Circulant Digraph Isomorphisms

We determine necessary and sufficient conditions for a Cayley digraph of the cyclic group of order n to have the property that any other Cayley digraph of a cyclic group of order n is isomorphic to the first if and only if an isomorphism between the two digraphs is a group automorphism of the cyclic group of order n

Self-Complementary Arc-Transitive Graphs and Their Imposters

This thesis explores two infinite families of self-complementary arc-transitive graphs: the familiar Paley graphs and the newly discovered Peisert graphs. After studying both families, we examine a result of Peisert which proves the Paley and Peisert graphs are the only self-complementary arc transitive graphs other than one exceptional graph. Then we consider other families of graphs which share many properties with the Paley and Peisert graphs. In particular, we construct an infinite family of self-complementary strongly regular graphs from affine planes. We also investigate the pseudo-Paley graphs of Weng, Qiu, Wang, and Xiang. Finally, we prove a lower bound on the number of maximal cliques of certain pseudo-Paley graphs, thereby distinguishing them from Paley graphs of the same order

Éléments explicites en théorie algébrique des nombres

This report consists in a synthesis of my research activities in algebraic number theory, between 2003 and 2013, on my own or with colleagues. The main goal is the study of the Galois module structure of modules associated to number field extensions, under various hypothesis, specifically about ramification. We also present results about other subjects which came into the way of the previous study: the construction of a certain type of Galois extensions of the field of rationals, the complexity of self-dual normal bases for multiplication in finite fields, and a bit of combinatorics. We stress the importance of an explicit knowledge of the objects under study.Ce mémoire présente une synthèse de mes travaux de recherche en théorie algébrique des nombres menés entre 2003 et 2013, seul ou en collaboration. Ils portent principalement sur l'étude de la structure galoisienne de modules associés à des extensions de corps de nombres, sous diverses hypothèses en particulier de ramification. Ils abordent aussi des thèmes rencontrés chemin faisant : construction d'un certain type d'extensions galoisiennes du corps des rationnels, complexité des bases normales auto-duales pour la multiplication dans les corps finis, un peu de combinatoire. Dans la présentation de tous ces travaux, l'accent est mis sur l'aspect explicite des objets étudiés

Constructions of Self-Complementary Circulants With No Multiplicative Isomorphisms

The main topic of the paper is the question of the existence of self-complementary Cayley graphs Cay(G; S) with the property S oe 6= G # n S for all oe 2 Aut(G). We answer this question in the positive by constructing an infinite family of selfcomplementary circulants with this property. Moreover, we obtain a complete classification of primes p for which there exist self-complementary circulants of order p 2 with this property. 1 Introduction All graphs we consider in this paper have neither loops nor multiple edges. For a graph \Gamma, denote by V \Gamma and E \Gamma the vertex-set and edge-set, respectively. Let \Gamma be the complement of \Gamma. A graph \Gamma is said to be self-complementary if it is isomorphic to its complement \Gamma. Denote by Aut \Gamma the full automorphism group of \Gamma. It easily follows that, for any graph \Gamma, Aut \Gamma = Aut \Gamma. A graph \Gamma is said to be vertex-transitive if Aut \Gamma is transitive on V \Gamma. For a finite group G, ..