On the character degree graph of finite groups

Given a finite group G, let cd (G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd (G) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd (G). In this paper, we consider the complement of the character degree graph, and we characterize the finite groups for which this complement graph is not bipartite. This extends the analysis of Akhlaghi et al. (Proc Am Math Soc 146:1505\u20131513, 2018), where the solvable case was treated

On the character degree graph of solvable groups

Let G be a finite solvable group, and let 06(G) denote the prime graph built on the set of degrees of the irreducible complex characters of G. A fundamental result by P. P. P\ue1lfy asserts that the complement 06(G) of the graph 06(G) does not contain any cycle of length 3. In this paper we generalize P\ue1lfy\u2019s result, showing that 06(G) does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of 06(G) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if n is the clique number of 06(G), then 06(G) has at most 2n vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous \u3c1-\u3c3 conjecture by B. Huppert

z~2: An Epoch of Disk Assembly

We explore the evolution of the internal gas kinematics of star-forming galaxies from the peak of cosmic star-formation at $z\sim2$ to today. Measurements of galaxy rotation velocity $V_{rot}$, which quantify ordered motions, and gas velocity dispersion $\sigma_g$, which quantify disordered motions, are adopted from the DEEP2 and SIGMA surveys. This sample covers a continuous baseline in redshift from $z=2.5$ to $z=0.1$, spanning 10 Gyrs. At low redshift, nearly all sufficiently massive star-forming galaxies are rotationally supported ($V_{rot}>\sigma_g$). By $z=2$, the percentage of galaxies with rotational support has declined to 50$\%$ at low stellar mass ($10^{9}-10^{10}\,M_{\odot}$) and 70$\%$ at high stellar mass ($10^{10}-10^{11}M_{\odot}$). For $V_{rot}\,>\,3\,\sigma_g$, the percentage drops below 35$\%$ for all masses. From $z\,=\,2$ to now, galaxies exhibit remarkably smooth kinematic evolution on average. All galaxies tend towards rotational support with time, and it is reached earlier in higher mass systems. This is mostly due to an average decline in $\sigma_g$ by a factor of 3 since a redshift of 2, which is independent of mass. Over the same time period, $V_{rot}$ increases by a factor of 1.5 for low mass systems, but does not evolve for high mass systems. These trends in $V_{rot}$ and $\sigma_g$ with time are at a fixed stellar mass and should not be interpreted as evolutionary tracks for galaxy populations. When galaxy populations are linked in time with abundance matching, not only does $\sigma_g$ decline with time as before, but $V_{rot}$ strongly increases with time for all galaxy masses. This enhances the evolution in $V_{rot}/\sigma_g$. These results indicate that $z\,=\,2$ is a period of disk assembly, during which the strong rotational support present in today's massive disk galaxies is only just beginning to emerge.Comment: 12 pages, 8 figures, submitted to Ap

The impact of vegan diet in the prevention and treatment of type 2 diabetes: A systematic review

A protective effect of vegan diets on health outcomes has been observed in previous studies, but its impact on diabetes is still debated. The aim of this review is to assess the relationship between vegan diets and the risk for type 2 diabetes (T2D) along with its effect on glycemic control and diabetes-related complications. In accordance with PRISMA (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) guidelines, Pubmed and Cochrane library databases were systematically searched for all relevant studies. Seven observational and eight randomized controlled (RCTs) studies were included. The methodological quality of studies was assessed using the National Institutes of Health quality assessment tool for observational cohort and cross-sectional studies and the Cochrane Risk of Bias Tool for RCTs. We found that a vegan diet is associated with lower T2D prevalence or incidence and in T2D patients decreases high glucose values and improves glucose homeostasis, as reported from the majority of included studies. This approach seems to be comparable to other recommended healthful eating models, but as it may have potential adverse effects associated with the long-term exclusion of some nutrients, appropriate nutritional planning and surveillance are recommended, particularly in specific groups of diabetic patients such as frail elderly, adolescents, and pregnant or breastfeeding women

Dietary and protective factors to halt or mitigate progression of autoimmunity, covid-19 and its associated metabolic diseases

COVID-19 is without any doubt the worst pandemic we have faced since the H1N1 virus outbreak. Even if vaccination against SARS-CoV-2 infection is becoming increasingly available, a more feasible approach for COVID-19 prevention and therapy is still needed. Evidence of a pathological link between metabolic diseases and severe forms of COVID-19 has stimulated critical reflection and new considerations. In particular, an abnormal immune response observed in certain patients with SARS-CoV-2 infection suggested possible common predisposing risk factors with autoimmune diseases such as Type 1 Diabetes (T1D). Correct supplementation with dietary factors may be key to preventing and counteracting both the underlying metabolic impairment and the complications of COVID-19. A set of agents may inhibit the cytokine storm and hypercoagulability that characterize severe COVID-19 infection: vitamin D3, omega-3 polyunsaturated fatty acids, polyphenols like pterostilbene, polydatin and honokiol, which can activate anti-inflammatory and antioxidant sirtuins pathways, quercetin, vitamin C, zinc, melatonin, lactoferrin and glutathione. These agents could be highly beneficial for subjects who have altered immune responses. In this review, we discuss the antiviral and metabolic effects of these dietary factors and propose their combination for potential applications in the prevention and treatment of COVID-19. Rigorous studies will be fundamental for validating preventive and therapeutic protocols that could be of assistance to mitigate disease progression following SARS-CoV-2 infection