Graphs and Hypergraphs

该项目的第一部分集中于通过与其他数学分支的联系来回答各种图论问题。首先，使用解析、代数和组合技术，我将根据n阶图中的最大模、实部和虚部来研究与这些多项式的根相关的问题。此外，已经出现了新的着色多项式的推广，考虑了顶点集上的禁止颜色集，并且一些困难的极值问题需要更多的关注。连接表明，在有限域上对代数进行更深入的探索，作为进一步约束多项式和定位其根的一种方式。此外，我计划研究一个著名的开放问题，关于两个图的某个积的着色数是否可以产生矩阵论的方法。在网络可靠性方面，我计划继续探索这些多项式的解析性质，范围从根和不动点的位置到可靠性多项式的最优性的新概念，使用积分。此外，我计划完成一种新的局部（而不是全局）鲁棒性度量理论的开发，这可能对社交网络更有用。我建议通过关联多项式从理论上和解析上研究一个抽象的凸性概念。我还计划探索Tutte五流问题的矩阵理论方法，利用计数原理，根据某些子空间的维度来限制图的五流数量。与超图相关的某些向量空间直到最近才被探索，并且在将超图的结构属性与相关空间的向量空间维度（和基）联系起来方面还有很多工作要做。结果可能有助于为超图族的构建块提供新的见解。

The first part of the project concentrates on answering a variety of graph theoretical questions via connections to other branches of mathematics. First, using analytic, algebraic and combinatorial techniques, I will investigate problems related to the roots of such polynomials in terms of the the largest modulus, real and imaginary parts among graphs of order n. As well, new generalizations of colouring polynomials have arisen that take into account forbidden sets of colours at the vertex sets, and some difficult extremal problems require more attention. Connections have suggested that a deeper exploration of the algebras over finite fields as a way to further bound the polynomials and locate their roots. Also, I plan to investigate whether a famous open problem on the colouring number of a certain product of two graphs might yield to a matrix-theoretic approach.In network reliability, I plan to continue to explore analytic properties of these polynomials, ranging from location of the roots and fixed points to new notions of optimality of reliability polynomials, using integrals. Also, I plan to complete development for the theory of a new local (rather then global) measure of robustness, that may be more useful for social networks. I propose to investigate an abstract notion of convexity both graph theoretically and analytically via associated polynomials. I also plan to explore a matrix-theoretic approach to Tutte's 5-flow problem, utilizing counting principles, to bound the number of 5-flows of a graph in terms of the dimensions of certain subspaces. Certain vector spaces associated with hypergraphs have only recently been explored, and there is much work to be done yet on connecting up structural properties of the hypergraphs with the vector space dimension (and bases) of the associated spaces. Results are likely to help provide new insights into the building blocks of families of hypergraphs.