Matching Items (3)
Description
A tiling is a collection of vertex disjoint subgraphs called tiles. If the tiles are all isomorphic to a graph $H$ then the tiling is an $H$-tiling. If a graph $G$ has an $H$-tiling which covers all of the vertices of $G$ then the $H$-tiling is a perfect $H$-tiling or an $H$-factor. A goal of this study is to extend theorems on sufficient minimum degree conditions for perfect tilings in graphs to directed graphs. Corrádi and Hajnal proved that every graph $G$ on $3k$ vertices with minimum degree $delta(G)ge2k$ has a $K_3$-factor, where $K_s$ is the complete graph on $s$ vertices. The following theorem extends this result to directed graphs: If $D$ is a directed graph on $3k$ vertices with minimum total degree $delta(D)ge4k-1$ then $D$ can be partitioned into $k$ parts each of size $3$ so that all of parts contain a transitive triangle and $k-1$ of the parts also contain a cyclic triangle. The total degree of a vertex $v$ is the sum of $d^-(v)$ the in-degree and $d^+(v)$ the out-degree of $v$. Note that both orientations of $C_3$ are considered: the transitive triangle and the cyclic triangle. The theorem is best possible in that there are digraphs that meet the minimum degree requirement but have no cyclic triangle factor. The possibility of added a connectivity requirement to ensure a cycle triangle factor is also explored. Hajnal and Szemerédi proved that if $G$ is a graph on $sk$ vertices and $delta(G)ge(s-1)k$ then $G$ contains a $K_s$-factor. As a possible extension of this celebrated theorem to directed graphs it is proved that if $D$ is a directed graph on $sk$ vertices with $delta(D)ge2(s-1)k-1$ then $D$ contains $k$ disjoint transitive tournaments on $s$ vertices. We also discuss tiling directed graph with other tournaments. This study also explores minimum total degree conditions for perfect directed cycle tilings and sufficient semi-degree conditions for a directed graph to contain an anti-directed Hamilton cycle. The semi-degree of a vertex $v$ is $min{d^+(v), d^-(v)}$ and an anti-directed Hamilton cycle is a spanning cycle in which no pair of consecutive edges form a directed path.
ContributorsMolla, Theodore (Author) / Kierstead, Henry A (Thesis advisor) / Czygrinow, Andrzej (Committee member) / Fishel, Susanna (Committee member) / Hurlbert, Glenn (Committee member) / Spielberg, Jack (Committee member) / Arizona State University (Publisher)
Created2013
Description
In this thesis, I investigate the C*-algebras and related constructions that arise from combinatorial structures such as directed graphs and their generalizations. I give a complete characterization of the C*-correspondences associated to directed graphs as well as results about obstructions to a similar characterization of these objects for generalizations of directed graphs. Viewing the higher-dimensional analogues of directed graphs through the lens of product systems, I give a rigorous proof that topological k-graphs are essentially product systems over N^k of topological graphs. I introduce a "compactly aligned" condition for such product systems of graphs and show that this coincides with the similarly-named conditions for topological k-graphs and for the associated product systems over N^k of C*-correspondences. Finally I consider the constructions arising from topological dynamical systems consisting of a locally compact Hausdorff space and k commuting local homeomorphisms. I show that in this case, the associated topological k-graph correspondence is isomorphic to the product system over N^k of C*-correspondences arising from a related Exel-Larsen system. Moreover, I show that the topological k-graph C*-algebra has a crossed product structure in the sense of Larsen.
ContributorsPatani, Nura (Author) / Kaliszewski, Steven (Thesis advisor) / Quigg, John (Thesis advisor) / Bremner, Andrew (Committee member) / Kawski, Matthias (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2011
Description
The Resource Description Framework (RDF) is a specification that aims to support the conceptual modeling of metadata or information about resources in the form of a directed graph composed of triples of knowledge (facts). RDF also provides mechanisms to encode meta-information (such as source, trust, and certainty) about facts already existing in a knowledge base through a process called reification. In this thesis, an extension to the current RDF specification is proposed in order to enhance RDF triples with an application specific weight (cost). Unlike reification, this extension treats these additional weights as first class knowledge attributes in the RDF model, which can be leveraged by the underlying query engine. Additionally, current RDF query languages, such as SPARQL, have a limited expressive power which limits the capabilities of applications that use them. Plus, even in the presence of language extensions, current RDF stores could not provide methods and tools to process extended queries in an efficient and effective way. To overcome these limitations, a set of novel primitives for the SPARQL language is proposed to express Top-k queries using traditional query patterns as well as novel predicates inspired by those from the XPath language. Plus, an extended query processor engine is developed to support efficient ranked path search, join, and indexing. In addition, several query optimization strategies are proposed, which employ heuristics, advanced indexing tools, and two graph metrics: proximity and sub-result inter-arrival time. These strategies aim to find join orders that reduce the total query execution time while avoiding worst-case pattern combinations. Finally, extensive experimental evaluation shows that using these two metrics in query optimization has a significant impact on the performance and efficiency of Top-k queries. Further experiments also show that proximity and inter-arrival have an even greater, although sometimes undesirable, impact when combined through aggregation functions. Based on these results, a hybrid algorithm is proposed which acknowledges that proximity is more important than inter-arrival time, due to its more complete nature, and performs a fine-grained combination of both metrics by analyzing the differences between their individual scores and performing the aggregation only if these differences are negligible.
ContributorsCedeno, Juan Pablo (Author) / Candan, Kasim S (Thesis advisor) / Chen, Yi (Committee member) / Sapino, Maria L (Committee member) / Arizona State University (Publisher)
Created2010