .
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