We first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best possible.

Tamoxifen is commonly prescribed for preventing recurrence in patients with breast cancer. However, the responses of the patients on tamoxifen treatment are variable. Cytochrome P450 genetic variants have been reported to have a significant impact on the clinical outcomes of tamoxifen treatment but no tangible conclusion can be made up till now. The present review attempts to provide a comprehensive review on the associative relationship between genetic polymorphisms in cytochrome P450 enzymes and survival in breast cancer patients on adjuvant tamoxifen therapy. The literature search was conducted using five databases, resulting in the inclusion of 58 studies in the review. An appraisal of the reporting quality of the included studies was conducted using the assessment tool from the Effective Public Health Practice Project (EPHPP). Meta-analyses were performed on CYP2D6 studies using Review Manager 5.3 software. For other studies, descriptive analyses were performed. The results of meta-analyses demonstrated that shorter overall survival, disease-free survival and relapse-free survival were found in the patients with decreased metabolisers when compared to normal metabolisers. The findings also showed that varying and conflicting results were reported by the included studies. The possible explanations for the variable results are discussed in this review.

Three-dimensional Ni3S2-reduced graphene oxide (rGO) nanosheets composite is directly grown on nickel foam (Ni3S2-rGO@NF) by a one-step hydrothermal process involving in situ sulfurization of NF and reduction of GO. The introduction of GO is found not only to control the aggregation and the growth of Ni3S2 nanosheets, but also to increase the number of active sites and improve conductivity of composite. The heterogeneous Ni3S2-rGO@NF electrode as electrocatalysts for hydrogen evolution reaction (HER) exhibits significantly enhanced catalytic activity in alkaline media. The onset potential of Ni3S2-rGO@NF can be as low as ∼0 mV, which is comparable to platinum, and only a small overpotential of ∼44 mV is needed to reach a benchmark current density of 10 mA/cm2. Moreover, it demonstrates a good stability. All evidences suggest that the in situ surfurization can be considered as an effective way to prepare metal sulfides as electrocatalysts for hydrogen generation.

Graphene nanoribbons as a quasi-one-dimensional form of graphene has attracted intensive attention in energy related devices. Upon oxidation and cutting of multiwall carbon nanotubes (MWCNTs), highly dispersive graphene oxide nanoribbons (GONRs) were obtained, on which Zn2+ and Sn4+ can be homogenously deposited. The reduced graphene oxide nanoribbons (rGONRs)/Zn2SnO4 composite with a homogeneous distribution of nanoparticles on the nanoribbons have been prepared through facile in situ chemical co-reduction process. It is worth noting that the size of Zn2SnO4 particles tightly dispersed on rGONRs is about 15 nm. Benefit from the introduction of rGONRs, the specific surface area and electrode conductivity of rGONRs/Zn2SnO4 can both be effectively enhanced. The as-prepared rGONRs/Zn2SnO4 as anode material for lithium-ion batteries displays desirable electrochemical performance (727.2 mA h/g after 50 cycles at the current density of 100 mA/g), which is mainly attributed to the uniformly distributed Zn2SnO4 nanoparticles and the immobilizing and conducting effects of rGONRs.