Graphs and digraphs

第一部分的建议的研究中心组合多项式，从图着色网络的可靠性和独立多项式。代数和分析技术不仅用于限制无向和有向情况下的网络可靠性，而且还用于检查底层组合结构。定位多项式的根将更清楚地揭示潜在系数的突出单峰问题，以及提供更多的洞察力到一个新定义的类分形与独立的图形和单纯复形。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度量。特别是，有一个相关的新的单纯复形，以前没有研究过，它的属性可能有助于有效地约束这个可靠性措施。一种新的方法将涉及压缩的有向和无向图首先与适当的小波。

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.