Research Doctorate - Doctor of Philosophy (PhD) / Directed graphs are combinatorial objects used to model networks like fluid-flow systems in which the direction of movement through the network is important. In 1980, Enomoto and Watatani used finite directed graphs to provide an intuitive framework for the Cuntz-Krieger algebras introduced by Cuntz and Krieger earlier in the same year. The theory of the C*-algebras of directed graphs has since been extended to include infinite graphs, and there is an elegant relationship between connectivity and loops in a graph and the structure theory of the associated C*-algebra. Higher-rank graphs are a higher-dimensional analogue of directed graphs introduced by Kumjian and Pask in 2000 as a model for the higher-rank Cuntz-Krieger algebras introduced by Robertson and Steger in 1999. The theory of the Cuntz-Krieger algebras of higher-rank graphs is relatively new, and a number of questions which have been answered for directed graphs remain open in the higher-rank setting. In particular, for a large class of higher-rank graphs, the gauge-invariant ideal structure of the associated C*-algebra has not yet been identified. This thesis addresses the question of the gauge-invariant ideal structure of the Cuntz-Krieger algebras of higher-rank graphs. To do so, we introduce and analyse the collections of relative Cuntz-Krieger algebras associated to higher-rank graphs. The first two main results of the thesis are versions of the gauge-invariant uniqueness theorem and the Cuntz-Krieger uniqueness theorem which apply to relative Cuntz-Krieger algebras. Using these theorems, we are able to achieve our main goal, producing a classification of the gauge-invariant ideals in the Cuntz-Krieger algebra of a higher-rank graph analogous to that developed for directed graphs by Bates, Hong, Raeburn and Szymañski in 2002. We also demonstrate that relative Cuntz-Krieger algebras associated to higher-rank graphs are always nuclear, and produce conditions on a higher-rank graph under which the associated Cuntz-Krieger algebra is simple and purely infinite.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Operations Research." Includes bibliographical references (p. 132-138).

Thesis (M.S.)--Ohio University, March, 2004. / Title from PDF t.p. Includes bibliographical references (leaves 51-55).

Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2008/05/29). Includes bibliographical references.

Thesis (M.Sc.). / Written for the Dept. of Mathematics and Statistics. Title from title page of PDF (viewed 2008/04/12). Includes bibliographical references.

Thesis (Ph.D.) - Carleton University, 2002. / Includes bibliographical references (p. 98-100). Also available in electronic format on the Internet.

Konstanz, Univ., Diplomarb., 2000.

Zugl.: Clausthal, Techn. Univ., Habil.-Schr., 2008

Techn. University, Diss., 2004--Clausthal.

Interval graphs admit elegant ordering and structural characterizations. A natural digraph analogue of interval graphs, called chronological interval digraphs, has recently been identified and studied. We introduce the class of chronological rectangle digraphs, and show that they are a higher dimensional analogue of chronological interval digraphs. A main goal of this thesis is to establish a foundation of knowledge about this class, including basic properties and an ordering characterization. Our most significant result is a forbidden induced subdigraph characterization for the series-parallel digraphs which are chronological rectangle. We also discuss obtaining chronological rectangle digraphs from orientations of graphs. In addition we introduce the related concept of the chronological interval dimension of a digraph, and determine the digraphs for which it is defined. Unit and proper chronological rectangle digraphs, defined analogously to unit and proper interval graphs, are also introduced and studied. / Graduate