Graphs and digraphs

研究的第一部分集中在组合多项式上，从图着色到网络可靠性和独立多项式。代数和分析技术将不仅被用来限定网络可靠性，无论是在无向和有向情况下，还将被用来检查潜在的组合结构。找到多项式的根将更多地揭示潜在系数的突出单峰性问题，并提供更多的洞察，以了解与图和单纯复形中的独立性相关的一类新定义的分形图。我计划继续研究强连通可靠性，这是一种衡量有向图健壮性的有用指标。具体地说，存在一个以前没有研究过的关联的新单纯复形，它的性质可能有助于有效地限制这种可靠性度量。一种新的方法将首先用适当的小波来压缩有向图和无向图。

The first part of the proposed research centers on combinatorial polynomials, from graph colourings to network reliability and independence polynomials. Algebraic and analytic techniques will be used to not only bound network reliability, both in the undirected and directed cases, but also to examine the underlying combinatorial structures. Locating the roots of the polynomials will shed more light on the outstanding unimodality problems of the underlying coefficients, as well as provide more insight into a newly defined class of fractals associated with independence in both graphs and simplicial complexes. I plan to continue to investigate strongly connected reliability, a useful measure of the robustness of a directed graph. In particular, there is an associated new simplicial complex that has not been previously studied, and its properties may help to efficiently bound this reliability measure. A new approach will involve compressing the directed and undirected graphs first with appropriate wavelets.