Combinatorial and spectral analysis of matrix patterns

提出的研究探索数学的两个不同领域，即线性代数和图论之间的相互作用。正在探索的一种特殊类型的问题是数字数组的数学如何受到定性信息的影响，例如数组中正、负和零条目的模式。有时，这个信息本身就足以观察到结果的显著限制，而与数组中单个数字的大小无关。这些符号模式可以使用图论建模，该理论可以提供对给定模式的限制的见解。这项研究的一部分需要确定这些模型可以发现哪些限制，同样，这些模型如何帮助证明缺乏限制。

The proposed research explores the interplay between two different areas of mathematics, namely linear algebra and graph theory. One particular type of problem being explored is how mathematics on arrays of numbers is affected by qualitative information, such as the pattern of positive, negative and zero entries in the array. Sometimes this information alone is sufficient to observe significant restrictions on results, independent of the magnitude of the individual numbers in the array. These sign patterns can be modeled using graph theory and this theory can provide insights into the restrictions for given patterns. Part of this research entails determining what restrictions can be discovered with these models, and likewise, how these models can help demonstrate lack of restrictions.