Graphs and Hypergraphs

点击翻译按钮获取中文摘要

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.