Inequivalence of difference sets: on a remark of Baumert

An often cited statement of Baumert in his book Cyclic difference sets asserts that four well known families of cyclic (4t - 1,2t - 1,t - 1) difference sets are inequivalent, apart from a small number of exceptions with t ≤ 8. We are not aware of a proof of this statement in the literature. Three of the families discussed by Baumert have analogous constructions in non-cyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases

On highly regular digraphs

We explore directed strongly regular graphs (DSRGs) and their connections to association schemes and finite incidence structures. More specically, we study flags and antiflags of finite incidence structures to provide explicit constructions of DSRGs. By using this connection between the finite incidence structures and digraphs, we verify the existence and non-existence of $1\frac{1}{2}$-designs with certain parameters by the existence and non-existence of corresponding digraphs, and vice versa. We also classify DSRGs of given parameters according to isomorphism classes. Particularly, we examine the actions of automorphism groups to provide explicit examples of isomorphism classes and connection to association schemes. We provide infinite families of vertex-transitive DSRGs in connection to non-commutative association schemes. These graphs are obtained from tactical configurations and coset graphs

Quantum entanglement: insights via graph parameters and conic optimization

In this PhD thesis we study the effects of quantum entanglement, one of quantum mechanics most peculiar features, in nonlocal games and communication problems in zero-error information theory. A nonlocal game is a thought experiment in which two cooperating players, who are forbidden to communicate, want to perform a certain task. Zero-error information theory is the mathematical field that studies communication problems where no error is tolerated. The unifying link among the various scenarios we consider is their combinatorial nature and in particular their reformulations as graph theoretical problems, mainly concerning the chromatic and stability numbers and some quantum generalizations thereof. In this thesis we propose a novel approach to the study of these quantum graph parameters using the paradigm of conic optimization. For that, we introduce and study the completely positive semidefinite cone, a new matrix cone consisting of all symmetric matrices that admit a Gram representation by positive semidefinite matrices. Furthermore, we investigate whether entanglement allows for better-than-classical communication schemes in some well-known problems from zero-error information theory. For example we study the channel coding problem, which asks a sender to transmit data reliably to a receiver in the presence of noise, as well as some of its generalizations

On 2-arc-transitivity of Cayley graphs

AbstractThe classification of 2-arc-transitive Cayley graphs of cyclic groups, given in (J. Algebra. Combin. 5 (1996) 83–86) by Alspach, Conder, Xu and the author, motivates the main theme of this article: the study of 2-arc-transitive Cayley graphs of dihedral groups. First, a previously unknown infinite family of such graphs, arising as covers of certain complete graphs, is presented, leading to an interesting property of Singer cycles in the group PGL(2,q), q an odd prime power, among others. Second, a structural reduction theorem for 2-arc-transitive Cayley graphs of dihedral groups is proved, putting us—modulo a possible existence of such graphs among regular cyclic covers over a small family of certain bipartite graphs—a step away from a complete classification of such graphs. As a byproduct, a partial description of 2-arc-transitive Cayley graphs of abelian groups with at most three involutions is also obtained