Counting dominating sets and related structures in graphs

We consider some problems concerning the maximum number of (strong) dominating sets in a regular graph, and their weighted analogues. Our primary tool is Shearer's entropy lemma. These techniques extend to a reasonably broad class of graph parameters enumerating vertex colorings satisfying conditions on the multiset of colors appearing in (closed) neighborhoods. We also generalize further to enumeration problems for what we call existence homomorphisms. Here our results are substantially less complete, though we do solve some natural problems

An entropy proof of the Kahn-Lovasz theorem

Bregman [2], gave a best possible upper bound for the number of perfect matchings in a balanced bipartite graph in terms of its degree sequence. Recently Kahn and Lovasz [8] extended Bregman’s theorem to general graphs. In this paper, we use entropy methods to give a new proof of the Kahn-Lovasz theorem. Our methods build on Radhakrishnan’s [9] use of entropy to prove Bregman’s theorem

The organisation and practices of mapping rural statistics: a case study of Wales

Working at the Wales Rural Observatory (2004 - 2014) provided first-hand experience of mapping rural statistics for policymaking. It was evident that representing social and economic data in population-sparse areas was not as straightforward as the technology permitted. Mapping could reveal rural issues but also caused others to be hidden or misrepresented. This was an issue worthy of further investigation. How was this understood by others attempting to represent social and economic statistics? Were mistakes made, could they be rectified, and what were the consequences? Literature linked to the topic was fragmented; split between the technical, theoretical and practical. This research has attempted a synthesis, helping develop concepts to guide this research and a lens for understanding mapping practices within organisations. A case study of Wales was used to investigate mapping practices used for policymaking and planning, applying qualitative methods to study quantitative practices. Studying mapping required more than technical knowledge and more than just critique, it required the study of mapping in context, and more so the detail of these processes in action. As such this research focussed on the experiences of those closest to these processes in an attempt to sensitise future studies to often overlooked interactions. Multiple barriers existed in Wales and included a lack of technical awareness, capacity, and appropriate training. To overcome these barriers the literature suggests that mapping practices become collaborative activities. However this should not be just in the formation or presentation stages but throughout the mapping process. As a collective all resources can be pooled and used many times, with common rules defined through a process of debate and learning, with all forms of knowing admissible. The technology is certainly in place to enable this to happen. The challenge going forward is raising awareness and creating frameworks that enable this to happen