Graphs and digraphs

第一部分的研究重点是组合多项式，从图的着色到网络的可靠性和独立性多项式。代数和分析技术将不仅用于约束网络的可靠性，无论是在无向和有向的情况下，而且还用于检查潜在的组合结构。找到多项式的根将更清楚地揭示潜在系数的突出单峰问题，并提供对新定义的分形类的更多见解，这些分形类在图和简单复形中都具有独立性。我计划继续研究强连接可靠性，这是有向图鲁棒性的一个有用的度量。特别是，有一个相关的新的简单复合体，以前没有研究过，它的性质可能有助于有效地约束这种可靠性措施。一种新的方法是先用合适的小波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.