Combinatorial Designs and Graph Theory

图、组合区组设计及其性质是本研究计划的中心研究领域。图通常可以被视为具有节点和节点之间的链接的网络。设计是一种组合结构，它展示了如何从一个较大的集合中形成几个子集，以便展示某些特性。设计通常应用于需要进行交互的情况。作为一个类比，老师可能会遇到这样的情况，即有一个班级的学生，每个学生都必须被分配到几个团队项目中，以便每对学生在一个团队中的次数与其他每对学生相同。不同的团队分配导致设计具有不同的属性，其中一些是可取的，但可能难以实现。此外，如何安排给定设计的团队，使他们满足某些约束条件（特别是连续安排的团队有特定数量的共同人员）是将研究的几个问题之一，通常通过研究与设计相关的图的数学性质。除了在抽象层面上发现新知识外，这项研究还将有助于在算法、调度、数字通信等领域开展工作。该研究还将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的经验。

Graphs, combinatorial block designs, and their properties are the central research areas of this research proposal. Graphs can often be viewed as networks with nodes and linkages between nodes. Designs are combinatorial structures that show how to form several subsets from a larger set so that certain properties are exhibited. Designs often have application in situations in which interactions need to take place. As an analogy, a teacher might encounter the scenario of having a class of students, each of whom must be assigned to several team projects so that each pair of students is on a team together the same number of times as every other pair of students. Different team assignments lead to designs with differing attributes, some of which are desirable but may be difficult to achieve. Moreover, how to go about scheduling the teams of a given design so that they satisfy certain constraints (especially so that consecutively scheduled teams have a specific number of people in common) is one of several questions that will be studied, often by investigating mathematical properties of graphs that are associated with designs. In addition to discovering new knowledge at the abstract level, the research will contribute to work being done in areas such as algorithms, scheduling, digital communications and others. The research will also provide advanced training opportunities for students, who will acquire experience in mathematical problem solving, technical writing, and programming with parallel computing.