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，我将利用解析、代数和组合的技巧，用n阶图的最大模、实部和虚部来研究这类多项式的根问题。同时，考虑到顶点集上的禁止色集，出现了着色多项式的新的推广，一些困难的极值问题需要引起更多的关注。有联系表明，对有限域上的代数进行更深入的探索，可以进一步限定多项式并找到它们的根。此外，我还计划调查关于两个图的某个乘积的着色数的一个著名的公开问题是否会导致矩阵理论的方法。

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.