Symmetries of Cayley Graphs

凯莱图的对称性

• 批准号: RGPIN-2017-04905
• 负责人: Morris, Joy
• 金额: $ 1.17万
• 依托单位: University of Lethbridge
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711288
• 关键词: Symmetries; Cayley; Graphs

My research program involves using permutation groups to study symmetries of Cayley graphs. A "graph" is a model of a network, with "vertices" (points) representing nodes or terminals, and edges representing connections. Cayley graphs are graphs with strong symmetry: a simple example is the pentagram (star), which has rotational and reflectional symmetries. Cayley graphs are interesting combinatorial objects to study for their own sakes. They also provide a class of graphs for which powerful techniques of group theory (abstract algebra) can be brought to bear in exploring difficult problems. I will continue to work with students, post-doctoral fellows, and collaborators, to explore open problems on symmetries of Cayley graphs, including: 1. There is a canonical way of colouring the edges of a Cayley graph so that the most natural symmetries of the graph preserve this canonical colouring. I will be exploring when it is the case that only the most natural symmetries preserve this colouring, and when extra symmetries might preserve this colouring. Edge-colourings are much studied since they can be used to represent properties that vary for different edges of the graph, such as capacity. However, most edge-colourings destroy most symmetry. 2. If a Cayley graph has the "Cayley Isomorphism property," algebraic techniques enable us to determine whether or not another Cayley graph is essentially the same network, just drawn differently. This has been much studied for finite graphs. I will be studying the extension of this problem to infinite graphs, which has attracted much less attention. I will also work to extend some previous results of mine on which finite graphs have this property. 3. Sometimes Cayley graphs are allowed to have directed edges, representing one-way connections in the network; if they do, we call them digraphs. It is known that (with some exceptions) any "nice" collection of symmetries (a symmetry group) has a "regular" representation by a Cayley graph, meaning that there is a Cayley graph with precisely these "regular" symmetries. There is a similar result for Cayley digraphs. However, whether or not a symmetry group can be regularly represented by an oriented Cayley graph (with at most one directed edge between any pair of vertices) is a long-standing open problem. A co-author and I recently solved this problem for "non-solvable" symmetry groups. We will try to extend our work to "solvable" symmetry groups. 4. Describing a graph as a Cayley graph provides information about a large number of its symmetries, but often not all of them. I will work to characterise graphs that can be represented as a Cayley graph in more than one way. The second representation significantly enhances our ability to understand all of the symmetries such graphs exhibit. These are steps toward solving recognition and isomorphism problems for Cayley graphs: extremely important open questions in algebraic graph theory.

我的研究项目包括使用置换群来研究Cayley图的对称性。“图”是网络的模型，“顶点”(点)代表节点或终端，边代表连接。Cayley图是具有强对称性的图：一个简单的例子是具有旋转对称性和反射对称性的五角星(星形)。 Cayley图是值得研究的有趣的组合对象。它们还提供了一类图，对于这种图，可以利用群论(抽象代数)的强大技术来探索困难的问题。 我将继续与学生、博士后研究员和合作者一起探索关于Cayley图对称性的公开问题，包括： 1.有一种给Cayley图的边着色的规范方法，使得该图的最自然的对称保持这种规范着色。我将探索只有最自然的对称才能保留这种颜色的情况，以及额外的对称何时可能保留这种颜色。边着色被广泛研究，因为它们可以用来表示图的不同边的不同属性，例如容量。然而，大多数边色破坏了大部分的对称性。 2.如果一个Cayley图具有“Cayley同构性质”，则代数技术使我们能够确定另一个Cayley图是否本质上是相同的网络，只是绘制的不同而已。对于有限图，这方面的研究已经很多了。我将研究如何将这个问题推广到无限大的图上，这方面的关注要少得多。我还将致力于推广我以前的一些结果，在这些结果上，有限图具有这种性质。 3.有时Cayley图被允许有向边，表示网络中的单向连接；如果有，我们称之为有向图。众所周知(除了某些例外)，任何“好的”对称集合(一个对称群)都有一个由Cayley图表示的“正则”表示，这意味着存在一个恰好具有这些“正则”对称的Cayley图。对于Cayley有向图也有类似的结果。然而，对称群是否可以用有向Cayley图(任意两个顶点之间至多只有一条有向边)来正则表示是一个长期未解决的问题。最近，我和一位合著者解决了“不可解”对称群的这个问题。我们将尝试将我们的工作扩展到“可解”对称群。 4.将一个图描述为Cayley图提供了关于它的大量对称性的信息，但通常不是所有的对称性。我将致力于刻画可以以多种方式表示为Cayley图的图。第二种表示法显著增强了我们理解这些图形所展示的所有对称性的能力。 这些是解决Cayley图的识别和同构问题的步骤：代数图论中非常重要的开放问题。