On product-one sequences over dihedral groups

Let $G$ be a finite group. A sequence over $G$ means a finite sequence of terms from $G$, where repetition is allowed and the order is disregarded. A product-one sequence is a sequence whose elements can be ordered such that their product equals the identity element of the group. The set of all product-one sequences over $G$ (with concatenation of sequences as the operation) is a finitely generated C-monoid. Product-one sequences over dihedral groups have a variety of extremal properties. This article provides a detailed investigation, with methods from arithmetic combinatorics, of the arithmetic of the monoid of product-one sequences over dihedral groups.Comment: to appear in Journal of Algebra and its Application

Diameter, Girth And Other Properties Of Highly Symmetric Graphs

We consider a number of problems in graph theory, with the unifying theme being the properties of graphs which have a high degree of symmetry. In the degree-diameter problem, we consider the question of finding asymptotically large graphs of given degree and diameter. We improve a number of the current best published results in the case of Cayley graphs of cyclic, dihedral and general groups. In the degree-diameter problem for mixed graphs, we give a new corrected formula for the Moore bound and show non-existence of mixed Cayley graphs of diameter 2 attaining the Moore bound for a range of open cases. In the degree-girth problem, we investigate the graphs of Lazebnik, Ustimenko and Woldar which are the best asymptotic family identified to date. We give new information on the automorphism groups of these graphs, and show that they are more highly symmetrical than has been known to date. We study a related problem in group theory concerning product-free sets in groups, and in particular those groups whose maximal product-free subsets are complete. We take a large step towards a classification of such groups, and find an application to the degree-diameter problem which allows us to improve an asymptotic bound for diameter 2 Cayley graphs of elementary abelian groups. Finally, we study the problem of graphs embedded on surfaces where the induced map is regular and has an automorphism group in a particular family. We give a complete enumeration of all such maps and study their properties