Combinatorial Designs and Graph Theory

图、组合区组设计及其性质是本研究方案的中心研究领域。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更大的集合中形成几个子集，以便展示某些属性。设计通常适用于需要进行交互的情况。以此类推，老师可能会遇到这样的场景：有一班学生，每个学生必须被分配到几个团队项目中，这样每对学生在一个团队中的次数与其他两个学生在一起的次数相同。不同的团队分配导致具有不同属性的设计，其中一些是可取的，但可能很难实现。此外，如何对给定设计的团队进行调度，以使它们满足特定的约束(特别是使连续调度的团队具有特定数量的共同人员)，这是将被研究的几个问题之一，通常是通过调查与设计相关的图的数学性质来进行的。除了在抽象层面发现新知识外，这项研究还将为算法、调度、数字通信等领域的工作做出贡献。这项研究还将为学生提供高级培训机会，他们将获得解决数学问题、技术写作和使用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.