Combinatorial Designs, Graphs, and Networks

派克博士的研究小组致力于解决组合设计理论和图论领域的问题。图是数学抽象，表示网络的核心元素，即它们的节点和它们之间的链接。网络的图论模型为研究不同的网络问题提供了统一的背景。令人感兴趣的问题包括抑制错误信息通过社交网络的传播，以及如何在计算机网络中隔离恶意软件。组合设计是一种数学结构，它展示了如何从一个主集形成一个子集的集合，从而使元素在子集中一起出现规定的次数。设计在日程安排中有着天然的应用。例如，一位老师可以使用一种称为斯坦纳三元系的设计，他希望在一学年的课程中将每个学生分配到几个团队项目中，这样每个团队有3个学生，而且每对学生在一个团队中恰好一起工作一次。设计在编码理论和密码学中也有应用。我们将研究具有很强结构属性的设计，这些设计反过来对应于具有额外限制的调度场景。我们还打算证明或驳斥Cioaba等人的一个猜想。每个偶数阶强连通强正则图都有一类边着色；这个猜想的其余公开情形是关于设计的区块交图。这项提议的主要目标之一是为学生提供高级培训机会，他们将获得数学问题解决、技术写作和并行计算编程方面的经验。学生们将准备好在就业市场上竞争，为加拿大经济做出贡献。

Dr. Pike's research group tackles problems in the domains of combinatorial design theory and graph theory. Graphs are mathematical abstractions that represent the core elements of networks, namely their nodes and links between them. Graph theoretic models of networks provide a unified context in which to study diverse network problems. Problems of interest include inhibiting the spread of misinformation through social networks, and how to isolate malware in a computer network. Combinatorial designs are mathematical structures that show how to form a collection of subsets from a master set so that elements occur together a prescribed number of times among the subsets. Designs have natural applications in scheduling. For example, a type of design called a Steiner triple system can be used by a teacher who wants to assign each student to several team projects over the course of the school year so that each team has 3 students and also so that each pair of students works together in a team exactly once. Designs also have applications in coding theory and cryptography. We will study designs with strong structural properties, and which in turn correspond to scheduling scenarios with extra constraints imposed on them. We also intend to prove or refute a conjecture by Cioaba et al. that every connected strongly regular graph of even order has a Class 1 edge colouring; the remaining open case of this conjecture is for block intersection graphs of designs. One of the main goals of this proposal is to provide advanced training opportunities for students, who will acquire experience in mathematical problem solving, technical writing, and programming with parallel computing. Students will be prepared to compete in the job market and contribute to the Canadian economy.