Flippable Pairs and Subset Comparisons in Comparative Probability Orderings and Related Simple Games

We show that every additively representable comparative probability order on n atoms is determined by at least n - 1 binary subset comparisons. We show that there are many orders of this kind, not just the lexicographic order. These results provide answers to two questions of Fishburn et al (2002). We also study the flip relation on the class of all comparative probability orders introduced by Maclagan. We generalise an important theorem of Fishburn, Peke?c and Reeds, by showing that in any minimal set of comparisons that determine a comparative probability order, all comparisons are flippable. By calculating the characteristics of the flip relation for n = 6 we discover that the regions in the corresponding hyperplane arrangement can have no more than 13 faces and that there are 20 regions with 13 faces. All the neighbours of the 20 comparative probability orders which correspond to those regions are representable. Finally we define a class of simple games with complete desirability relation for which its strong desirability relation is acyclic, and show that the flip relation carries all the information about these games. We show that for n = 6 these games are weighted majority games

On the limitations of the use of solvable groups in Cayley graph cage constructions

AbstractA (k,g)-cage is a (connected) k-regular graph of girth g having smallest possible order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, there appears to be no general theory of the relationship between the girth of a Cayley graph and the structure of the underlying group. We attempt to fill this gap by focusing on the girth of Cayley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive notion that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girth. Specifically, we establish the existence of upper bounds on the girth of Cayley graphs with respect to the nilpotency class and/or the derived length of the underlying group, when this group is nilpotent or solvable, respectively

Resolution of a conjecture about linking ring structures

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Poto\v{c}nik and S. Wilson, titled `Linking rings structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7 (2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper