Super- and sub-additive envelopes of aggregation functions: interplay between local and global properties, and approximation

Super- and sub-additive transformations of aggregation functions have been recently introduced by Greco, Mesiar, Rindone and Sipeky [The superadditive and the subadditive transformations of integrals and aggregation functions, Fuzzy Sets and Systems 291 (2016), 40{53]. In this article we give a survey of the recent development regarding the existence of aggregation functions with a preassigned super- and sub-additive transformation, and address approximation of these transformations. The underpinning feature of the presented results is dependence of global properties of super- and sub-additive transformations on local properties of aggregation functions

Injectivity radius of representations of triangle groups and planar width of regular hypermaps

We develop a rigorous algebraic background for representations of triangle groups in linear groups over algebras arising from factor rings of multivariate polynomial rings. This is then used to substantially improve the existing bounds on the order of epimorphic images of triangle groups with a given injectivity radius and, analogously, the size of the associated hypermaps of a given type with a given planar width

Aggregation functions with given super-additive and sub-additive transformations

Aggregation functions and their transformations have found numerous applications in various kinds of systems as well as in economics and social science. Every aggregation function is known to be bounded above and below by its super-additive and sub-additive transformations. We are interested in the “inverse” problem of whether or not every pair consisting of a super-additive function dominating a sub-additive function comes from some aggregation function in the above sense. Our main results provide a negative answer under mild extra conditions on the super- and sub-additive pair. We also show that our results are, in a sense, best possible

Small vertex-transitive graphs of given degree and girth

We investigate the basic interplay between the small k-valent vertex-transitive graphs of girth g and the (k, g)-cages, the smallest k-valent graphs of girth g. We prove the existence of k-valent Cayley graphs of girth g for every pairof parameters k ≥ 2 and g ≥ 3, improve the lower bounds on the order of the smallest (k, g) vertex-transitive graphs forcertain families with prime power girth, and generalize the construction of Bray, Parker and Rowley that has yielded several of the smallest known (k, g)-graphs

Large graphs of diameter two and given degree

Let r(d, 2), C(d, 2), and AC(d, 2) be the largest order of a regular graph, a Cayley graph, and a Cayley graph of an Abelian group, respectively, of diameter 2 and degree d. The best currently known lower bounds on these parameters are r(d, 2) ≥ $d^2$ − d + 1 for d − 1 an odd prime power (with a similar result for powers of two), C(d, 2) ≥ (d + 1)$^2$/2 for degrees d = 2q − 1 where q is an odd prime power, and AC(d, 2) ≥ (3/8)($d^2$ − 4) where d = 4q − 2 for an odd prime power q. Using a number theory result on distribution of primes we prove, for all sufficiently large d, lower bounds on r(d, 2), C(d, 2), and AC(d, 2) of the form c · $d^2$ − O($d^1.525$) for c = 1, 1/2, and 3/8, respectively. We also prove results of a similar flavour for vertex transitive graphs and Cayley graphs of cyclic groups.Peer Reviewe