A finite simple graph is called a
k
-multicirculant if its automorphism group contains a cyclic semiregular subgroup having
k
orbits on the vertex set. It was shown by Giudici et al. that, if
k
is ...squarefree and coprime to 6, then a cubic arc-transitive
k
-multicirculant has at most
6
k
2
vertices (J. Combin. Theory Ser. B, 2017). In this paper, we classify the latter graphs under the assumption that their semiregular cyclic subgroups are contained in a soluble group of automorphisms acting transitively on the arc set of the graphs. As an application, cubic arc-transitive
p
-multicirculants are completely described for each odd prime
p
.
A graph X is said to be an m-Cayley graph on a group G (|G|≠1) if its automorphism group contains a semiregular subgroup isomorphic to G having m orbits on the vertex set of X. If G is cyclic and ...m=1,2,3,4, or 5 then X is said to be a circulant, a bicirculant, a tricirculant, a tetracirculant, or a pentacirculant, respectively.
A graph is said to be symmetric if its automorphism group acts transitively on the set of its arcs. All cubic symmetric circulants, bicirculants and tricirculants are known, and in this paper we give complete classifications of cubic symmetric tetracirculants and pentacirculants. In particular, it is shown that there are infinitely many connected cubic symmetric tetracirculants whereas there are only two connected cubic symmetric pentacirculants.
Edge Regular Graph Products Frelih, Boštjan; Miklavič, Štefko
The Electronic journal of combinatorics,
03/2013, Letnik:
20, Številka:
1
Journal Article
Recenzirano
A regular nonempty graph $\Gamma$ is called edge regular, whenever there exists a nonegative integer $\lambda_{\Gamma}$, such that any two adjacent vertices of $\Gamma$ have precisely ...$\lambda_{\Gamma}$ common neighbours. An edge regular graph $\Gamma$ with at least one pair of vertices at distance 2 is called amply regular, whenever there exists a nonegative integer $\mu_{\Gamma}$, such that any two vertices at distance 2 have precisely $\mu_{\Gamma}$ common neighbours. In this paper we classify edge regular graphs, which can be obtained as a strong product, or a lexicographic product, or a deleted lexicographic product, or a co-normal product of two graphs. As a corollary we determine which of these graphs are amply regular.
An
n-
bicirculant is a graph having an automorphism with two orbits of length
n and no other orbits. Symmetry properties of
p-bicirculants,
p a prime, are extensively studied. In particular, the ...actions of their automorphism groups are described in detail in terms of certain algebraic representation of such graphs.
The relationship between anti-SARS-CoV-2 humoral immune response, pathogenic inflammation, lymphocytes and fatal COVID-19 is poorly understood.
A longitudinal prospective cohort of hospitalised ...patients with COVID-19 (n=254) was followed up to 35 days after admission (median, 8 days). We measured early anti-SARS-CoV-2 S1 antibody IgG levels and dynamic (698 samples) of quantitative circulating T-, B- and natural killer lymphocyte subsets and serum interleukin-6 (IL-6) response. We used machine learning to identify patterns of the immune response and related these patterns to the primary outcome of 28-day mortality in analyses adjusted for clinical severity factors.
Overall, 45 (18%) patients died within 28 days after hospitalisation. We identified six clusters representing discrete anti-SARS-CoV-2 immunophenotypes. Clusters differed considerably in COVID-19 survival. Two clusters, the anti-S1-IgG
T
B
NK
IL-6
and the anti-S1-IgG
T
B
NK
IL-6
had a high risk of fatal COVID-19 (HR 3.36-21.69; 95% CI 1.51-163.61 and HR 8.39-10.79; 95% CI 1.20-82.67; p≤0.03, respectively). The anti-S1-IgG
T
B
NK
IL-6
and anti-S1-IgG
T
B
NK
IL-6
cluster were associated with moderate risk of mortality. In contrast, two clusters the anti-S1-IgG
T
B
NK
IL-6
and anti-S1-IgG
T
B
NK
IL-6
clusters were characterised by a very low risk of mortality.
By employing unsupervised machine learning we identified multiple anti-SARS-CoV-2 immune response clusters and observed major differences in COVID-19 mortality between these clusters. Two discrete immune pathways may lead to fatal COVID-19. One is driven by impaired or delayed antiviral humoral immunity, independently of hyper-inflammation, and the other may arise through excessive IL-6-mediated host inflammation response, independently of the protective humoral response. Those observations could be explored further for application in clinical practice.
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