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Symmetries of the Woolly Hat graphs
No full textA graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length. While the tetravalent edge-transitive graphs admitting a semiregular automorphism with only one orbit are easy to determine, those that admit a semiregular automorphism with two orbits took a considerable effort and were finally classified in 2012. Of the several possible different "types" of potential tetravalent edge-transitive graphs admitting a semiregular automorphism with three orbits, only one "type" has thus far received no attention. In this paper we focus on this class of graphs, which we call the Woolly Hat graphs. We prove that there are in fact no edge-transitive Woolly Hat graphs and classify the vertex-transitive ones
Prehod od ploskovnega zapiranja do standardnega zapiranja pletenic v R^3, v ročajnih telesih in v odebeljenih ploskvah
No full textGiven a knot or link in the form of plat closure of a braid, we describe an algorithm to obtain a braid representing the same knot or link with the standard closure, and vice-versa. We analyze the three cases of knots and links in ℝ^3, in handlebodies and in thickened surfaces, and we give for each one a detailed construction of the relative algorithm, with proofs of the results and the calculation of the computational complexity. Indeed, we show that the algorithm has quadratic computational complexity in the number of crossings and loop generators of the braid when passing from plat to standard closure, while it has linear computational complexity in the number of crossings generators of the braid when passing from standard to plat closure. The article includes numerous illustrations to facilitate the understanding of the results, as well as an example of implementation of the algorithm for the general case
Ehrhartove limite
No full textWe introduce the definition of an Ehrhart limit, that is, a formal power series with integer coefficients that is the limit in the ring of formal power series of a sequence of Ehrhart h*-polynomials. We identify a variety of examples of sequences of polytopes that yield Ehrhart limits, with a focus on reflexive polytopes and simplices.Vpeljemo definicijo Ehrhartove limite, formalne potenčne vrste s celoštevilskimi koeficienti, ki je v kolobarju potenčnih vrst limita zaporedja Ehrhartovih h∗-polinomov. Identificiramo različne primere zaporedij politopov, ki imajo Ehrhartove limite, s poudarkom na refleksivnih politopih in simpleksih
Zvesti in tanki nepolitopski manipleksi
No full textManiplexes are coloured graphs that generalise maps on surfaces and abstract polytopes. Each maniplex uniquely defines a partially ordered set that encodes information about its structure. When this poset is an abstract polytope, we say that the associated maniplex is polytopal. Maniplexes that have two properties, called faithfulness and thinness, are completely determined by their associated poset, which is often an abstract polytope. We show that all faithful thin maniplexes of rank three are polytopal. So far only one example, of rank four, of a thin maniplex that is not polytopal was known. We construct the first infinite family of maniplexes that are faithful and thin but are non-polytopal for all ranks greater than three.Manipleksi so barvni grafi, ki predstavljajo posplošitev tako zemljevidov na ploskvah kot tudi abstraktnih politopov. Vsak manipleks enolično določa delno urejeno množico, ki vsebuje informacijo o njegovi strukturi. Kadar je ta delno urejena množica abstrakten politop, pravimo, da je ustrezen manipleks politopski. Manipleksi, ki imajo dve lastnosti, imenovani zvestost in tankost, so povsem določeni z njim pridruženo delno urejeno množico, ki je pogosto abstrakten politop. Dokažemo, da so vsi zvesti tanki manipleksi ranga tri politopski. Doslej je bil znan samo en primer, in sicer ranga štiri, tankega manipleksa, ki ni politopski. Konstruiramo prvo neskončno družino manipleksov, ki so zvesti in tanki, niso pa politopski, in sicer za vse range večje kot tri
The genetic trail of the invasive mosquito species Aedes koreicus from the east to the west of Northern Italy
No full textBackground Aedes koreicus is native to Far East Asia and recorded in Europe since 2008. In Italy, Ae. koreicus is widespread throughout the Northern part of the peninsula, highlighting its invasive potential and spread. However, no clear clues about the dispersal patterns of the species have been collected so far. Methodology/Principal findings Population genetic analyses were performed to assess the genetic structure of populations of Ae. koreicus and to make hypotheses about its dispersal patterns in Northern Italy. Ten microsatellite markers specific for Ae. koreicus were used to genotype 414 individuals from 13 populations in the pre-alpine area of Italy, and neighboring Slovenia. Basic and Bayesian population genetic analyses were performed to evaluate patterns of genetic variation, genetic structure, and demography of selected mosquito populations. While presenting a certain degree of structuring, the Italian and Slovenian populations of Ae. koreicus were poorly differentiated. Moreover, demographic analysis supports the expansion of a single population propagule of Ae. koreicus in Italy and Slovenia and provides evidence of the presence of overwintering populations in the studied area. Conclusions/Significance Our results highlight a common origin, and stable colonization of Northern Italy and Slovenia, as a probable consequence of the expansion of a unique population. This stresses out the importance of continuous monitoring of Ae. koreicus, to finally uncover the geographic origins and entrance pathways of invasive populations and to prevent or limit further introductions
