Thesis (Ph. D.)--Auburn University. / Abstract. Vita. Includes bibliographical references (p. 42-43).

Les graphes définis à partir de structures algébriques possèdent d’excellentes propriétés de symétries particulièrement intéressantes. L’exemple le plus flagrant est la notion de graphe de Cayley qui s’est révélée très riche non seulement du point de vue théorique mais aussi pratique par ses applications à de nombreux domaines incluant l’architecture des réseaux ou les machines parallèles. Néanmoins, la régularité des graphes de Cayley se révèle parfois être une limite étant donné qu’ils sont toujours sommet-transitifs et donc en particulier non pertinents pour générer des réseaux semiréguliers.Cette observation a motivé, en 2005, la définition d’une nouvelle classe de graphes définis à partir d’un groupe, appelés G-graphes. Ils possèdent aussi de nombreuses propriétés de régularité mais de manière moins restrictive.Cette thèse propose un nouveau regard sur cette classe de graphes par une approche plutôt orientée recherche opérationnelle alors que la grande majorité des études précédentes est dominée par des approches essentiellement algébriques. Nous-nous sommes alors intéressés à plusieurs questions :— La caractérisation des G-graphes : nous proposons des améliorations par rapport aux précédents résultats.— Identifier des classes de graphes comme des G-graphes grâce à des isomorphismes ou en utilisant le théorème de caractérisation.— Etudier la structure et les propriétés de ces graphes, en particulier pour de possibles applications aux réseaux : colorations semi-régulières, symétries et robustesse.— Une approche algorithmique pour la reconnaissance de cette classe avec notamment un premier exemple de cas polynomial lorsque le groupe est abélien. / Interactions between graph theory and group theory have already led to interesting results for both domains. Graphs defined from algebraic groups have highly symmetrical structure giving birth to interesting properties. The most famous example is Cayley graphs, which revealed to be particularly interesting both from a theoretical and a practical point of view due to their applications in several domains including network architecture or parallel machines. Nevertheless, the regularity of Cayley graphs is also a limit as they are always vertex-transitive and therefore not relevant to generate semi-regular networks. This observation motivated the definition, in 2005, of a new family of graphs defined from a group, called G-graphs. They also have many regular properties but are less restrictive. These graphs are in particular semi-regular k-partite, with a chromatic number k directly given in the group representation and they can be either transitive or not.This thesis proposes a new insight into this class of graphs using an approach based on operational research while most of previous studies have been so far dominated by algebraic approaches. Then, the thesis addresses different kind of questions:— Characterizing G-graphs: we propose improvements of previous results.— Identifying some classes of graphs as G-graphs through isomorphism or using the characterization theorem.— Studying the structure and properties of these graphs, in particular for possible applications to networks: semi-regular coloring, symmetries and robustness.— Algorithmic approach for recognizing this class with a first example of polynomial case when the group is abelian.

The programming languages have always been a useful tool for sustainability and continuity of advancements in multilayered fields as like education and common industries. PHP is a widespread instrument which has an important place in this practices just like many other programming languages. This research is conducted to understand PHP language better and to increase academical familiarity about it. In the end of this practice a basic PHP library for further use has been produced as a representation of the research which has been conducted. This library allows the user to create multivariate basic graphs for visualization of variable different data and it’s hoped that it will be used by further educational and business purposes by various individuals.

In this dissertation several results in Steinhaus graphs are investigated. First under some further conditions imposed on the induced cycles in steinhaus graphs, the order of induced cycles in Steinhaus graphs is at most [(n+3)/2]. Next the results of maximum clique size in Steinhaus graphs are used to enumerate the Steinhaus graphs having maximal cliques. Finally the concept of jumbled graphs and Posa's Lemma are used to show that almost all Steinhaus graphs are Hamiltonian.

<p> Personalized PageRank with high teleportation probability enables exploring the environment of a seed. With this insight, one can use an orthogonal factorization of a set of personalized PageRank vectors, like SVD, to derive a 2-dimensional representation of the network. This can be done for the whole network or a smaller piece. The power of this method lies in the fact that only a few columns, compared to the size of the networks, can be used to generate a local representation of the part of the network we are interested in. This technique has the potential to be seamlessly used for higher order structures, such as hypergraphs which have found a great deal of use for real-world data. This work investigates the characteristics of personalized PageRank and how it compares to the transition probabilities on the graph in terms of their ability to develop low dimensional representations. A key focus of the thesis are the similarities between the embeddings generated due to PageRank and those generated by spectral methods.</p>

<p> Graphs are defined as a special kind of relational system and an analogue of Birkhoff's Representation Theorem for Universal Algebras is proved. The notion of ultrapower, a specialization of the ultraproducts introduced into Mathematical Logic by Tarski, Scott and others, is demonstrated to provide a unifying framework within which various problems of graph theory and infinite combinatorial mathematics can be formulated and solved. Thus, theorems extending to the infinite case results of N.G.de Bruijn and P.Erdős in graph colouring, and of P. and M. Hall in combinatorial set theory are proved via the method of ultrapowers. Finally, the problem of embedding graphs in certain topological spaces is taken up, and a characterization of infinite connected planar graphs is derived (see Introduction). </p> / Thesis / Doctor of Philosophy (PhD)

<p> Extremal problems for asymmetric graphs and asymmetries of graphs are jointly investigated using function A(p,q). Values 0,1 of function are characterized.</p> / Thesis / Master of Science (MSc)

It is shown that the cardinal product of graphs does not satisfy unique prime factorization even for a very restrictive class of graphs. It is also proved that every connected graph has a decomposition as a weak cartesian product into indecomposable factors and that this decomposition is unique to within isomorphisms. This latter result is established by considering a certain class of equivalence relations on the edge set of a graph and proving that this collection is a principal filter in the lattice of all equivalences. The least element of this filter is then used to decompose the graph into a weak cartesian product of prime graphs that is unique to within isomorphisms. / Thesis / Doctor of Philosophy (PhD)

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