Dominating sets in Kneser graphs

This thesis investigates dominating sets in Kneser graphs as well as a selection of other topics related to graph domination. Dominating sets in Kneser graphs, especially those of minimum size, often correspond to interesting combinatorial incidence structures. We begin with background on the dominating set problem and a review of known bounds, focusing on algebraic bounds. We then consider this problem in the Kneser graphs, discussing basic results and previous work. We compute the domination number for a few of the Kneser graphs with the aid of some original results. We also investigate the connections between Kneser graph domination and the theory of combinatorial designs, and introduce a new type of design that generalizes the properties of dominating sets in Kneser graphs. We then consider dominating sets in the vector space analogue of Kneser graphs. We end by highlighting connections between the dominating set problem and other areas of combinatorics. Conjectures and open problems abound

2012 - The Seventeenth Annual Symposium of Student Scholars

The full program book from the Seventeenth Annual Symposium of Student Scholars, held on April 10, 2012. Includes abstracts from the presentations and posters.https://digitalcommons.kennesaw.edu/sssprograms/1011/thumbnail.jp

Fast Scramblers, Horizons and Expander Graphs

We propose that local quantum systems defined on expander graphs provide a simple microscopic model for thermalization on quantum horizons. Such systems are automatically fast scramblers and are motivated from the membrane paradigm by a conformal transformation to the so-called optical metric.Comment: 22 pages, 2 figures. Added further discussion in section 3. Added reference

Markov Properties and almost Markov properties in one, two or more directions

In this review-type paper written at the occasion of the Oberwolfach workshop {\em One-sided vs. Two-sided stochastic processes} (february 22-29, 2020), we discuss and compare Markov properties and generalisations thereof in more directions, as well as weaker forms of conditional dependence, again either in one or more directions. In particular, we discuss in both contexts various extensions of Markov chains and Markov fields and their properties, such as $g$-measures, Variable Length Markov Chains, Variable Neighbourhood Markov Fields, Variable Neighbourhood (Parsimonious) Random Fields, and Generalized Gibbs Measures.Comment: arXiv admin note: substantial text overlap with arXiv:1812.0615