Combinatorial Designs and Graph Theory

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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. **