A 2-arc Transitive Hexavalent Nonnormal Cayley Graph on A119

A Cayley graph Γ=Cay(G,S) is said to be normal if the base group G is normal in AutΓ. The concept of the normality of Cayley graphs was first proposed by M.Y. Xu in 1998 and it plays a vital role in determining the full automorphism groups of Cayley graphs. In this paper, we construct an example of a 2-arc transitive hexavalent nonnormal Cayley graph on the alternating group A119. Furthermore, we determine the full automorphism group of this graph and show that it is isomorphic to A120.

Geometry of the Wiman–Edge monodromy

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

Neighbor Connectivity of the Alternating Group Graph

Given a graph [Formula: see text], its neighbor connectivity is the least number of vertices whose deletion along with their neighbors results in a disconnected, complete, or empty graph. The edge neighbor connectivity is the least number of edges whose deletion along with their endpoints results in a disconnected, complete, or empty graph. In this paper, we determine the neighbor connectivity [Formula: see text] and the edge neighbor connectivity [Formula: see text] of the alternating group graph. We show that [Formula: see text], where [Formula: see text] is the [Formula: see text]-dimensional alternating group graph.

COMMUTING PROBABILITY OF COMPACT GROUPS

For any (Hausdorff) compact group G, denote by $\mathrm{cp}(G)$ the probability that a randomly chosen pair of elements of G commute. We prove that there exists a finite group H such that $\mathrm{cp}(G)= {\mathrm{cp}(H)}/{|G:F|^2}$ , where F is the FC-centre of G and H is isoclinic to F with $\mathrm{cp}(F)=\mathrm{cp}(H)$ whenever $\mathrm{cp}(G)>0$ . In addition, we prove that a compact group G with $\mathrm{cp}(G)>\tfrac {3}{40}$ is either solvable or isomorphic to $A_5 \times Z(G)$ , where $A_5$ denotes the alternating group of degree five and the centre $Z(G)$ of G contains the identity component of G.

The Bounds of Generalized 4-Connectivity of Alternating Group Graphs

Generalized connectivity is a parameter of evaluating the reliability of a network. Let [Formula: see text] be a vertex set of graph [Formula: see text] and [Formula: see text], there is a tree in [Formula: see text] as it connects all vertices of [Formula: see text] which is called a [Formula: see text]-tree. Let [Formula: see text] and [Formula: see text] be two [Formula: see text]-trees in [Formula: see text]. If [Formula: see text] and [Formula: see text] are given, we define that two [Formula: see text]-trees [Formula: see text] and [Formula: see text] are internally disjoint. Then the maximum number of internally disjoint [Formula: see text]-trees in [Formula: see text] is denoted by [Formula: see text]. For an integer [Formula: see text] with [Formula: see text], the generalized [Formula: see text]-connectivity of a graph [Formula: see text] is denoted as [Formula: see text]. In this paper, we study the generalized [Formula: see text]-connectivity of alternating group graphs, denoted by [Formula: see text]. We show the upper bound and the lower bound of [Formula: see text], [Formula: see text] for [Formula: see text], and that [Formula: see text] is given.

Determinants of the varied profiles of Plasmodium falciparum infections among infants living in Kintampo, Ghana

Background Understanding why some infants tolerate infections, remaining asymptomatic while others succumb to repeated symptomatic malaria is beneficial for studies of naturally acquired immunity and can guide control interventions. This study compared demographic, host and maternal factors associated with being either parasite negative or having asymptomatic infections versus developing symptomatic malaria in the first year of life. Methods A birth cohort (n = 1264) was monitored longitudinally over two years for malaria infections in Kintampo, Ghana. Symptomatic and asymptomatic infections were detected actively through monthly home visits, complemented by passive case detection. Light microscopy was used to detect parasitaemia. Based on data from a minimum of eight monthly visits within the first year of life, infants were classified into one of four groups: “parasite negative”, “only-asymptomatic”, “only-symptomatic” or “alternating” i.e., sometimes symptomatic and other times asymptomatic. The host and maternal characteristics and demographic factors in relation to these four groups were compared. Results The parasite negative group formed 36% of the cohort, whilst the only-symptomatic were 35%. The alternating group were 22% and the only-asymptomatic were 7% of the cohort. There were significant associations between residence, socio-economic status (SES), parity, IPTp doses, delivery place of infant and having or not having malaria parasites. Maternal factors such as early commencement and frequency of ante-natal care (ANC) were significantly higher in the parasite negative group compared to all others. ITN use in pregnancy increased the odds of infant having only asymptomatic infections (“protected against disease”). Placental malaria was more common in the groups of infants with symptomatic malaria. Urban residence was significantly higher in the parasite negative group, while birth in the malaria transmission season were significantly more common in the alternating and parasite negative groups. Risk factors for infants with symptomatic malaria included low SES, birth in private maternity homes, sickle cell normal variant, lower MUAC, reported intake of anti-malarials and increased morbidity before the first microscopic infection was detected. Conclusion Strengthening ANC by encouraging early and regular attendance, the use of IPTp, maternal bed nets and improving the nourishment of infants help reduce the frequency of symptomatic malaria over the first year of life.

On Self-Mullineux and Self-Conjugate Partitions

The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.